Degrees of Iterates of Rational Maps on Normal Projective Varieties
Nguyen-Bac Dang

A negative answer to a conjecture arising in the study 
 of selection-migration models in population geneticst1
    Elisa Sovrano
    18 november 2016
===============================================================================================================

 Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. 
We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh, Sibony <cit.>, and by Truong <cit.>. 
Precisely, we give a new proof of the submultiplicativity properties of these degrees and of their birational invariance.
Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. 
In particular, we prove an algebraic version of an inequality first obtained by Xiao <cit.> and Popovici <cit.>, which generalizes Siu's inequality (see <cit.>) to algebraic cycles of arbitrary codimension.  This allows us to show that 
the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back
on the space of numerical classes in X.






§ INTRODUCTION



Let f : X  X be any dominant rational self-map of a normal projective variety X of dimension n defined over an algebraically closed field  of arbitrary characteristic. 
If X is not normal then one can always consider its normalization. Moreover, if the field is not algebraically closed, then we shall take its algebraic closure.
 
 Given any big and nef (e.g ample) Cartier divisor H_X on X, and any integer 0 ⩽ i ⩽ n, one defines the i-th degree of f as the integer:

    _i, H_X(f) = (π_1^* H_X^n-i·π_2^* H_X^i),

where π_1 and π_2 are the projections from the normalization of the graph of f in X × X onto the first and the second factor respectively and where ( · ) denotes the intersection product on this graph. 
  

Our main theorem can be stated as follows.

  Let X be a normal projective variety of dimension n and let H_X be a big and nef Cartier divisor on X. 


  (i) There is a positive constant C>0 
such that for any dominant rational self-maps f, g on X, one has:

    _i,H_X(f ∘ g ) ⩽ C _i,H_X (f) _i,H_X(g).


  (ii) For any big nef Cartier divisor H_X' on X, there exists a constant C>0 such that for any rational self-map f on X, one has:

    1C⩽_i,H_X(f)_i,H_X' (f)⩽ C.






Observe that Theorem <ref>.(ii) implies that the degree growth of f is a birational invariant, in the sense that there is a positive constant C such that for any birational map g: X'  X with X' projective, and any big nef Cartier divisor H_X' on X', one has

    1C⩽_i, H_X (f^p)_i,H_X' (g^-1∘ f^p ∘ g)≤ C,

for any p ∈ℕ. 
Indeed, by applying Theorem <ref>.(ii) for the induced action by f on the normalization of the graph of g, one deduces that the growth of the degrees on the graph of g and on X and X' are controlled by a strictly positive constant. 
Fekete's lemma and Theorem <ref>.(i) also imply the existence of the dynamical degree (first introduced in <cit.> for rational maps of the projective space) as the following quantity:

    λ_i(f):=lim_p → + ∞_i,H_X(f^p)^1/p .
 
The independence of λ_i(f) under the choice of H_X, and its birational invariance are the consequence of Theorem <ref>.(ii) .



When = ℂ, Theorem <ref> was proved by Dinh and Sibony in <cit.>, and further generalized to compact Kähler manifolds in <cit.>. 
 The core of their argument relied on a procedure of regularization for closed positive currents of any bidegree (<cit.>) and was therefore transcendental in nature.
When  is a field of characteristic zero, there exists an inclusion of the field  in ℂ by Lefschetz principle (<cit.>) and Dinh and Sibony's argument  proves that the i-th dynamical degree of any rational dominant map is well-defined. 
 Recently, Truong <cit.> managed to get around this problem and proved Theorem <ref> for arbitrary smooth varieties using an appropriate Chow-type moving lemma. 
 He went further in <cit.> and obtained Theorem <ref> for any normal variety in all characteristic by applying de Jong's alteration theorem (<cit.>). 
 Note however that he had to deal with correspondences since a rational self-map can only be lifted as a correspondence through a general alteration map.
 Our approach avoids this technical difficulty.     



To illustrate our method, let us explain the proof of Theorem <ref>, when X is smooth, i=1 and f, g are regular following the method initiated in <cit.>. 
Recall that a divisor α on X is pseudo-effective and one writes α⩾ 0 if for any ample Cartier divisor H on X, and any rational ϵ >0, a suitable multiple of the ℚ-divisor α + ϵ H is linearly equivalent to an effective one. 

Recall also the fundamental Siu inequality[this inequality is also referred to as the weak transcendantal holomorphic Morse inequality in <cit.>] (<cit.>, <cit.>, <cit.>) which states:

    α⩽ n (α·β^n-1)(β^n)β,

for any nef divisor α, and any big and nef divisor β. 
 
Since the pullback by a dominant morphism of a big nef divisor remains big and nef, 
we may apply (<ref>) to the big nef divisors α = g^* f^* H_X and β = f^* H_X, and we get

    g^*f^*  H_X ⩽ n _1,H_X(f)(H_X^n)  g^* H_X .

Intersecting with the cycle H_X^n-1 yields the submultiplicativity of the degrees with the constant C = n/(H_X^n). 

We observe that the previous inequality (<ref>) can be easily extended to complete intersections by cutting out by suitable ample sections. 
In particular, we get a positive constant C such that for any big nef divisors α and β, one has:

    α^i ⩽ C (α^i ·β^n-i)(β^n)β^i.

Such inequalities have been obtained by Xiao (<cit.>) and Popovici (<cit.>) in the case = ℂ. 
Their proof uses the resolution of complex Monge-Ampère equations and yields a constant C = n  i. 
On the other hand, our proof applies in arbitrary characteristic and in fact to more general classes than complete intersection ones. 
We refer to Theorem <ref> below and the discussion preceding it for more details.
Note however that we only obtain C = (n-i+1)^i, far from the expected optimal constant C =  n  i of Popovici.
Once (<ref>) is proved, Theorem <ref> follows by a similar argument as in the case i = 1.





Going back to the case where X is a complex smooth projective variety, recall that the degree of f is controlled up to a uniform constant by the norm of the linear operator f^∙,i, induced by pullback on the de Rham cohomology space H^2i_dR (X)_ℝ (<cit.>).
One way to construct f^∙,i is to use the Poincaré duality isomorphisms ψ_X: H^2i_dR(X,ℝ) →H_2n-2i(X,ℝ), ψ_Γ_f : H^2i_dR(Γ_f,ℝ) →H_2n-2i(Γ_f,ℝ) where H_i(X,ℝ) denotes the i-th simplicial homology group of X.
 The operator f^∙,i is then defined following the commutative diagram below:

    H_dR^2i(Γ_f, ℝ)[r]^-ψ_Γ_f   H_2n-2i(Γ_f,ℝ) [r]^π_1_*   H_2n-2i(X,ℝ) [d]^ψ_X^-1
    
    H_dR^2i(X, ℝ) [rr]_f^∙,i[u]^π_2^*         H_dR^2i(X,ℝ),

where Γ_f is a desingularization of the graph of f in X × X, and π_1, π_2 are the projections from Γ_f onto the first and second factor respectively. 

In order to state an analogous result in our setting, we need to find a replacement for the de Rham cohomology group H^2i_dR(X)_ℝ and define suitable pullback operators. 
When X is smooth, one natural way to proceed  is to consider the spaces ^i(X)_ℝ of algebraic ℝ-cycles of codimension i modulo numerical equivalence. 
The operator f^∙, i is then simply given by the composition π_1_* ∘π_2^* : ^i(X)_ℝ→^i(X)_ℝ. 

When X is singular, then the situation is more subtle because one cannot intersect arbitrary cycle classes in general [an arbitrary curve can only be intersected with a Cartier divisor, not with a general Weil divisor.]. 
One can consider two natural spaces of numerical cycles ^i(X)_ℝ and _i(X)_ℝ on which pullback operations and pushforward operations by proper morphisms are defined respectively. 
More specifically, the space of numerical i-cycles _i(X)_ℝ is defined as the group of ℝ-cycles of dimension i modulo the relation z ≡ 0 if and only if (p^* z · D_1 ·…· D_e+i) = 0 for any proper flat surjective map p : X' → X of relative dimension e and any Cartier divisors D_j on X'.
One can prove that _i(X)_ℝ is a finite dimensional vector space and one defines ^i(X)_ℝ as its dual (_i(X)_ℝ, ℝ).


Note that our presentation differs slightly from Fulton's definition (see Appendix <ref> for a comparison), but we also recover the main properties of the numerical groups. This approach is more suitable to compare cycles using positivity estimates on complete intersections. 


As in the complex case, we are able to construct Poincaré duality  maps ψ_X : ^i(X)_ℝ→_n-i(X)_ℝ and ψ_Γ_f :  ^i(Γ_f)_ℝ→_n-i(Γ_f)_ℝ, but they are not necessarily isomorphisms due to the presence of singularities. As a consequence, we are only able to define a linear map f^∙,i as f^∙,i := π_1_* ∘ψ_Γ_f∘π_2^* : ^i(X)_ℝ→_n-i(X)_ℝ between two distinct vector spaces. 
Despite this limitation, we prove a result analogous to one of Dinh and Sibony. The next theorem was obtained by Truong for smooth varieties (<cit.>).

 Let X be a normal projective variety of dimension n. Fix any norms on ^i(X)_ℝ and _n-i(X)_ℝ, and denote by · the induced operator norm on linear maps from ^i(X)_ℝ to _n-i(X)_ℝ. Then there is a constant C>0 such that for any rational selfmap f: X  X, one has:

    1C⩽|| (f)^∙,i||_i,H_X(f)⩽ C.
 




Our proof of Theorem <ref> exploits a natural notion of positive classes in ^i(X)_ℝ combined with a strengthening of (<ref>) to these classes that we state below (see Theorem <ref>).

To simplify our exposition, let us suppose again that X is smooth. 
As in codimension 1, one can define the pseudo-effective cone ^i(X) as the closure in ^i(X)_ℝ of the cone generated by effective cycles of codimension i. Its dual with respect to the intersection product is the nef cone ^n-i(X), which however does not behave well when i ⩾ 2 (see <cit.>). 
Some alternative notions of positive cycles have been introduced by Fulger and Lehmann in <cit.>, among which the notion of basepoint free classes emerges. Basepoint free classes have many good properties such as being both pseudo-effective and nef, being invariant by pull-backs by morphisms and by intersection products, and forming a salient convex cone with non-empty interior. The terminology comes from the fact that the basepoint free classes always have a cycle representing them with intersects any subvariety with the expected dimension. 
 Denote by ^i(X) the cone of basepoint free classes. It is defined as the closure in ^i(X)_ℝ of the cone generated by ℝ-cycles of the form p_* (D_1 ·…· D_e+i) where D_j are ample Cartier ℝ-divisors and p : X' → X is a flat surjective proper morphism of relative dimension e. 

For basepoint free classes, we are able to prove the following generalization of (<ref>). 

  Let X be a normal projective variety of dimension n. Then there exists a constant C>0 such that for any basepoint free class α∈^i(X), for any big nef divisor β, one has in ^i(X)_ℝ:
  
    α⩽ C (α·β^n-i)(β^n)×β^i.
  
 
 
Theorem <ref> follows from (<ref>) by observing that  f^∙,i^i(X) ⊂^i(X), so that the operator norm ||f^∙,i|| can be computed by evaluating f^∙,i only on basepoint free classes. 
 


In the singular case, the proof of Theorem <ref> is completely similar but the spaces  ^i(X)_ℝ and _n-i(X)_ℝ are not necessarily isomorphic in general. 
As a consequence, several dual notions of positivity appear in ^i(X)_ℝ and _i(X)_ℝ
that make the arguments more technical.

















Finally, using the techniques developed in this paper, we give a new proof of the product formula of Dinh, Nguyen, Truong (<cit.>,<cit.>) which they proved when = ℂ and which was later generalized by Truong (<cit.>) to normal projective varieties over any field. 


The setup is as follows. 
Let q : X → Y be any proper surjective  morphism between normal projective varieties, and
fix two big and nef divisors  H_X, H_Y  on X and Y respectively.   
Consider  two dominant rational self-maps f: X  X, g : Y  Y, which are semi-conjugated by 
q, i.e. which satisfy q ∘ f = g ∘ q. To simplify notation we shall write
X/_qY fg X/_q Y when these assumptions hold true.

Recall that the i-th relative degree of X/_q Y fg X/_q Y is given by the intersection product

    _i (f) := (π_1^*( H_X^ X - Y - i· q^* H_Y^ Y) ·π_2^* H_X^i),

where π_1 and π_2 are the projections from the graph of f in X × X onto the first and the second component respectively. 
One can show a relative version of Theorem <ref>  (see Theorem <ref>), and define as in the absolute case, the i-th relative dynamical degree λ_i(f, X/Y) 
as the limit lim_p→ +∞_i(f^p)^1/p. 
It is also a birational invariant in the sense that if φ : X'  X, ψ: Y'  Y such that q' = ψ^-1∘ q∘φ is regular, then λ_i( φ^-1∘ f ∘φ , X'/Y') = λ_i(f, X/Y), and does not depend on the choices of H_X and H_Y. 
When q : X Y is merely rational and dominant, then we define (see Section <ref>) the i-th relative degree of f by replacing X with the normalization of graph of q. 
We prove the following theorem.
 
Let X,Y be normal projective varieties.
For any dominant 
rational self-maps f: X  X, g : Y  Y which are  semi-conjugated by 
a dominant rational map q: X  Y, we have

    λ_i(f)  = max_max(0, i- l ) ⩽ j ⩽min(i,e) ( λ_i-j(g) λ_j (f, X/Y) ) .







Our proof follows closely Dinh and Nguyen's method from <cit.> and relies on a fundamental inequality (see Corollary <ref> below) which follows from Künneth formula at least when = ℂ.
To state it precisely, consider π : X' → X a surjective generically finite morphism and q : X → Y a surjective morphism where X', X and Y are normal projective varieties such that   n=  X= X' and such that l =  Y. 
We prove that for any basepoint free classes α∈^i(X') and β∈^n-i(X'), one has:

    (β·α) ⩽ C ∑_max(0 , i-l) ≤ j ≤min(i,e) U_j(α) × ( β·π^* (q^* H_Y^i-j· H_X^j)),
  
where H_Y and H_X are big and nef divisors on Y and X respectively, 
and U_j(α) is the intersection product given by U_j(α) = (π^*(q^*H_Y^l-i+j· H_X^e-j) ·α).

In the singular case, Truong has obtained this inequality using Chow's moving intersection lemma. 
We replace this argument by a suitable use of Siu's inequality and Theorem <ref> in order to prove a positivity property for a class given by the difference between a basepoint free class in X' × X' and the fundamental class of the diagonal of X' in X' × X' (see Theorem <ref>).
Inequality (<ref>) is a weaker version of <cit.> proved by Dinh-Nguyen when Y is a complex projective variety, and was extended to a field arbitrary characteristic by Truong when Y is smooth (<cit.>). 
 




 







 §.§ Organization of the paper
 



In the first Sections <ref> and <ref>, we review the background on the Chow groups and recall the definitions of the spaces of numerical cycles and provide their basic properties. 
In <ref>, we discuss the various notions of positivity of cycles and prove Theorem <ref>. 
In <ref>, we define relative numerical cycles and canonical morphisms which are the analogous to the Poincaré morphisms ψ_X in a relative setting. 
In <ref>, we prove Theorem <ref>, Theorem <ref> and Theorem <ref>. 
Finally we give an alternate proof of Dinh-Sibony's theorem in the Kähler case (<cit.>) in <ref> using Popovici <cit.> and Xiao's inequality <cit.>. Note that these inequalities allow us to avoid regularization techniques of closed positive currents but rely on a deep theorem of Yau. 
In Section <ref>, we prove that our presentation and Fulton's definition of numerical cycles are equivalent, hence proving that any numerical cycles can be pulled back by a flat morphism. 



Firstly, I would like to thank my advisor C. Favre for his patience and our countless discussions on this subject. I thank also S. Boucksom for some helpful discussions and for pointing out the right argument for the appendix, S. Cantat, L. Fantini, M. Fulger, T. Truong,  B. Lehmann, R. Mboro and J. Xie for their precious comments on my previous drafts and for providing me some references.
The author is supported by the ERC-starting grant project "Nonarcomp" no.307856, and is supported by ANR project “Lambda” ANR-13-BS01-0002














§ CHOW GROUP
 



 §.§ General facts


Let X be a normal projective variety of dimension n defined over an algebraically closed field  of arbitrary characteristic.

The space of cycles Z_i(X) is the free abelian group generated by irreducible subvarieties of X of dimension i,   
 and Z_i(X)_ℚ, Z_i(X)_ℝ will denote the tensor products Z_i(X) ⊗_ℤℚ and Z_i(X) ⊗_ℤℝ.



Let q: X → Y be a morphism where Y is a normal projective variety. Since X and Y are respectively projective, the map q is proper. Following <cit.>, we define the proper pushforward of the  cycle [V] ∈ Z_i(X) as the element of Z_i(Y) given by: 

    q_* [V] = {[                          0                         if                (q(V)) <  V; [ (η) : (q(η))  ] × [q(V)]                         if                V = (q(V)), ] .
 
where V is an irreducible subvariety of X of dimension i, η is the generic point of V and (η), (q(η)) are the residue fields of the local rings 𝒪_η and 𝒪_q(η) respectively.
We extend this map by linearity and obtain a morphism of abelian groups q_* : Z_i(X) → Z_i(Y).




Let C be any closed subscheme of X of dimension i and denote by C_1, …, C_r its i-dimensional irreducible components. Then C defines a fondamental class [C]∈ Z_i(X) by the following formula: 

    [C] := ∑_j=1^r l_𝒪_C_j, C(𝒪_C_j,C)[C_j],

where l_A(M) denotes the length of an A-module M (<cit.>).


For any flat morphism q: X → Y of relative dimension e between normal projective varieties, we can define a flat pullback of cycles q^*: Z_i(Y) → Z_i+e(X) (see <cit.>). If C is any subscheme of Y of dimension i, the cycle q^* [C] is by definition the fundamental class of the scheme-theoretic inverse by q:
 
    q^* [C] := [q^-1(C)] ∈ Z_i+e(X).
  



Let W be a subvariety of X of dimension i+1 and φ be a rational map on W. Then we define a cycle on X by:

    [(φ)]:= ∑_V(φ) [V],

where the sum is taken over all irreducible subvarieties V of dimension i of W ⊂ X. 
A cycle α defined this way is rationally equivalent to 0 and in that case we shall write α 0. 


The i-th Chow group A_i(X) of X is the quotient of the abelian group Z_i(X) by the free group generated by the cycles that are rationally equivalent to zero. We denote by A_∙(X) the abelian group ⊕ A_i(X). 


We recall now the functorial operations on the Chow group, which result from the intersection theory developped in <cit.>.


 Let q: X → Y be a morphism between normal projective varieties. Then we have:


  (i) 
The morphism of abelian groups q_* : Z_i(X) → Z_i(Y) induces a morphism of abelian groups q_* : A_i(X) → A_i(Y). 

  (ii) 
If the morphism q is flat of relative dimension e, then the morphism q^* : Z_i(Y) → Z_i+e(X) induces a morphism of abelian groups q^* : A_i(Y) → A_i+e(X). 



Assertion (i) is proved in <cit.> and assertion (ii) is given in <cit.>.


 Let q: X → Y is a flat morphism of normal projective varieties. Suppose α∈ A_i(Y) is represented by an effective cycle α∑ n_j [V_j] where the  n_j are positive integers. Then q^*α is also represented by an effective cycle. 


Any cycle α∈ Z_0(X)_ℤ is of the form ∑ n_j [p_j] with p_j ∈ X() and n_j ∈ℤ. We define the degree of α to be (α) := ∑ n_j and we shall write:

    (α ):= (α) = ∑ n_j.


The morphism of abelian groups : Z_0(X)_ℤ→ℤ induces a morphism of abelian groups : A_0(X) →ℤ. 




 §.§ Intersection with Cartier divisors



Let X be a normal projective variety and D be a Cartier divisor on X. Let V be a subvariety of of dimension i in X and denote by j : V ↪ X the inclusion of V in X. We define the intersection of D with [V] as the class:

    D · [V] := j_* [D'] ∈ A_i-1(X),

where D' is a Cartier divisor on V such that the line bundles j^* 𝒪_X(D) and 𝒪_V(D') are isomorphic.
Observe that D' exists since the exact sequence 0 [r]    𝒪^*_V [r]    ℳ_V^* [r]    ℳ_V^* / 𝒪_V^*[r]     0 induces a  surjective map from the divisor subgroups H^0(V, ℳ_V^* / 𝒪_V^*) of V onto the Picard group (V) = H^1(V, 𝒪_V^*) where ℳ_V^* is the sheaf of non-zero rational functions on V.

We extend this map by linearity into a morphism of abelian groups D · :Z_i(X) → A_i-1(X).  

  Let X be a normal projective variety and D be a Cartier divisor on X. 
The map D · : Z_i(X) → A_i-1(X) induces a morphism of abelian groups D · : A_i(X) → A_i-1(X). 
Moreover, the following properties are satisfied:
 

  * For all Cartier divisors D and D' on X, for all class α∈ A_i(X), we have: 

    (D' + D) ·α = D' ·α + D ·α.


  * (Projection formula) Let q: X → Y be a morphism between normal projective varieties. Then for all class β∈ A_i(X) and all Cartier divisor D on Y, we have in A_i-1(Y):

    q_* (q^*D ·β) = D · q_*(β).









 §.§ Characteristic classes
 


 Let X be a normal projective variety of dimension n and L be a line bundle on X. There exists a Cartier divisor D on X such that the line bundles L and 𝒪_X(D) are isomorphic. We define the first Chern class of L as:

    c_1(L) := [D] ∈ A_n-1(X).




 For all normal projective varieties X, the group ^i(X) is the free group generated by elements of the form D_1 ·…· D_i where D_1, …, D_i are Cartier divisors on X.



 Let X be a normal projective variety and E be a vector bundle of rank e+1 on X. 
Given any vector bundle E on X, we shall denote by (E) the projective bundle of hyperplanes in E following the convention of Grothendieck.
Let p be the projection from (E^*) to X and ξ = c_1(𝒪_(E^*) (1)). 
We define the i-th Segré class s_i (E) as the morphism s_i(E) ·: A_∙(X) → A_∙-i(X) given by:

    s_i(E) α := p_* (ξ^e+i· p^*α).




 When X is smooth of dimension n, we can define an intersection product on the Chow groups A_i(X) × A_l(X) → A_n-i-l(X) (see <cit.>) which is compatible with the intersection with Cartier divisors and satisfies the projection formula (see <cit.>).
Applying the projection formula to (<ref>), we get 

    s_i(E) α = p_* (ξ^e+i)·α,

so that  s_i(E) is represented by an element in A_n-i(X). To simplify we shall also denote s_i(E) this element.




As Segré classes of vector bundles are operators on the Chow groups A_∙(X), the composition of such operators defines a product. 

  (cf <cit.>) Let q: X → Y be a morphism between normal projective varieties. 
 For any vector bundle E and F on Y, the following properties hold. 


  (i) For all α∈ A_i(Y) and all j< 0, we have s_j(E) α = 0. 

  (ii) For all α∈ A_i(Y), we have s_0(E) α = α. 

  (iii) For all integers j,m, we have s_j(E) ( s_m(F) α) = s_m(F)  (s_j(E) α).

  (iv) (Projection formula) For all β∈ A_i(X) and any integer j, we have q_*(s_j(q^* E) β) = s_j(E)  q_* β. 
 
  (v) If the morphism q: X → Y is flat, then for all α∈ A_i(Y) and any integer j, we have s_j(q^*E)  q^* α = q^* (s_j(E) α)). 

 
 
 
The j-th Chern class c_j(E) of a vector bundle E on X is an operator c_j(E) : A_∙(X) → A_∙ -j defined formally as the coefficients in the inverse power series:

    (1+  s_1(E) t + s_2(E)t^2 + …   )^-1 = 1 + c_1(E) t + … + c_r+1(E)t^r+1.

A direct computation yields for example
 c_1(E) = - s_1(E), c_2(E)  =  (s_1(E)^2 - s_2(E)).


 Let X be a normal projective variety. 
The abelian group A^i(X) is the subgroup of (A_∙(X), A_∙-i(X)) generated by product of Chern classes c_i_1 (E_1) ·…· c_i_p(E_p) where i_1, …, i_p are integers satisfying i_1 + … + i_p = i and where E_1, …, E_p are vector bundles over X. We denote by A^∙(X) the group ⊕ A^i(X).

Observe that by definition, A^i(X) contains the image of ^i(X).






Recall that the Grothendieck group K^0(X) is the free group generated by vector bundles on X quotiented by the subgroup generated by relations of the form [E_1] + [E_3] - [E_2] where there is an exact sequence of vector bundles:

    0 [r]    E_1 [r]     E_2 [r]     E_3 [r]     0  .

Moreover, the group K^0(X) has a structure of rings given by the tensor product of vector bundles.



Recall also that the Chern character is the unique morphism of rings : (K^0(X), +, ⊗) → (A^∙(X), + , · ) satisfying the following properties (see <cit.>).


  * If L is a line bundle on X, then one has:

    (L)= ∑_i ⩾ 0c_1(L)^ii!.


  * For any morphism q: X' → X and any vector bundle E on X, we have q^* (E) = (q^*E).
 

For any vector bundle E on X, we will denote by _i(E) the term in A^i(X) of (E).  



We recall Grothendieck-Riemann-Roch's theorem for smooth varieties.
  (see <cit.>) Let X be a smooth variety. Then the Chern character induces an isomorphism:

    [X] : E ∈ K^0(X)⊗ℚ→(E)  [X] ∈ A_∙(X)⊗ℚ.






We also recall the definition of Schur polynomials.

  Consider a vector bundle E of rank e on X.
Fix two integers e,i and a decreasing partition λ =(λ_1, …, λ_i) of i with terms lower or equal than e.
The Schur class s_λ(E) is the class given by:

    s_λ(E) =  | [        c_λ_1(E)    c_λ_1 + 1(E)               … c_λ_1 + i -1(E);     c_λ_2 -1(E)       c_λ_2 (E)               … c_λ_2 +i - 2(E);                                               …;  c_λ_i -i +1(E) c_λ_i - i +2(E)               …        c_λ_i(E) ] | .

If E is a vector bundle of rank e on X, then the Schur class s_λ(E) ∈ A^i(X) is the Schur polynomial in the variables given by the Chern classes c_1(E), … , c_e(E). 


When the vector bundle E is globally generated, then the Schur classes can be interpreted as degeneracy loci (see <cit.>). 







§ SPACE OF NUMERICAL CYCLES
 



 


 §.§ Definitions
 

In all this section, X, Y,X_1,X_2,X_3 and X' are  normal projective varieties and X is of dimension n.
 Two cycles α and β in Z_i(X) are said to be numerically equivalent and we will denote by α≡β if for all flat morphisms p_1 : X_1 → X of relative dimension e and all Cartier divisors D_1 , … , D_e+i in X_1, we have:

    ( D_1 ·…· D_e+i· q^*α ) = ( D_1 ·…· D_e+i· q^*β) .





The group of numerical classes of dimension i is the quotient _i(X) = Z_i(X) / ≡. 





By construction, the group _i(X) is torsion free and there is a canonical surjective morphism A_i(X) →_i(X) for any integer i. 


Observe also that for i = 0, two cycles are numerically equivalent if and only if they have the same degree. Since smooth points are dense in X (see <cit.>) and are of degree 1, this proves that the degree realizes the isomorphism _0(X) ≃ℤ.



We set _i(X)_ℚ and _i(X)_ℝ the two vector spaces obtained by tensoring by ℚ and ℝ respectively.
 
     
  This definition allows us to pullback numerical classes by any flat morphism q : X → Y of relative dimension e. 
Our presentation is slightly different from the classical one given in <cit.>. We refer to Appendix <ref> for a proof of the equivalence of these two approaches.

    


 Let q: X → Y a morphism. Then the morphism of groups q_* : Z_i(X) → Z_i(Y) induces a morphism of abelian groups q_*:_i(X) →_i(Y). 




  Let n be the dimension of X and l be the dimension of Y, and
let α be a cycle in Z_i(X) such that α is numerically trivial. 
We need to prove that q_* α is also numerically trivial.

Take p_1 : Y_1 → Y a flat morphism of relative dimension e_1.
Let X_1 be the fibred product X ×_Y Y_1 and let p_1' and q' be the natural projections from X_1 to X and Y_1 respectively. 

    X_1 [d]^q'[r]^p_1'    X [d]^q 
    
    Y_1 [r]^p_1    Y

Since flatness is preserved by base change (<cit.>), the morphism p_1' is flat and q' is proper. 
Pick any cycle γ whose class is in ^e_1 +i(Y_1). 
We want to prove that ( γ·p_1^* q_*α) = 0.
By <cit.>, we have that p_1^*q_* α = q'_*p_1'^* α in Z_e_1+i(Y_1). Applying the projection formula, we get: 
 
    γ· p_1^* q_* α = γ· q'_* p_1'^* α q'_*(q'^* γ· p_1'^* α).
 
Because p_1' is flat and q'^*γ∈^e_1 + i (X_1), we have (q'^* γ· p_1'^* α ) = 0 so that (γ· p_1^* q_* α )= 0 as required. 


The numerical classes defined above are hard to manipulate, we want to define a pullback of numerical classes by any proper morphism. 
We proceed and define dual classes.


We denote by Z^i(X) = _ℤ( Z_i(X) , ℤ) the space of cocycles. If p_1 : X_1 → X is a flat morphism of relative dimension e_1, then any element γ∈^e_1+i(X_1) induces an element [γ] in  Z^i(X) by the following formula: 

    [γ] : α∈ Z_i(X) → (γ·  p_1^* α) ∈ℤ.


 
The abelian group ^i(X) is the subgroup of Z^i(X) generated by elements of the form [γ] where γ∈^e_1+i(X_1) and X_1  is flat over X of relative dimension e_1.


  By definition, the map : Z_0(X) →ℤ is naturally an element of Z^0(X).
 Moreover, one has using Theorem <ref>.(ii) that:

    z ∈ Z_0(X) → (s_0(E)  z) = (z) ∈ℤ,

for any vector bundle E on X.
Hence,  defines an element of ^0(X) by definition of Segré classes (Definition <ref>).







 By definition of the numerical equivalence relation, any element of ^i(X) induces an element of the dual _ℤ(_i(X), ℤ). Hence, we can define a natural pairing between ^i(X) and _i(X).  For any normal projective variety, the pairing ^i(X) ×_i(X) →ℤ is non degenerate (i.e the canonical morphism from ^i(X) to _ℤ(_i(X), ℤ) is injective). 


 It follows directly from the definition of ^i(X) and _i(X). 





A priori, an element of ^i(X) is a combination of elements [γ_1] + [γ_2] + … + [γ_j]. 
The following proposition proves one can always take j=1 at least if we tensor all spaces by ℚ.

 
 Any element of ^i(X) is induced by γ∈^e_1+i(X_1)_ℚ where p_1: X_1 → X is a flat morphism of relative dimension e_1.


 
By an immediate induction argument, we are reduced to prove the assertion for the sum of two elements [γ_1] +[γ_2] where  γ_j ∈^e_j + i(X_i)_ℚ and p_j : X_j → X are flat morphisms of relative dimension e_1 and e_2 respectively.  


Let us consider X' the fibre product X_1 × X_2 over X and p_j' the flat projections from X' to X_j for j=1,2. 
By linearity , we only need to show that there exists an element γ_1' ∈^e_1+e_2 + i (X') such that [γ_1'] = [γ_1] in ^i(X). 

    X_1 × X_2  [ld]^p_2'[rd]^p_1'   
    
     X_1  [rd]^p_1       X_2  [ld]^p_2
        X
 
Take an ample Cartier divisor H_X_2 on X_2 and λ_2 an integer such that p_2_* H_X_2^e_2λ_2 [X]. 
Setting γ_1' = 1λ_2 p_1'^* H_X_2^e_2· p_2'^* γ_1, we need to prove that for any α∈ Z_i(X), one has (γ_1 · p_1^* α )= (γ_1' · p_2'^* p_1^* α).
By <cit.>, we have the equality p_2'_*p_1'^* H_X_2^e_2 = p_1^*p_2_* H_X_2^e_2 in Z^e_2(X_2), hence:

    p_2'_*p_1'^* H_X_2^e_2 = λ_2 p_1^* [X].

Since X_1 is reduced and p_1^* [X] is a cycle of codimension 0 in X_1, we have p_1^* [X] = [X_1]. 
Hence by the projection formula, we have:

    [ 1λ_2p_2'_* ( p_2'^* (γ_1 · p_1^* α) · p_1'^* H_X_2^e_2)                                                       =              1λ_2 (p_1^* α·γ_1) ·p_2_* p_1'^* H_X_2^e_2;                                                                                                               =                            1λ_2 (p_1^* α·γ_1) ·λ_2[X_1];                                                                                                               =                                            p_1^* α·γ_1. ]

In particular, the degrees are equal and [γ_1]= [γ_1'] ∈^i(X) as required. 


By the same argument, there exists a class γ_2' ∈^e_1 + e_2 + i (X_1 × X_2) such that [γ_2] = [γ_2'] ∈^i(X), hence [γ_1] + [γ_2] = [γ_1'] + [γ_2'] = [γ_1' + γ_2'] ∈^i(X) as required.



We define _∙(X) (resp. ^∙(X)) by ⊕_i _i(X) (resp. ⊕_i ^i(X)). 



 §.§ Algebra structure on the space of numerical cycles


We now define a structure of algebra on ^∙(X), and prove that _∙(X) has a structure of ^∙(X) module.




Pick γ∈^e_1+i(X_1)_ℚ where p_1: X_1 → X is a flat morphism of relative dimension e_1. The element γ induces a morphism in the Chow group:

    γ· :α∈ A_l(X) →p_1_* (γ· p_1^* α) ∈ A_l-i(X).

The morphism γ· :A_l(X) → A_l-i(X) induces a morphism of abelian groups from _l(X) to _l-i(X). 


 
Any element α∈^i(X) 
induces a morphism α· : N_∙(X) →_∙-i(X) such that the following conditions are satisfied.


  (i) If α is induced by γ∈^e_1 + i(X_1)_ℚ where p_1: X_1 → X is a flat morphism of relative dimension e_1, 
then for any integer l and any z ∈_l(X), one has in _l-i(X):

    α z = γ z .



  (ii)  For any α, β∈^i(X) and any z ∈_l(X), we have:

    (α + β)  z = α z + β z.





 
Let us consider α∈^i(X) and suppose it is induced by γ_1∈^e_1 +i(X_1)_ℚ where p_1 : X_1 → X is a flat morphism of relative dimension e_1. 
We define the map α· as :

    α z = γ_1  z,
 
for any z ∈_l(X). 
We show that the morphism does not depend on the choice of the class γ_1 and (i) is follows from Proposition <ref>. Assertion (ii) follows from the linearity of the intersection product whose proof follows closely the proof of  Proposition <ref>.






 Suppose that [γ_1] = [γ_2] ∈^i(X) where γ_2 ∈^e_2 + i(X_2)_ℚ and p_2: X_2 → X is a flat morphism of relative dimension  e_2,  then we need to prove that:

    p_1_* ( γ_1 ·  p_1^* z) ≡p_2_* (γ_2 · p_2^*z),

for any fixed z ∈ Z_l(X). 
Take β∈^e_3 + l-i(X_3) where p_3: X_3 → X is flat morphism of relative dimension e_3, we only need to show that:

    (β· p_3^* p_1_* (γ_1 · p_1^* z) ) = (β· p_3^* p_2_* (γ_2 · p_2^* z)).
 
Let X_1' and X_2' the fibre products X_1 × X_3 and X_2 × X_3, and p_1': X_1' → X_3, p_3': X_1' → X_1, q_2 : X_2' → X_3, q_3: X_2' → X_2 be the corresponding flat projection morphisms such that we obtain the following commutative diagrams:

    X_1' [rd]^p_1'[ld]^p_3'          X_2' [rd]^q_2[ld]^q_3   
    
    X_1 [rd]^p_1       X_3 [ld]^p_3    X_2 [rd]^p_2       X_3  [ld]^p_3
        X           X .
 
As above, we have p_3^*p_1_* = p_1'_* p_3'^*, hence:


    [         (β· p_3^* p_1_* (γ_1 · p_1^* z) )                                         =         (β·p_1'_* p_3'^* (γ_1 · p_1^* z));                                                                                   =        (p_1'^* β· p_3'^* (γ_1 · p_1^* z));                                                                                   = (p_3'^* γ_1 · p_1'^* p_3^* z · p_1'^* β );                                                                                   =     ( γ_1 ·p_3'_* p_1'^* ( p_3^* z ·β ) );                                                                                   =          (γ_1 · p_1^*p_3_* (p_3^* z ·β ));                                                                                   =        (γ_2 · p_2^* p_3_* (p_3^* z ·β )). ]

By a similar argument, we show that (β· p_3^* p_2_* (γ_2 · p_2^* z)) = (γ_2 · p_2^* p_3_* (p_3^* z ·β)) which implies the desired equality:

    (β· p_3^* p_1_* (γ_1 · p_1^* z) ) = (β· p_3^* p_2_* (γ_2 · p_2^* z)).





 
 There exists a unique structure of commutative graded ring with unit () on ^∙(X) given by (α,β) ∈^∙(X) ×^∙(X) ↦α·β∈^∙(X)  which satisfies the following properties:
 
 
  (i) For any α,β∈^∙(X) and any z ∈_∙(X), one has:
 
    ( α·β)  z = ( α ( β z) ) = ( β ( α z) ).

 
  (ii) For any z∈_∙(X), we have ()  z = z.
 
  (iii) The morphism of abelian groups given by 
    (α, z) ∈^∙(X) ×_∙(X) ↦α z ∈_∙(X)
 is bilinear.
 

  Hence, the abelian group _∙(X) has the structure of a graded ^∙(X)-module.

 
Take α_1 ∈^i(X) and α_2 ∈^l(X) and define φ∈ Z^i+l(X) by the formula:

    φ : z ∈ Z_i+l(X) →  (α_1  ( α_2 z)).

We prove that φ is an element of ^i+l(X). 


By linearity, we can suppose that α_i is induced by γ_i ∈^i+e_j(X_j) where p_j :X_j → X is a flat morphism of relative dimension e_j for j=1,2.
 Let X' = X_1 ×_X X_2 be the fibre product, let p_1' and p_2' be the projections from X' to X_1 and X_2 respectively such that we have the commutative diagram:
 
    X' [rd]^p_1'[ld]^p_2'   
    
     X_1 [rd]^p_1       X_2  [ld]^p_2
        X.

By the projection formula, we obtain for all z ∈ Z_i+l(X):

    φ(z) = ( p_1'^* γ_2 ·  p_2'^* γ_1 ·  p_2'^* p_1^*z).
 
In particular, we have shown that φ is induced by p_1'^* γ_2 · p_2'^* γ_1∈^e_1 + e_2 +i+l(X'), hence φ is an element of ^i+l(X). 
Moreover, the commutativity of the intersection product in (<ref>) proves that (α_2  (α_1  z))= (α_1  (α_2  z)) for any z ∈_i+l(X), hence α_1 ·α_2 = α_2 ·α_1.







Pick a vector bundle E on X. 
As the element ∈^0(X) is equal to z → (s_0(E)  z) in ^0(X) (see Remark <ref>), we get using Theorem <ref>.(ii) that:

    (α z) =(α (s_0(E)  z)) = (s_0(E)  (α z)) = ((α·)  z) = (·α)  z,

for any z ∈_l(X) and any α∈^l(X).
Hence,  is a unit of ^∙(X).
 











 



 §.§ Pullback on dual numerical classes




Let us consider q: X → Y a proper morphism. 
We define for any integer i the pullback q^* : ^i(Y) →_ℤ(_i(X) , ℤ) as the dual of the pushforward operation q_* : _i(X ) →_i(Y) with respect to the pairing ^i(X) ×_i(X) →ℤ defined in Proposition <ref>. 



 Let q: X → Y be a proper morphism.
The morphism q^* induces a morphism of graded rings q^* :^∙(Y) →^∙ (X) which satisfies the projection formula:

    ∀α∈^i(Y), ∀ z ∈_l(X), q_* (q^* α z ) = α q_* z.



 


We only need to prove that the image q^* (^i(Y)) is contained in ^i(X) and that the projection formula is satisfied as it directly implies that q^*: ^∙ (Y) →^∙(X) is a morphism of rings since:

    (α·β)  q_*z =  q^* (α·β)  z = α q_* (q^* β z) = (q^* α· q^*β )  z,
 
for any α∈^i(Y), β∈^l(Y) and any z ∈_i+l(X).





Consider a class α∈^i(Y) which is induced by γ∈^e_1 + i(Y_1) where p_1 : Y_1  → Y is a flat proper morphism of relative dimension e_1. 
Setting X_1 to be the fibre product Y_1 × X and p_1', q' the projections from X_1 to X and X_1 respectively, one remarks using the equality q'_*p_1'^* = p_1^*q_* (<cit.>) that q^* α is induced by q'^* γ, hence q^* α∈^i(X) as required. 
And the projection formula follows easily from the projection formula on divisors (Theorem <ref>.(ii)).

 








Let us sum up all the properties of numerical classes proven so far :
  Let q: X → Y be a proper morphism. For any integer 0 ⩽ i⩽ X and 0 ⩽ l ⩽ Y: 


  (i) The pushforward morphism q_* : Z_i(X) → Z_i(Y) induces a morphism of abelian groups q_* : _i(X) →_i(Y).

  (ii) The dual morphism q^* : Z^l(Y) → Z^l(X) maps ^l(Y) into ^l(X). 

  (iii) The induced morphism q^* : ^∙(Y) →^∙ (X) preserves the structure of graded rings.

  (iv) (Projection formula)For all α∈^l(Y) and all z ∈_i(X), we have q_*( q^*α z) ≡α q_* z in _i-l(Y). 

 




 §.§ Canonical morphism


  
The morphism ψ_X: α∈^i(X) ↦α [X] ∈_n-i(X) is the unique morphism which satisfies the following properties. 


  (i) The image of the morphism : Z_0(X) →ℤ seen as an element of Z^0(X) is given by ψ_X() = [X].

  (ii) The morphism ψ_X is ^i(X)-equivariant, i.e for all α∈^i(X) and all β∈^l(X), we have: 

    ψ_X( α·β) = αψ_X(β).
 

  (iii) Suppose q : X → Y is a generically finite morphism where Y is of dimension n, then we have the following identity:

    q_* ∘ψ_X ∘  q^* = (q) ×ψ_Y .
 



 







Recall that  is the unit in ^∙(X), hence ψ_X() = [X] and (ii) follows directly from the definition and Proposition <ref>. 
 
Assertion (iii) is then a consequence of the projection formula (see Theorem <ref>.(iv)) and the fact that q_*[X] = (q) [Y].


Let us prove that ψ_X is unique. Suppose that φ : ^i(X) →_n-i(X) satisfies the hypothesis of the theorem. 
Since φ() = [X] and since  is the unit element of the ring ^∙(X), we have that for any α∈^i(X), α = α·. By (ii), 

    φ( α ) = φ (α·) = αφ() = α [X] = ψ_X(α),

as required.




Now we prove some properties of ψ_X in some particular cases.
  
The following properties are satisfied.


  (i) If X is smooth, then for all integers 0 ⩽ i ⩽ n,  the induced morphism ψ_X : ^i(X)_ℚ→_n-i(X)_ℚ is an isomorphism.



  (ii) If X is smooth and q: X → Y is a surjective generically finite morphism where Y is a normal projective variety. Then we have for all integer i:

    q^* ( ψ_Y(^n-i(Y)_ℚ)^) =  q^*( ^i(Y)_ℚ) ∩( q_* ∘ψ_X: ^i(X)_ℚ→_n - i(Y)_ℚ ).
 





 (i) Let us show that ψ_X is surjective.
 By the Grothendieck-Riemann-Roch's theorem (Theorem <ref>), the Chern character induces an isomorphism:

    [X] : E ∈ K^0(X) ⊗ℚ→(E)  [X] ∈ A_∙(X) ⊗ℚ.

This implies that the morphism ψ_X: ^i(X)_ℚ→_n-i(X)_ℚ is surjective because any Chern class is the image of a product of Cartier divisors by a flat map (see Remark <ref>). 


We now prove that ψ_X : ^i(X)_ℚ→_n-i(X)_ℚ is injective. 
  Take α_1∈^i(X)_ℚ such that ψ_X(α_1) = 0. By Proposition <ref>, the class α_1 is induced by γ_1 ∈^e_1 + i(X_1)_ℚ where p_1 : X_1 → X is a flat morphism of relative dimension e_1.
 The condition ψ_X(α_1) =0 is equivalent to the equality p_1_* γ_1 = 0 ∈_n-i(X). 
 We need to show that (γ_1 · p_1^*z)= 0 for any cycle z ∈ Z_i(X).
 
As X is smooth, we may compute intersection products inside the Chow group A_∙(X) directly by Remark <ref> and we get:

    (γ_1 · p_1^* z) = ( p_1_* (γ_1 · p_1^* z) ) = (p_1_* γ_1 · z ) = 0

as the class z ∈_i(X) is the image of an element of ^n-i(X)_ℚ by surjectivity of ψ_X.








(ii) We have the following series of equivalence:

    [                   β∈ψ_Y ( ^n- i (Y)_ℚ)^                                       ⇔            ∀α∈^n-i(Y)_ℚ, (βψ_Y(α) ) = 0;                                                                               ⇔  ∀α∈^n-i(Y)_ℚ ,( β (q_* ψ_X q^* α)) = 0;                                                                               ⇔       ∀α∈^n-i(Y)_ℚ , (q^* β· q^* α) = 0;                                                                               ⇔   ∀α∈^n-i(Y)_ℚ , (α q_* ψ_X q^* β ) = 0;                                                                               ⇔ q^* β∈(q_* ∘ψ_X :  ^i(X)_ℚ→_n - i(Y)_ℚ, ]

where the second equivalence follows from Theorem <ref>.(iii), the third and the fourth equivalence from the projection formula, and the last equivalence is a consequence of the fact that ψ_X is self-adjoint :

    ( βψ_Y(α)  ) = (β (α [Y]))=(α( β [Y ])) = (αψ_Y( β)),

where α∈^i(Y) and β∈^n-i(Y).



  The proof of Theorem <ref>.(i) shows that when X is smooth, _i(X)_ℚ is the quotient of Z_i(X)_ℚ by cycles z ∈ Z_i(X)_ℚ such that for any cycle z' ∈ Z_n-i(X)_ℚ, one has (z · z') = 0.


 When X is smooth and when = ℂ, denote by ^i(X) the subgroup of the de Rham cohomology H^2i(X, ℂ) generated by algebraic cycles of dimension i in X. Then there is a surjective morphism ^i(X) →^i(X)_ℚ 













 §.§ Numerical spaces are finite dimensional



 
Both ℚ-vector spaces _i(X)_ℚ and ^i(X)_ℚ are finite dimensional.


 
If X is smooth, then using Remark <ref>, _i(X)_ℚ is the quotient of Z_i(X)_ℚ by the equivalence relation which identifies cycles α and β in Z_i(X)_ℚ if for any cycle z ∈ Z_n-i(X)_ℚ, (z ·α) = ( z ·β). 
In particular, the vector-space _i(X)_ℚ is finitely generated (see <cit.> for a reference), and so is ^i(X)_ℚ using Theorem <ref>.(i). 



If X is not smooth, by DeJong's alteration theorem (cf <cit.>), there exists a smooth projective variety X' and a generically finite surjective morphism q: X' → X. We only need to show that the pushforward q_* : _i(X')_ℚ→_i(X)_ℚ is surjective. 
Indeed this first implies that _i(X)_ℚ is finite dimensional. Since the natural pairing ^i(X)_ℚ×_i(X)_ℚ→ℚ is non degenerate we get an injection of ^i(X)_ℚ onto _ℚ(_i(X)_ℚ, ℚ) which is also finite dimensional.


We take V an irreducible subvariety of codimension i in X. If q^-1(V) =  V, then the class q_* [q^-1(V)] in _ V(X)_ℚ is represented by a cycle of dimension V which is included in V. As V is irreducible, we have q_*[q^-1(V)] ≡λ [V] for some λ∈ℕ^*. 

 If the dimension of q^-1(V) is strictly greater than V, we take W an irreducible component of q^-1(V) such that its image by q_|W : W → V is dominant. 
 We write the dimension of W as V + r where r>0 is an integer. 
Fix an ample divisor H_X on X. 
 The class H_X^r  [W] ∈_ V(X')_ℚ is represented by a cycle of dimension V in W. So the image of the class q_*(H_X^r  [W]) ∈_ V(X)_ℚ is a multiple of [V] which implies the surjectivity of q_*. 






 
For any integer 0 ⩽ i ⩽ n, the pairing ^i(X)_ℝ×_i(X)_ℝ→ℝ is perfect (i.e the canonical morphism from ^i(X)_ℝ to _ℝ( _i(X)_ℝ, ℝ) is an isomorphism).





 Suppose that the dimension of X is 2n, then the morphism ψ_X: ^n(X)_ℚ→_n(X)_ℚ is an isomorphism.



We apply (<ref>) to an alteration X' of X where q : X' → X is a proper surjective morphism and where X' is a smooth projective surface. This proves that ψ_X : ^n(X)_ℚ→_n(X)_ℚ is surjective. By duality, this gives that ψ_X : ^n(X)_ℚ→_n(X)_ℚ is injective. As a consequence, we have that ψ_X:^n(X)_ℚ→_n(X)_ℚ is an isomorphism.




  Let X be a complex normal projective variety with at most rational singularities. We suppose that X is numerically ℚ-factorial in the sense of <cit.>. Then the morphisms ψ_X : ^1(X)_ℚ→_n-1(X)_ℚ and ψ_X:^n-1(X)_ℚ→_1(X)_ℚ are isomorphisms.


 Using <cit.>, then any Weil divisor which is numerically ℚ-Cartier is ℚ-Cartier. In particular, ψ_X : ^1(X)_ℚ →_n-1(X)_ℚ is surjective. Using (<ref>) to an alteration of X' applied to i=1, we have that ψ_X: ^1(X)_ℚ→_n-1(X)_ℚ is injective. Hence ^1(X)_ℚ and _n-1(X)_ℚ are isomorphic and by duality ^n-1(X)_ℚ and _1(X)_ℚ are also isomorphic.








 When X = X(Δ) be a toric variety associated to a complete fan Δ. 
The map ψ_X: ^1(X)_ℚ→_n-1(X)_ℚ is an isomorphism if and only if Δ is a simplicial fan. Indeed, denote by N the lattice containing Δ and M = (N, ℤ) its dual. 

For any cone σ∈ N, we denote by M(σ) the vector space defined by M(σ) = { l ∈ M  | ⟨ l , v ⟩ = 0,  ∀  v ∈σ}. 
The proposition in <cit.> implies that any class in _n-1(X)_ℚ is represented by a torus-invariant Weil ℚ-divisor D = ∑ a_i [V_i] in X(Δ). Since every maximal cone σ in the fan Δ⊂ N is full-dimensional, one has M(σ) = { 0} and there exists an element u(σ) ∈ M / M(σ) = M such that for any 1-dimensional ray v_i ∈σ, one has:

    ⟨ u(σ) , v_i ⟩ = -a_i.

The element u(σ) is uniquely determined if and only if the family of rays v_i ∈σ are linearly independent (i.e Δ is simplicial).












§ POSITIVITY
 


The notion of positivity is relatively well understood for cycles of codimension 1 and of dimension 1. 
For cycles of intermediate dimension this situation is however more subtle and was only recently seriously considered (see <cit.>, <cit.>, <cit.> and the recent series of papers by Fulger and Lehmann (<cit.>, <cit.>).

For our purpose, we will first review the notions of pseudo-effectivity and numerically effective classes. Then we generalize the construction of the basepoint free cone introduced by <cit.> to normal projective varieties. This cone is suitable for stating generalized Siu's inequalities (see Section <ref>).




 §.§ Pseudo-effective and numerically effective cones


As in the previous section, X is a normal projective variety of dimension n. 
To ease notation we shall also write ^i(X) and _i(X) for the real vector spaces ^i(X)_ℝ and _i(X)_ℝ.

 
A class α∈_i(X) is pseudo-effective if it is in the closure of the cone generated by effective classes. This cone is denoted _i(X).  


When i=1, _1(X) is the Mori cone (see e.g <cit.>), and when i = n-1, _n-1(X) is the classical cone of pseudo-effective divisors, its interior being the big cone.

 A class β∈^i(X) is numerically effective (or nef) if for any class α∈_n-i(X), (βα)  ⩾ 0. We denote this cone by ^i(X). 
 


When i= 1, the cone ^1(X) is the cone of numerically effective divisors, its interior is the ample cone.


We can define a notion of effectivity in the dual ^i(X). 
 
A class α∈^i(X) is pseudo-effective if ψ_X(α) ∈_n-i(X). We will write this cone as ^i(X). 


 
A class z ∈_i(X) is numerically effective if for any class α∈^i(X), one has ( α z) ⩾ 0. This cone is denoted _i(X).





By convention, we will write α⩽β (resp. α⩽β) for any α, β∈_i(X) (resp. α, β∈^i(X) ) if β  - α∈_i(X) (resp. β -α∈^i(X)). 

When X is smooth, the morphism ψ_X induces an isomorphism between ^i(X) and _n-i(X), and we can identify these cones:

    [ ^i(X) = _n-i(X) ,;  ^i(X) = _n-i(X). ]





 §.§ Pliant classes


We recall the definition of pliant classes introduced in <cit.> and their main properties. Their definition involve Schur classes which were introduced in Section <ref>.


The pliant cone ^∙(X) is defined as the convex cone generated by product of Schur classes of globally generated vector bundle.

We denote by ^i(X) the set of pliant classes of codimension i in X.

 (see <cit.>) 
The pliant cone ^i(X) satisfies the following properties.


  (i) The cone ^i(X) is a closed convex salient cone with non-empty interior in ^i(X)_ℝ.

  (ii) The cone ^i(X) contains product of ample Cartier divisors in its interior.

  (iii) For all integer i, l, we have ^i(X) ·^l(X) ⊂^i+l(X).

  (iv) For any (proper) morphism q: X → Y, one has that q^* ^i(Y) ⊂^i(X).



We recall another proposition which we will reuse in our proofs.
(cf <cit.>) Let 𝔾 be a Grassmannian variety. Then ^i( 𝔾) = ^i(𝔾).




 §.§ Basepoint free cone on normal projective varieties


In this section, we define a cone ^i(X) and prove in Corollary <ref>  that this cone is equal to the basepoint free cone defined by Fulger-Lehmann when X is smooth. 
This generalizes <cit.> to normal projective varieties and our proof follows closely Fulger-Lehmann's approach.




Recall that a complete intersection γ∈^i+e(X') on X' where p:X' → X  is a flat morphism of relative dimension e and where X' is an equidimensional projective scheme induces naturally (see Definition <ref>) an element  [γ] ∈^i(X)_ℝ = _ℝ(_i(X)_ℝ, ℝ) by intersecting the class γ with the pullback by p of a i-dimensional cycle in X. We also refer to Proposition <ref> for the definition of the product ^i(X)_ℝ×^l(X)_ℝ→^i+l(X)_ℝ.  

 The cone ^i(X) is the closure of the convex cone in ^i(X)_ℝ generated by products of the form [γ_1] ·…· [γ_l] where each γ_j is a product of e_j + i_j ample Cartier divisors on an equidimensional projective scheme X_j which is flat over X of relative  dimension e_j and where i_j are integers satisfying i_1 + … + i_l = i. 


 By definition, the cone ^i(X) contains the products of ample Cartier divisors and Segré classes of anti-ample vector bundles.



Recall also that if q : X → Y is a flat morphism of relative dimension e between projective schemes, then the pushforward is well-defined on numerical cycles 
q_* : ^i(X)_ℝ→^i-e(Y)_ℝ (see Corollary <ref>).





  The cone ^i(X) is  satisfies the following properties.


  (i) The cone ^i(X) is a salient, closed, convex cone with non-empty interior in ^i(X)_ℝ.

  (ii) The cone ^i(X) contains products of ample Cartier divisors in its interior.

  (iii) For all integer i and l, we have  ^i(X) ·^l(X) ⊂^i+l(X). 

  (iv) For any (proper) morphism q: X → Y, we have q^* ^i(Y) ⊂^i(X). 

  (v) For any integer i, we have ^i(X) ⊂^i(X) ∩^i(X). 

  (vi) In codimension 1, one has ^1(X) = ^1(X).

  (vii) For any flat morphism q: X → Y between equidimensional projective schemes of relative dimension e and any integer i ⩾ e, we have q_* ^i(X) ⊂^i - e(Y).



Moreover, (X) is the smallest cone satisfying properties (iii),(vi) and (vii).



We prove successively the items (iii), (vii), (v), (vi), (iv), (ii) and (i). 

(iii), (vii) This follows from the definition of ^i(X).



(v) It is sufficient to prove that for any effective cycle z ∈ Z_n-l(X) and any basepoint free class α∈^i(X), then α z ∈_n-i-l(X). Indeed, apply this successively to z =[X] and z ∈_i(X) give the inclusions ^i(X) ⊂^i(X) and ^i(X)⊂^i(X). 
 By definition of basepoint free classes and by linearity, we can suppose that α is equal to a product [γ_1] ·…· [γ_p] where γ_i ∈^e_j + i_j(X_j)_ℝ are products of ample Cartier divisors on X_j where p_j : X_i → X is a flat proper morphism of relative dimension e_j and where i_j are integers such that i_1 + … + i_p = i.
 By definition, one has 
[γ_1 ]  z = p_1_* (γ_1 ·  p_1^* z). 
Because the cycle z is pseudo-effective, the cycle p_1^* z remains pseudo-effective as p_1 is a flat morphism. As γ_1 is a positive combination of products of ample Cartier divisors, we deduce that the cycle γ_1 · p_1^* z is pseudo-effective. Hence, [γ_1]  z ∈_n-l_1-l(X). Iterating the same argument, we get that α z ∈_n-i-l(X) as required.





(vi) The interior of ^1(X) is equal to the ample cone of X so by definition: 

    (^1(X))  ⊂^1(X).
 
As the closure of the ample cone is the nef cone by <cit.>, one gets ^1(X)  ⊂^1(X). Conversely, the cone ^1(X) is included in the cone ^1(X), so we get ^1(X) = ^1(X). 





 (iv) By linearity and stability by products, we are reduced to treat the case of a class [D] induced by an ample Cartier divisor on Y_1 where  p_1 : Y_1 → Y is a flat proper morphism, and prove that q^*[D] is a limit of ample Cartier divisors on a flat variety over X. 
Let X_1 be the fibre product of Y_1 and X and let q' be the natural projection from X_1 to Y_1, observe that q^*[D] is induced by q'^* D which remains nef on X_1 as q' is proper.
In particular, it is the limit of ample divisors on ^1(X_1). 




 

(i) 
Take α∈^i(X) such that -α∈^i(X). Then for all z ∈_i(X), one has that (α z)= 0 as α is nef by (v). 
Since effective classes of dimension i generate Z_i(X), it follows that  (α z) = 0 for any z ∈_i(X)_ℝ which implies by definition that α  = 0.  
This shows ^i(X) is salient.




(ii) We show now that ^i(X) contains product of ample divisors in its interior.
To do so we prove that ^i(X) ⊂^i(X) for any integer i ⩾ 1.






For i = 1, ^1(X) = ^1(X), and by definition, the divisor h is ample so it is in the interior of the nef cone and we are done.
Take a globally generated vector bundle E of rank r on X and consider the induced morphism ϕ given by:

    ϕ : X →𝔾= G(r, ℙ(H^0(X, E)^*)).

 Since ^i(X) ⊃ϕ^* ^i( 𝔾) and since these cones are preserved by pullbacks, we are then reduced to proving that
 ^i(𝔾) ⊂^i(𝔾).
 Denote by G = (H^0(X,E)^*) the projective special orthogonal group of the vector space H^0(X,E)^* and consider a class α∈^i(𝔾)_ℝ. 
Since 𝔾 is smooth, ψ_𝔾 :^i(𝔾)_ℝ→_n-i(𝔾)_ℝ is an isomorphism by Theorem <ref> and α is represented by an effective cycle z ∈ Z_n-i(𝔾)_ℝ.
 

Consider W the Zariski closure in G ×𝔾 given by:

    W =  { (g, g · x)}_g∈ G, x∈ z⊂ G×𝔾.

By construction, W is a quasi-projective scheme and the projection p : W →𝔾 onto 𝔾 is a flat morphism. 
Denote by q: W → G the projection onto G.  
Fix H a very ample divisor on G and denote by M the dimension of the group (H^0(X,E)^*). Then there exists an open embedding j : W →^M_𝔾 such that one has the following diagram:

    G [r]    ^M     ℙ^M_𝔾[l]_h[rd]^π   
       W [ru]^j[lu]^q[rr]^p       𝔾

where π : ℙ^M_𝔾→𝔾 is the projection onto 𝔾 and h: ℙ^M_𝔾→ℙ^M is the projection onto ^M

  
By construction the general fiber of q over an element g ∈ G is numerically equivalent to α and since we can choose H to be a hyperplane of ^M, we have:

    1(H^M) p_* q^* H^M =  α∈_n-i(𝔾)_ℝ.

Moreover <cit.> implies that p_* j^*  = π_* in Z_n-i(^M_𝔾), hence:

    p_* q^* H^M = π_* h^* H^M = (H^M) α∈_n-i(𝔾)_ℝ.

Since H is ample, h^* H is nef and the class h^* H^M belongs to _n-i(^M_𝔾). 
Assertion (vii) thus implies that the class π_* h^* H^M / (H^M) = α belongs to _n-i(𝔾)_ℝ, as required. 



Since ^i(X) has non-empty interior in ^i(X)_ℝ by Theorem <ref>.(ii), we have proved (ii).


Let us prove that the cone ^i(X) is the smallest cone satisfying properties (iii),(vi) and (vii).
Denote by ' the minimal cone satisfying these conditions. We have that '^i(X) ⊂^i(X) by minimality. 
Take q: X_1 → X a flat morphism of relative dimension e where X_1 is an equidimensional projective scheme and consider α∈^i+e(X_1) a product of ample Cartier divisors on X_1. 
Since q_* : ^i(X_1)_ℝ→^i-e(X)_ℝ and since α∈'^i+e(X_1), we have that q_* α∈'^i(X) by (vii), hence ^i(X) ⊂'^i(X) as required.



We recall Fulger-Lehmann's construction of the basepoint free cone. 
A class α∈_n-i(X)_ℝ is strongly basepoint free if there is: 


  ∙ an equidimensional quasi-projective scheme U of finite type over ,

  ∙ a flat proper morphism s: U → X 

  ∙ and a proper morphism p : U → W of relative dimension n-i to a quasi-projective scheme W such that each component of U surjects onto W

such that 

    s_| F_p_* ([F_p]) = α,
 
where [F_p] is the fundamental class of a general fiber of p. 
We denote by '^i(X) the closure of the convex cone generated by strongly basepoint free classes in this sense. 
The cone '(X) as above was defined by Fulger-Lehmann and they proved that this cone satisfies Theorem <ref> when X is smooth (<cit.>). The following result proves that the cones '(X) and (X) are equal in this case. 

  Suppose X is smooth, then the cone ^i(X) is equal to the basepoint free cone '^i(X). 


 Our construction of the cone (X) allows us to generalize Fulger-Lehmann's result for normal varieties. This improvement is due to the fact that we are able to pushforward dual numerical classes by flat morphism. 



By <cit.>, the cone '(X) satisfies the conditions of Theorem <ref>, hence ^i(X) ⊂'^i(X). 
Let us prove the reverse inclusion '^i(X) ⊂^i(X). 
Take p : U → W a projective morphism onto an equidimensional quasi-projective variety W where U is a quasi-projective scheme and a flat map s: U → X such that s_* [F_p] = α where F_p is a general fiber of p. 
Take H_W an ample divisor on W, then the class α satisfies:

    α = s_* p^* H_W^i+e∈_n-i(X)_ℝ.

Choose an ample divisor H on  U, since the class p^* H_W is nef, for any ϵ >0, the divisor p^* H_W + ϵ H is ample.

Since the morphism s: U → X is also quasi-projective and there exists an integer l (which depends on ϵ) such that the following diagram is commutative

    ℙ^l_X [rd]^π   
    
    U [ru]^f_ϵ[rr]^s        X

where f_ϵ : U →ℙ^l_X is an immersion induced by p^* H_W + ϵ H and π: ℙ^l_X → X is the flat projection onto X. 

Let ξ be the relative class c_1(𝒪_ℙ^l_X(1)) on ℙ^l_X, then one has that for any cycle z ∈ Z_i(X)_ℝ:

    ((p^* H_W + ϵ H)^i+e· s^*z) = ( ξ^i+e·π^* z),

since f_ϵ^* ξ = p^* H_W + ϵ H. 
Hence, we obtain:

    ( s_* (p^* H_W + ϵ H)^i+e· z) = (π_* ξ^i+e· z).

Since the class ξ^i+e is nef and since these cones are stable by flat pushforward, we have π_*(ξ^i+e) ⊂^i(X). 
Taking the limit as ϵ→ 0, we have that s_* (p^* H_W + ϵ H)^i+e→α = s_* p^* H_W^i+e, hence α∈^i(X) since each class s_* (p^* H_W + ϵ H)^i+e) ∈^i(X)_ℝ belongs to ^i(X).












We give here a detailed proof of the fact that the pseudo-effective cone is salient (see also <cit.>). The proof uses a useful proposition that we will use later on.  


 

Let α∈_n-i(X) be a pseudo-effective class on X and γ∈^n-i(X) be class lying in the interior of the basepoint free cone. 
Then we have ( γα) = 0 if and only if α = 0.  




Let us fix two basepoint free classes β and γ in ^n-i(X), and a norm || · || on ^n-i(X)_ℝ. As γ is in the interior of ^n-i(X) by Theorem <ref>.(ii), there exists a positive constant C > 0 such that for any β∈^n-i(X), one has:

    C || β ||_^n-i(X)_ℝγ  -β∈^n-i(X).
  
Intersecting with α∈_n-i(X) and using Theorem <ref>.(v), we have that (β·α )= 0.
 Since the basepoint free cone ^n-i(X) generates all ^n-i(X)_ℝ by Theorem <ref>.(i), we have proved that (β' α) = 0 for any β' ∈^n-i(X), hence α= 0 as required.



  
The pseudo-effective cone _n-i(X) is a closed, convex, full dimensional salient cone in _n-i(X)_ℝ.



We take u ∈_n-i(X) such that -u ∈_n-i(X), then for any ample Cartier divisor H_X on X, the products (H_X^n-i· u) and (-u · H_X^n-i) are non-negative hence (u · H_X^n-i) = 0. This implies that u = 0 by Proposition <ref>.








 §.§ Siu's inequality in arbitrary codimension
 











We recall Siu's inequality:

  (<cit.>) Let V be a closed subscheme of dimension r in X and
let A,B be two -divisors nef on X such that A_|V is big, then we have in _i-1(X),

    B  [V] ⩽r ( (A^r-1· B ) [V] )(A^r  [V])  A  [V].





 The case V= X is a consequence of the bigness criterion given in <cit.>, however we will need the result for possibly non-reduced subschemes of X. 


 The proof of the previous proposition implies that B_|V⩽ r (A^r-1· B  [V])/ (A^r  [V]) × A_|V in the Chow group A^1(V). However, since we want to work in the numerical group, we compare these classes in X (we look at their pushforward by the inclusion of V in X).


 The proof is the same as in <cit.>, that is to find a section of the line bundle 𝒪_V(m(A -B)). 
Up to some small pertubations of A and B of the form  A + ϵ H and B + ϵ H of A and B where ϵ→ 0, we can suppose that A and B are ample. 
Moreover, by taking a high multiple of A and B, we can suppose that they are also both very ample.  
Since B is very ample, we choose m general elements E_j of the linear system |B| and consider the exact sequence:

    0 [r]    𝒪_V(mA - mB) [r]   𝒪_V(mA) [r]   𝒪_∪ E_j(mA)[r]     0 .

Taking long exact sequence associated, one obtains the minoration:

    h^0(V, 𝒪_V(mA-mB) ⩾ h^0(V , 𝒪_V(mA)) -  h^0( ∪_j=1^m E_j, 𝒪_∪_j=1^m E_j(mA)).

Observe that [∪ E_j] = ∑_j=1^m [E_j] = m B  [V].
 Applying <cit.> to the nef divisor A, we get h^0(V, 𝒪_V(mA))= m^r/(r!) (A^r  [V]) + o(m^r) and 
 
    h^0(∪ E_j , 𝒪_∪_j=1^m E_j(mA))=  ∑_j=1^m m^r-1(r-1)! A^r-1· B  [V] + o (m^r).
  
 Hence,
 
    h^0(V, 𝒪_V(mA-mB))⩾m^rr! (A^r - r A^r-1· B)  [V] + o(m^r).

 In particular, this implies the required inequality.




The next result is a key for our approach to controlling degrees of dominant rational maps. 


   
Let i be an integer and V be a closed subscheme of dimension r in X. For any Cartier divisors α_1, …, α_i and β which are big and nef on V, then there exists a constant C>0 depending only on r and i such that:

    (α_1 ·…·α_i)  [V] ⩽ (r-i+1)^i (α_1 ·…·α_i ·β^r-i [V])(β^r  [V])×β^i  [V] ∈_r-i(X).


Observe that (β^n) >0 since β is big.



By continuity, we can suppose that α_i and β are ample Cartier divisors. 
We apply successively Siu's inequality by restriction to subschemes representing the classes α_2 ·…·α_i  [V] , β·α_3 ·…·α_i  [V], …, β^i-1·α_i  [V]:  


    [                                                             α_1 ·α_2 ·…·α_i  [V]                                                                                ⩽    (r-i+1) (α_1 ·…·α_i ·β^r-i [V]) (β^r-i+1·α_2 ·…·α_i  [V]) ×β·α_2 ·…·α_i  [V],;                                                                β·α_2 ·…·α_i  [V]                                                                                ⩽ (r-i+1) (β^r-i+1·α_2 ·…·α_i [V]) (β^r-i+2·α_3 ·…·α_i [V]) ×β^2 ·α_3 ·…·α_i [V] ,;                                                                                …                                                                                                                                                                 …;                                                                    β^i-1·α_i [V]                                                                                ⩽                                        (r-i+1) (β^r-1·α_i [V])(β^r [V])×β^i [V]. ]

This gives the required inequality:

    α_1 ·…·α_i [V] ⩽ (n-i+1)^i (α_1 ·…·α_i ·β^r-i [V]) (β^r [V])×β^i [V].





  Let i be an integer, then for any a ∈^i(X) and any big nef Cartier divisor β on X, one has:

    a ⩽ (n-i+1)^i ( a ·β^n-i)(β^n)×β^i.

  


 By linearity and stability by product, we just need to prove the inequality for a = D_1 ·…· D_e_1 + i∈^e_1 + i(X_1), where D_i are ample Cartier divisors X_1, where p_1: X_1 → X is a flat proper morphism of relative dimension e_1. We apply Theorem <ref> to a'= D_e_1 + 1·…· D_e_1 + i· Z and β' = p_1^* β_|Z where Z = D_1 ·…· D_e_1. We obtain:

    a ⩽  (n-i+1)^i  (a · p_1^*β^n-i )( p_1^* β^n · Z)×  p_1^* b^i· Z.

As the restriction of p_1 on Z is generically finite, by the projection formula, we get:

    a ⩽ (n-i + 1)^i (a·β^n-i)(β^n)×β^i.
 



The previous inequality can be applied when we have positivity hypothesis on a birational model as follows. 

 Let X,Y be two normal projective varieties of dimension n. Let  β be a class in ^i(Y), we suppose there exists a birational morphism q: X → Y and an ample Cartier divisor A on X such that A^i  ⩽ q^* β. Then there exists a class β^* ∈_i(X)_ℝ∩_i(X) such that for any class α∈^i(X), we have:

    α⩽ ( αβ^* ) ×β.
   


 We just have to set β^* = (n-i+1)^i  (A^n) q_* ψ_X( A^n-i). 



 We conjecture that for any basepoint free class a ∈^i(X) and any big nef divisor b, one has 

    a ⩽ ( [ n; i ] )    (a · b^n-i)(b^n)b^i.

One can show that this inequality (if true) is optimal since equality can happen when X is an abelian variety.




































 §.§ Norms on numerical classes


In this section, the positivity properties combined with Siu's inequality allows us to define some norms on _i(X)_ℝ and on ^i(X)_ℝ. 




  §.§.§ Norms on _i(X)_ℝ



Let i ⩽ n be an integer and let γ∈^i(X) be a basepoint free class on X. Any cycle z ∈_i(X)_ℝ can be written z =z^+ - z^- where z^+ and z^- are pseudo-effective. We define :


    F_γ(z) := inf_ z = z^+ - z^- 
    
    z^+, z^- ∈_i(X)
    { (γ z^+)  + (γ z^-)  } .



  
For any class γ∈^i(X) lying in the interior of the basepoint free cone, the function F_γ defines a norm on _i(X)_ℝ. In particular, if we fix a norm ||· ||__i(X)_ℝ on _i(X)_ℝ, there exists a constant C> 0 such that for any pseudo-effective class z ∈_i(X), one has:

    1C || z||__i(X)_ℝ⩽ (γ z) ⩽ C || z ||__i(X)_ℝ.





The only point to clarify is that F_γ(z) = 0 implies z = 0. 
Observe that Proposition <ref> implies the result for z ∈_i(X).
In general, pick any two sequences
(z_p^+)_p∈ℕ and (z_p^-)_p ∈ℕ in _i(X) such that z = z_p^+ - z_p^- and such that 
γ· z_p^+  + γ· z_p^-  ⟶ 0.
Since z_p^+ and z_p^- are pseudo-effective and γ is basepoint free, it follows from Theorem <ref>.(v) that 

    lim_p→ + ∞ (γ· z_p^+)  = lim_p → + ∞ (γ·  z_p^-) = 0 .

As γ lies in the interior of ^i(X), given any β in ^i(X), one has that C γ - β is still in ^i(X) for some sufficently large constant C>0. 
Intersecting with the pseudo-effective classes z_p^+ and z_p^- and using Theorem <ref>.(v), we have lim_p →∞(β z_p^+)=lim_p →∞(β z_p^-)= 0, thus (β z)= 0. Since the basepoint free cone ^i(X) generates all ^i(X) by Theorem <ref>.(i), we conclude that z = 0 as required.










  §.§.§ Norms on ^i(X)_ℝ



 
We define the subcone _0^i(X) of ^i(X) as the classes α∈^i(X) such that for any birational map q : X' → X, there exists an ample Cartier divisor A on X' such that q^* α⩾ A^i.   



 When X is smooth, the cone _0^1(X) is equal to the big nef cone. In particular _0^i is neither closed nor open in general.



Take α∈^1(X)_ℝ a big nef divisor. Then for any birational map q : X' → X and any ample Cartier divisor A, one has by Theorem <ref> applied to A and q^*α:

    A ⩽ n (A · q^*α^n-1)(α^n) q^* α.

Hence, α∈^1_0(X). 
Conversely, take a class α∈^1_0(X), then there exists an ample divisor A on X such that α⩾ A. 
Since ample divisors are big, we have that α is big. 
Moreover, since ^1(X) = ^1(X) ∩^1(X), we have that α is big and nef as required.






 
The cone _0^i(X) is a convex open subset of ^i(X) that contains the classes induced by products of big nef divisors.



The cone _0^i(X) contains the products of big and nef Cartier divisors. 
The fact that _0^i(X) is convex is a consequence of Siu's inequality. We take two elements α and β in _0^i(X) and any birational map q: X' → X. 
By definition, there exists some ample Cartier divisors A and B on X' such that q^*α⩾ A^i and q^*β⩾ b^i. 
As A and B are ample, there is a constant C>0 such that A^i ⩾ C b^i using the generalization of Siu's inequality (Theorem <ref>). This proves that q^* (t ×α + (1-t)×β) ⩾ ( t C + (1-t) )× b^i for any t ∈ [ 0, 1]. Hence t×α + (1-t) ×β∈_0^i(X) and the cone _0^i(X) is convex.


We prove that _0^i(X) is an open subset of ^i(X). We take α∈_0^i(X). We take any ample Cartier divisor H_X on X such that α - t H_X^i is in ^i(X) for small t> 0. We just need to show that α - tH_X^i stays in _0^i(X) when t is small enough. 
Let q: X' → X be a birational map where X' is projective and normal. By definition of α, there exists an ample Cartier divisor A on X' such that q^*α⩾  A^i. By Siu's inequality, there exists a constant C such that:

    q^* H_X^i ⩽ C (A^i · q^* H_X^n-i)(H_X^n)× A^i.
 
This implies the inequality:

    q^* β - t q^* H_X^i ⩾  ( 1 - t C A^i · H_X^n-iH_X^n ) × A^i .

As A^i ⩽ q^* α, we have the following upper bound:

    (A^i · H_X^n-i )⩽ (q^* α· q^* H_X^n-i).

We get the following minoration which depends only on α and H_X:

    1- t C (α· H_X^n-i) (H_X^n)⩽  1 - tC (A^i · H_X^n-i)(H_X^n).
 
Using (<ref>) and (<ref>), one gets that for t <  (H_X^n) C (α· H_X^n-i), the class α - t H_X^i is in _0^i(X).




 The cone _0^i(X) is not always equal to the cone generated by complete intersections. Following <cit.>, there exists a smooth toric threefold such that the cone generated by complete intersections in _1(X)_ℝ is not convex, so it cannot be equal to _0^2(X) using the following proposition. 










Let X be a normal projective variety of dimension n. Any class α∈^i(X)_ℝ can be decomposed as α^+ - α^- where α^+ and α^- are basepoint free classes. For any γ∈_0^n-i(X), we define the function:

    G_γ (α) :=  inf_α = α^+ - α^- 
    α^+, α^- ∈^i(X)
    { (γ·α^+)  + (γ·α^-)  } .



 
For any γ∈_0^n-i(X), the function G_γ defines a norm on ^i(X)_ℝ. In particular, for any norm || · ||_^i(X)_ℝ on ^i(X)_ℝ, there is a constant C>0 such that for any class α∈^i(X):

    1C || α ||_^i(X)_ℝ⩽ (γ·α) ⩽ C || α ||_^i(X)_ℝ.
 


 
 The only fact which is not immediate is the fact that G_γ(α) = 0 implies α = 0. We are reduced to treat the case where α∈^i(X).



Suppose first that X is smooth.
Since γ belongs to the interior of the basepoint free cone by Proposition <ref>, one has that for any basepoint free class β∈^n-i(X), there exists a constant C>0 such that:

    C || β || γ - β∈^n-i(X).

In particular, since α is nef, one has:

    0= G_γ(α) = C || β|| (γ·α) ⩾ ( β·α ) ⩾ 0.

Hence (β·α) = 0 for any basepoint free class β∈^n-i(X) and α = 0 ∈^i(X)_ℝ since the basepoint free cone generates all ^n-i(X)_ℝ by Theorem <ref>.(i). 



Suppose that X is not smooth. Fix an ample Cartier divisor H_X on X.
Take an alteration π : X' → X of X. Since the morphism π^* : ^i(X)_ℝ→^i(X')_ℝ is injective, we are reduced to prove that π^* α = 0. 
Consider β∈^n-i(X), we have by the projection formula that:

    (π^* γ·π^* α) = (α·γ).

Since γ belongs to the interior of the basepoint free cone, there exists a constant C>0 such that:

    H_X^n-i⩽ C γ.
 
In particular, this implies that:

    (π^*H_X^n-i·π^*α ) = ( H_X^n-i·α)=0.

Since π^*H_X is a big nef Cartier divisor, the class π^* H_X^n-i belongs to ^n-i_0(X') by Proposition <ref>, hence π^* α = 0 by the previous argument. 




 In fact, the above proof gives a stronger statement:
for any generically finite morphism q : X' → X and any γ∈_0^n-i(X), the function G_q^* γ defines a norm on ^i(X')_ℝ. 





§ RELATIVE NUMERICAL CLASSES
 




 §.§ Relative classes



In this section, we fix q:X → Y a surjective proper morphism between normal projective varieties where X=n, Y = l and we denote by e =  X -  Y the relative dimension of q.

 
 
 
The abelian group _i(X/Y) is the subgroup of _i(X) generated by classes of subvarieties V of X such that the image q(V) is a point in Y. 
 
Observe that by definition, there is a natural injection from _i(X/Y) into _i(X):

    0 [r]    _i(X/Y)  [r]   _i(X) .

 
 
The abelian group ^i(X/Y) is the quotient of Z^i(X) by the equivalence relation ≡_Y where α≡_Y 0 if for any cycle z ∈ Z_i(X) whose image by q is a collection of points in Y, we have (α z)= 0.
 

 Therefore, one has the following exact sequence:

    ^i(X)  [r]   ^i(X/Y) [r]     0 .

  
  
As before, we write _i(X/Y)_ℝ = _i(X/Y) ⊗_ℤℝ, ^i(X/Y)_ℝ = ^i(X/Y) ⊗ℝ, _∙(X/Y) = ⊕_i(X/Y) and ^∙(X/Y) = ⊕^i(X/Y).    

 
The abelian groups _i(X/Y) and ^i(X/Y) are torsion free and of finite type. Moreover, the pairing _i(X/Y)_ℚ×^i(X/Y)_ℚ→ℚ induced by the pairing _i(X)_ℚ×^i(X)_ℚ→ℚ is perfect.
 


Since _i(X/Y) is a subgroup of _i(X), it is torsion free and of finite type. The group ^i(X/Y) is also torsion free.
Indeed pick α∈ Z^i(X) such that p α≡_Y 0 for some integer p, then for any cycle z whose image by q is a union of points, we have (p α z) = p (α z) = 0 hence α≡_Y 0.  
Finally, since there is a surjection from ^i(X) to ^i(X/Y), the group ^i(X/Y) is also of finite type.

Let us show that the pairing is well defined and non degenerate.
Take a cycle z ∈ Z_i(X)_ℚ such that q(z) is a finite number of points in Y, then if α∈^i(X) such that its image is 0 in ^i(X/Y), then (α z)= 0 and the pairing ^i(X/Y) ×_i(X/Y) →ℤ is well-defined.
Let us suppose that for any α∈^i(X/Y)_ℚ, (α z)=0. This implies that for any β∈^i(X), the intersection product (β z) = 0, thus z ≡ 0. 
Conversely, suppose that (α z)=0 for any z ∈_i(X/Y), then by definition α≡_Y 0.  




 

 When Y is a point, we have _i(X/Y) = _i(X) and ^i(X/Y) = ^i(X).
  

 
 If the morphism q : X → Y is finite, then we have ^0(X/Y)_ℚ = _0(X/Y)_ℚ = ℚ and ^i(X/Y) = _i(X/Y) = {0 } for i⩾ 1 since X is irreducible.
 
 

When i=1, the group _1(X/Y) is generated by curves contracted by q so that ^1(X/Y) is the relative Neron-Severi group and its dimension is the relative Picard number (see <cit.>). 
  
 
 When i is greater than the relative dimension, the relative classes might not be trivial. For example if q : X → Y is a birational map, then e=0 but the space ^1(X/Y)_ℝ is generated by classes of exceptional divisors of q.  








 
The intersection product on ^∙(X) induces a structure of algebra on ^∙(X/Y). Moreover, the action from ^∙(X) on _∙(X) induces an action from ^∙(X/Y) on _∙(X/Y), so that the vector space _∙(X/Y)_ℝ becomes  a ^∙(X/Y)_ℝ-module. 


 
Observe that if z ∈ Z_i(X) such that q(z) is a union of points in Y and α∈^l(X), then α z lies in _i-l(X/Y). Indeed, by definition, the class α z is represented by a cycle supported in z, so its image by q is a collection of points in Y.

Let us now prove that the product is well-defined in ^∙(X/Y). Take α∈^i(X) such that α = 0 in ^i(X/Y) and β∈^l(X), we must prove that α·β =0 in ^i(X/Y). Pick a cycle z ∈ Z_i+l(X) whose image by q is a collection of points, by the properties of the intersection product, ((α·β ) z)= ( α (β z)). As β z is in _i(X/Y), we get that ( (α·β)  z )= 0 as required.



As an illustration, we give an explicit description of these groups in a particular example. 
  
 Suppose q : X= (E) → Y where E  is a vector bundle of rank e+1 on Y. Then for any integer 0  ⩽ i ⩽ e, one has:
 
    _i(X/Y)_ℚ = ℚ ξ^e-i q^* [pt] ,
 

    ^i(X/Y)_ℚ = ℚ ξ^i,
 
where ξ = c_1(𝒪_(E)(1)).
 

 Since the pairing ^i(X/Y)_ℚ×_i(X/Y)_ℚ→ℚ is non degenerate and since (ξ^i ( ξ^e-i q^*[pt])) = 1, the second equality is an immediate consequence of the first one.
We suppose first that i > 0. 
Pick α∈ Z_i(X) which defines a class in _i(X/Y)_ℚ. 
Using <cit.>, α is rationally equivalent to ∑_ e-i ⩽ j ⩽ eξ^j   q^* α_j where α_j is an element of the Chow group A_i-e+j(Y)_ℚ. 
Since the image of α by q is a union of points in Y, we have that q_* α = 0 in A_i(Y)_ℚ. 
Observe that 

    q_* (ξ^e  q^* α_e) = α_e,

and that for any j < e, one has that

    q_* (ξ^j  q^* α_j ) = 0

since the support of the cycle α_i is of dimension i-e+j < i and q_*(ξ^e  q^* α_j) belongs to A_i(Y). 
 Hence the conditions q_* α = 0 implies that α_e = 0 in A_i(Y)_ℚ. 
 Since ξ^j α defines also a class in _i-j(X/Y)_ℚ, this implies also that α_e-j = 0 in A_i-e+j(Y)_ℚ for any j < i. We have finally that in _i(X/Y)_ℚ:

    α = ξ^e-i q^* α_e-i.

Since α_e-i belongs to A_0(Y)_ℚ and _0(Y) = ℚ [pt], the ℚ-module _i(X/Y) is generated by ξ^e-i q^*[pt] for i> 0.



For i = 0, the groups _0(X)_ℚ and _0(X/Y)_ℚ are isomorphic to ℚ, so we get the desired conclusion.





 §.§ Pullback and pushforward


In this section, we fix any two (proper) surjective morphisms q_1 : X_1 → Y_1, q_2 : X_2 → Y_2 between normal projective varieties.
To simplify the notation, we write X_1/_q_1 Y_1 fg X_2/_q_2 Y_2 when we have two regular maps f : X_1 → X_2 and g : Y_1 → Y_2 such that q_2 ∘ f = g ∘ q_1 and we shall say that X_1/_q_1 Y_1  fg X_2/_q_2Y_2 is a morphism. 
When f : X_1  X_2 and g : Y_1  Y_2 are merely rational maps, then we write X_1/_q_1 Y_1 fg X_2/_q_2 Y_2 and we shall call it a rational map.

  Let X_1/_q_1 Y_1  fg X_2/_q_2Y_2 be a morphism. Then the morphism of abelian groups f_* : _i(X_1) →_i(X_2) induces a morphism of abelian groups f_* : _i(X_1/Y_1) →_i(X_2/Y_2).  
 
 
 Take a cycle z ∈ Z_i(X_1) such that q_1(z) is a union of points of Y_1.
Then the image of the cycle z by q_2 ∘ f is also a union of points of Y_2 due to the fact that q_2 ∘ f = g ∘ q_1. Hence f_* maps _i(X_1/Y_1) to _i(X_2/Y_2).  

 
  Let X_1/_q_1 Y_1  fg X_2/_q_2Y_2 be a morphism. Then the morphism of graded rings f^* : ^∙(X_1) →^∙(X_2) induces a morphism of graded rings f^* : ^∙(X_1/Y_1)_ℚ→^∙(X_2/Y_2)_ℚ. 
 
  This results follows immediately by duality from the previous proposition since the pairing ^i(X_i/Y_i)_ℚ×_i(X_i/Y_i)_ℚ→ℚ is non degenerate. 
 


 §.§ Restriction to a general fiber and relative canonical morphism










Recall that X = n, Y = l and that the relative dimension of q: X→ Y is e. 

There exists a unique class α_X/Y∈^l(X)_ℚ satisfying the following conditions.


  * The image ψ_X(α_X/Y) belongs to the subspace _e(X/Y)_ℚ of _e(X)_ℚ.

  * For any class β∈_l(X)_ℚ, q_* β = (α_X/Yβ)   [Y].

Moreover, for any open subset V of Y such that the restriction q to U=q^-1(V) is flat, and for all y ∈ V and any irreducible component F of the scheme-theoretic fiber X_y, we have:

    ψ_X(α_X/Y) = [X_y] = r [F],

where r is a rational number which only depends on F and 
where [X_y] (resp. [F]) denotes the fundamental class of X_y (resp. F) viewed as an element of _e(X/Y).

More explicitly, the class α_X/Y is given by

    α_X/Y = 1(H_Y^l) q^* H_Y^l ∈^l(X/Y)_ℚ,
 
 where H_Y is an ample divisor on Y.


 Recall that by generic flatness (see <cit.>), one can always find an open subset V of Y such that the restriction of q to q^-1(V) is flat over V.


Fix an ample Cartier divisor H_Y on Y, we set

    α_X/Y := 1(H_Y^l) q^* H_Y^l ∈^l(X)_ℚ.


Write the class H_Y^l in A_0(Y) as:

    H_Y^l = ∑ a_j [p_j]

where p_j ∈ V() are points in V and a_j are positive integers satisfying ∑ a_j = (H_Y^l).
By the projection formula (Theorem <ref>.(iv)), the class α_X/Y satisfies (i) and (ii)  . 
Let us show that any class satisfying (i) and (ii) is unique. 
Suppose there is another one α' ∈^l(X)_ℚ. 
Then for any class β∈_l(X)_ℚ, ((α_X/Y - α') β ) =0 so that α = α' since the pairing ^l(X)_ℚ×_l(X)_ℚ→ℚ is non degenerate.


Let us prove the last assertion. 
 By generic flatness <cit.>, 
 Let V be an open subset of Y such that the restriction q_|q^-1(V) : q^-1(V) → V is flat and such that the dimension of every fiber is e. 
Since H_Y is ample, we can find some hyperplanes of H_i ⊂ Y such that H_1 ∩…∩ H_l represents the class H_Y^l and such that H_1 ∩…∩ H_l ⊂ V. 
In particular, by <cit.>, the pullback q^* H_Y^l is represented by a cycle in the fiber of H_1 ∩…∩ H_l. Denote by u : V → Y and g: U → X the inclusion maps of V and U into Y and X respectively. 
The morphisms u and g are open embedding hence are flat. Moreover we have the following commutative diagram. 

    U [d]^q_|U[r]^u    X [d]^q
    
     V [r]^g    Y

 
Using <cit.>, one has that for any β∈ A_l(X): 

    (q^*  H_Y^l β)  =   ( q_|U^* g^*(H_Y^l)  u^*β).
 
Using (<ref>), one obtains in A_e(X):

    q_|U^* g^* H_Y^l =  q_|U^* g^*(∑ a_j [p_j]) = ∑ a_j [q^-1(p_j)],

which is well-defined since the restriction of q on U is flat. 
By <cit.>, we have that [X_p_j] = [X_y] ∈_e(X) for any p_j, y ∈ V. In particular, we have:

    ψ_X(q^* H_Y^l) = (∑ a_j)  [X_y] = (H_Y^l)  [X_y] ∈_e(X),

where y is a point in V, which proves that ψ_X(α_X/Y) = [X_y] in _e(X)_ℚ for any point y in V.
By the Stein factorization theorem, there exists a morphism q' : X → Y' with connected fibres and a finite morphism f : Y' → Y such that q' = q ∘ f. 
Since (H_Y^l)[X_y] = q^* H_Y^l = q'^*f^* H_Y^l and since f^* H_Y^l ∈^l(Y')_ℝ which is canonically isomorphism to ℝ, we have that f^* H_Y^l = p · [y'] ∈^l(Y')_ℝ where p is an integer and where [y'] is a general point in f^-1(y).
 We have thus proven that:
 
    [X_y] = p(H_Y^l) [q'^-1(y')] ∈_e(X),

 and q'^-1(y') is an irreducible component of X_y as required.


The class previously constructed allows us to define a restriction morphism.
 Suppose that Y = l and that H_Y is an ample Cartier divisor on Y, then we define _X/Y: _∙(X)_ℚ→_∙ -l(X/Y)_ℚ by setting:

    _X/Y (β) := 1(H_Y^l) q^* H_Y^l β =  α_X/Yβ.

This morphism does not depend on the choice of H_Y.

We shall denote by _X/Y^* : β∈^∙(X/Y)_ℚ→α_X/Y·β∈^∙ + l(X)_ℚ the dual morphism induced by _X/Y with respect to the pairing ^∙(X/Y)_ℚ×_∙(X/Y)_ℚ→ℚ. 


 Recall that Y=l. The following properties are satisfied.


  * For any class α∈^∙(X)_ℚ, one has:

    ψ_X ∘_X/Y^* (α) = _X/Y∘ψ_X (α).


  * For any morphism X'/_q'Y' fg X/_q Y where X' =  X=n and Y' =  Y = l such that the topological degree of g is d, we have for any α∈^i-l(X/Y)_ℚ:

    d ×_X'/Y'^* ∘ f^* α = f^* ∘_X/Y^* α.
 



The definition of the restriction morphism gives a natural way to generalize the definition of the canonical morphism ψ_X : ^i (X) →_n-i(X) to the relative case.

 
Recall that the relative dimension of the morphism q: X → Y is e. For any integer i⩾ 0, we define the canonical morphism ψ_X/Y by:

    ψ_X/Y := ψ_X ∘_X/Y^* :  β∈^i(X/Y)_ℚ→ψ_X(α_X/Y·β) ∈_e-i(X/Y)_ℚ.



 When i> e by convention the map ψ_X/Y is zero.



We give here a situation where this map is an isomorphism.
 Suppose q: X → Y is a smooth morphism of relative dimension e, then for any integer 0 ⩽ i ⩽ e, the map ψ_X/Y: ^i(X/Y)_ℚ→_e-i(X/Y)_ℚ is an isomorphism.



Since the pairing ^i(X/Y)_ℚ×_i(X/Y)_ℚ→ℚ is perfect by Proposition <ref>, we have that the dual morphism ψ_X/Y^* : ^e-i(X/Y)_ℚ→_i(X/Y)_ℚ of ψ_X/Y is surjective whenever ψ_X/Y: ^i(X/Y)_ℚ→_e-i(X/Y)_ℚ is injective. 
We are thus reduced to prove the injectivity of ψ_X/Y : ^i(X/Y)_ℚ→_e-i(X/Y)_ℚ. 
Take a ∈^i(X/Y)_ℚ such that ψ_X/Y(a) = 0, and choose a class α∈^i(X)_ℚ representing a.  
 We fix a subvariety V of dimension i in a fiber X_y of q where y is a point in Y. 
 We need to prove that (α [V]) = 0.
By Proposition <ref>, the condition ψ_X/Y(α) = 0 implies that:

    α [X_y] = 0 ∈_e-i(X)_ℚ.
 
As the morphism q: X → Y is smooth, the fiber X_y over y is smooth. 
By Theorem <ref>, there exists a class β∈^e-i(X_y)_ℚ such that:

    β [X_y] = [V].

In particular, we get:

    (α [V] )= (α (β [X_y]))= (β (α [X_y])) = 0

as required.



 If X = (E) where E is a vector bundle on Y, then Proposition <ref> implies that ψ_X/Y : ^i(X/Y)_ℚ→_e-i(X/Y)_ℚ is an isomorphism for any integer 0 ⩽ i ⩽ e.


 
If X is the blow-up of ^1 ×^1 at a point and q is the projection from ^1×^1 to the first component Y=^1 composed with the blow-down from X to ^1 ×^1. Then the morphism ψ_X/Y : ^0(X/Y)_ℚ→_1(X/Y)_ℚ is not surjective and ψ_X/Y: ^1(X/Y)_ℚ→_0(X/Y)_ℚ is not injective. 

















§ APPLICATION TO DYNAMICS
 


In this section, we shall consider various normal projective varieties X_j and Y_j respectively of dimension n and l and we write e = n-l
 Recall from Section <ref> that the notation X_j /_q_j Y_j means that q_j : X_j → Y_j is a surjective morphism of relative dimension e and that X/_q Y  fg X'/_q' Y' means that f: X  X' and g : Y  Y' are dominant rational maps such that q' ∘ f = g ∘ q.
We shall also fix H_X_j and H_Y_j big and nef Cartier divisors on X_j and Y_j respectively.


 In this section we prove  Theorem <ref> and Theorem <ref>. 
 They will follow from Theorem <ref> and Theorem <ref> respectively.





 §.§ Degrees of rational maps





  Let us consider a rational map 
X_1/_q_1 Y_1 fg X_2/_q_2 Y_2 and let Γ_f (resp. Γ_g) be the  normalization of the graph of f (resp. g) in X_1 × X_2 (resp. Y_1 × Y_2).
We denote by Γ̃_̃f̃ the normalization of the graph of the map induced by q ∘ f from Γ_f to Γ_g, we thus have the following diagram. 

    Γ̃_̃f̃[ld]_π_1[rd]^π_2@/^1pc/[ddd]^ϖ   
    
    X_1 [d]^q_1@–>[rr]^f      X_2 [d]^q_2
    
    Y_1 @–>[rr]^g       Y_2
       Γ_g [ul]^π_1'[ur]_π_2'

The  i-th relative degree of f is defined by the formula:

    _i(f) : = ( π_1^*(H_X_1^e-i· (q_1^*H_Y_1)^l) ·π_2^*(H_X_2)^i).

When Y_1 and Y_2 are reduced to a point, we simply write _i(f) = _i(f).

 If e = 0, then _i(f) = (q_1^*H_Y_1^l) if i = 0 and _i(f) = 0 for i >0.


 Observe that in the above diagram, the ϖ : Γ̃_̃f̃→Γ_g is a regular surjective morphism.


Note that the degrees always depend on the choice of the big nef divisors, but to simplify the notations, we deliberately omit it. 

We now explain how to associate to any rational map X_1 /_q_1 Y_1 fg X_2/_q_2 Y_2 a pullback operator (f,g)^∙,i.

 Let X_1/_q_1 Y_1 fg X_2/_q_2 Y_2 be a rational map and let π_1 and π_2 be the projections from the graph of f in X_1 × X_2 onto the first and the second factor respectively. 
  We define the linear morphisms (f,g)^∙,i and (f,g)_∙,i by the following formula:

    (f,g)^∙,i : α∈^i(X_2/Y_2)_ℝ⟶  (π_1_* ∘ψ_Γ̃_̃f̃/Γ_g∘π_2^*  )(α) ∈_e-i(X_1/Y_1)_ℝ.



    (f,g)_∙,i : β∈^i(X_1/Y_1)_ℝ⟶  (π_2_* ∘ψ_Γ̃_̃f̃/Γ_g∘π_1^*  )(β) ∈_e-i(X_2/Y_2)_ℝ.



When Y_1 and Y_2 are reduced to a point, then we simply write f^∙,i (α) := (f, _{pt})^∙,i (α) and f_∙,i (β) := (f, _{pt})_∙,i (β).



Since ^i(X/Y) = 0 and _e-i(X) = 0 when i >e, it implies that
 (f,g)^∙,i and (f,g)_∙,i are identically zero for any i > e.









 §.§ Sub-multiplicativity



  Let us consider the composition X_1/_q_1Y_1 f_1g_2 X_2/_q_2 Y_2 f_2g_2 X_3/_q_3Y_3 of dominant rational maps.
Then for any integer 0 ⩽ i ⩽ e, there exists a constant C>0 which depends only on the choice of H_X_2, H_Y_2, i, l and e such that:

    _i(f_2 ∘ f_1 ) ⩽ C _i(f_1) _i (f_2).
  
More precisely, C= (e-i+1)^i/ (H_X_2^e · q_2^* H_Y_2^l).



 We denote by Γ̃_̃f̃_̃1̃ (resp. Γ̃_̃f̃_̃2̃, Γ_g_1, Γ_g_2) the normalization of the graph of q_2 ∘ f_1 (resp. q_3∘ f_2,g_1,g_2) and π_1, π_2 (resp.π_3,π_4, π_1', π_2' and π_3', π_4') the projections onto the first and the second factor respectively. We set Γ as the graph of the rational map π_3^-1∘ f_1 ∘π_1 : Γ̃_̃f̃_̃1̃Γ̃_̃f̃_̃2̃, u and v the projections from Γ onto Γ̃_̃f̃_̃1̃ and Γ̃_̃f̃_̃2̃ and ϖ_i the restriction on Γ̃_̃f̃_̃ĩ of the projection from X_i × X_i+1 to Y_i × Y_i+1 for each i=1,2.  We have thus the following diagram.



    Γ[ld]_u [rd]^v        
       Γ̃_̃f̃_̃1̃@/^1pc/[ddd]^ϖ_1[ld]_π_1[rd]^π_2      Γ̃_̃f̃_̃2̃[ld]_π_3[rd]^π_4@/^1pc/[ddd]^ϖ_2   
    
     X_1 [d]_q_1@–>[rr]_f_1        X_2[d]_q_2@–>[rr]_f_2       X_3 [d]_q_3
    
     Y_1@–>[rr]^g_1       Y_2@–>[rr]^g_2       Y_3 
       Γ_g_1[ul]^π'_1[ru]_π'_2      Γ_g_2[ul]^π'_3[ru]_π'_4

By Proposition <ref> applied to q_2 ∘π_2 ∘ u : Γ→ Y_2, the class ψ_Γ( u^* π_2^* q_2^* H_Y_2^l) is represented by the fundamental class [V] where V is a subscheme of dimension e in Γ which is a general fiber of q_2 ∘π_2 ∘ u. 
We apply Theorem <ref> by restriction to V to the class a = v^* π_4^* H_X_3^i    [V] and b=  u^* π_2^* H_X_2 [V]. We obtain: 


    v^* π_4^* H_X_3^i [V] ⩽  (e-i+1)^i ( v^* π_4^* H_X_3^i·u^* π_2^* H_X_2^e-i [V]  )( u^* π_2^* H_X_2^e [V] ) u^* π_2^* H_X_2^i [V] ∈_e-i(Γ) .

 Let us simplify the right hand side of inequality (<ref>).
Since π_2 ∘ u = π_3 ∘ v, ψ_Γ( u^* π_2^* q_2^* H_Y_2^l) = [V] ∈_e(Γ) and since the morphism v is generically finite, one has that:

    ( v^* π_4^* H_X_3^i·u^* π_2^* H_X_2^e-i [V]  ) = (v^* (π_4^* H_X_3^i ·π_3^* H_X_2^e-i·π_3^* q_2^* H_Y_2^l)  )= d ×_i(f_2),
 
where d is the topological degree of v. 
The same argument gives:

    (u^* π_2^* H_X_2^e  [V]) = d × (H_X_2^e · q_2^* H_Y_2^l).

Using (<ref>), (<ref>), inequality (<ref>) can be rewritten as:
 
    u^* π_2^* q_2^*H_Y_2^l· v^* π_4^* H_X_3^i ⩽ C _i( f_2)   u^* π_2^* H_X_2^i· u^* π_2^* q_2^*H_Y_2^l∈^l+i(Γ),

where C = (e-i+1)^i   /  (H_X_2^e · q_2^* H_Y_2^l). Since the class u^*π_1^* H_X_1^e-i∈^e-i (Γ) is nef, we can intersect this class in the previous inequality to obtain:

    (u^*(π_1^* H_X_1^n-l-i·π_2^* q_2^*H_Y_2^l) · v^* π_4^* H_X_3^i) ⩽ C' _i(f_2)   (u^* π_2^* H_X_2^i· u^* π_2^* q_2^*H_Y_2^l· u^*π_1^* H_X_1^n-l-i).

Let us simplify the expressions in inequality (<ref>). 
Because π_2^* q_2^* H_Y_2^l = ϖ_1^* π_2'^* H_Y_2^l and  
_l(g_1) = (π_2'^* H_Y_2^l), we deduce that: 

    π_2^* q_2^* H_Y_2^l = _l(g_1)(H_Y_1^l)ϖ_1^* π'_1^* H_Y_1^l =_l(g_1)(H_Y_1^l)π_1^* q_1^* H_Y_1^l .

Applying (<ref>), the inequality (<ref>) can be translated as:

    _l(g_1)(H_Y_1^l)  (u^*π_1^*( H_X_1^n-l-i·  q_1^*H_Y_1^l) · v^* π_4^* H_X_3^i) ⩽ C _l(g_1)(H_Y_1^l)_i( f_2)  ( u^*( π_2^* H_X_2^i·π_1^* q_1^*H_Y_1^l·π_1^* H_X_1^n-l-i)) .

We obtain thus:

    _l(g_1)(H_Y_1^l)_i(f_2 ∘ f_1) ⩽ C _l(g_1)(H_Y_1^l)_i(f_1) _i(f_2) .

This concludes the proof of the inequality after dividing by _l(g_1)/ (H_Y_1^l).



















 §.§ Norms of operators associated to rational maps






The proof of Theorem <ref> relies on an easy but crucial lemma which is as follows.
  Let us consider (V,|| · ||) a finite dimensional normed ℝ-vector space and let 𝒞 be a closed convex cone with non-empty interior in V. Then there exists a constant C> 0 such that any vector u ∈ V can be decomposed as v = v^+ - v^- where u^+ and u^- are in 𝒞 such that:

    || v^+/-|| ⩽ C ||v||.

 

 
 Let us define the map f : V →ℝ^+ given by:

    f (v) = inf{  ||v'|| +|| v' - v||  |  v' ∈𝒞  ,  v' - v ∈𝒞}.

We check easily that f defines a norm on V which is similar to the proof of Proposition <ref>. 
Since V is finite dimensional, there exists a constant C such that for any v ∈ V, one has:

    f(v) ⩽ C ||v||,
  
Hence ||v^+|| ⩽ C ||v || and ||v^-|| ⩽ C || v ||.



 
 Let X/_q Y fg X/_qY be a rational map.
We fix an integer i ⩽ e, some norms on ^i(X/Y)_ℝ, on _e-i(X/Y)_ℝ.
Then there is a constant C >0 such that for any rational map X/_q Y fg X/_q Y, we have:

    1C⩽|| (f, g)^∙,i ||_i(f)⩽ C.

 In particular, the i-th relative dynamical degree of f satisfies the following equality:

    λ_i( f,X/Y) = lim_p→ + ∞ || (f^p, g^p)^∙,i ||^1/p.

Moreover, when Y is reduced to a point, we obtain:


    λ_i(f)=lim_p → +∞ ||(f^p)^∙,i ||^1/p.



 The proof of Theorem <ref> follows directly from Theorem <ref> since ^i(X/Y) = ^i(X) and _e-i(X/Y) = _e-i(X) when Y is reduced to a point.


 We denote by π_1 and π_2 the projections from the normalization of the graph Γ̃_̃f̃ of q ∘ f  onto the first and the second component respectively as in Definition <ref>. 
Since we want to control the norm of f^∙,i by the i-th relative degree of f, we first find an appropriate norm to relate the norm on _e-i(X)_ℝ with an intersection product.
 As _e-i(X/Y)_ℝ is a subspace of _e-i(X)_ℝ, we can extend the norm || · ||__e-i(X/Y)_ℝ into a norm on _e-i(X)_ℝ.
As _e-i(X)_ℝ is a finite dimensional vector space and since H_X^e-i is a class in the interior of the basepoint free cone ^e-i(X), we can suppose by equivalence of norms that the norm on _e-i(X)_ℝ  given by 

    || z|| = inf_ z = z^+ - z^- 
    
    z^+, z^- ∈_e-i(X)
    { (H_X^e-i z^+)  + (H_X^e-i z^-)  }

 as in Proposition <ref>. 
Let us prove that the lower bound of || (f,g)^∙,i|| / _i(f) is 1. 
  We denote by φ : ^i(X) →^i(X/Y) the canonical surjection. Since H_X^i is basepoint free, it implies that the class (f,g)^∙( φ(H_X^i)) ∈_e-i(X/Y)_ℝ⊂_e-i(X)_ℝ is pseudo-effective. In particular, this implies that its norm is exactly _i(f). We have thus by definition: 

    || (f,g )^∙,i ||_i(f) =   (
     || (f,g )^∙,i || || (f,g)^∙,iφ(H_X^i)||  )  ⩾ 1 ,

as required. 

Let us find an upper bound for || (f,g )^∙,i ||/|| (f,g)^∙,iφ(H_X^i)||. 
First we fix a morphism s: ^i(X/Y)_ℝ→^i(X)_ℝ such that φ∘ s =. 
Take α∈^i(X/Y)_ℝ of norm 1, then the class u = s(α) ∈^i(X)_ℝ is a representant of α. By construction, the norm of u is bounded by || u ||_^i(X)_ℝ⩽ C_1 || α||_^i(X/Y)_ℝ = C_1 where C_1 is the norm of the operator s. Since by Proposition <ref>.(ii), _Γ_f/Γ_g^* ∘π_2^* = (1/_l(g)) ×π_2^* ∘_X/Y^*, we have therefore:

    (f,g)^∙,iα = 1_l(g)×π_1_* ∘ψ_Γ_f∘π_2^* ∘_X/Y^* (α) = _X/Y f^∙,i u.
 
By Theorem <ref>, the pliant cone ^i(X) has a non-empty interior in ^i(X)_ℝ and we can apply Lemma <ref>.
There exists a constant C_2>0 which depends only on ^i(X) and the choice of the norm on ^i(X)_ℝ such that the class u can be decomposed as u = u_1 - u_2 where u_i ∈^i(X) such that ||u_i||_^i(X)_ℝ⩽ C_2 || u||_^i(X)_ℝ for i=1,2. We set α_i = φ(u_i) for all i ∈{ 1,2}.
By the triangular inequality, we have:

    ||(f,g)^∙,iα ||__e-i(X/Y) || (f,g)^∙,iφ(H_X^i)||⩽|| (f,g)^∙,iα_1 ||__e-i(X)_ℝ|| (f,g)^∙,iφ(H_X^i)|| + 
    || (f,g)^∙,iα_2 ||__e-i(X)_ℝ|| (f,g)^∙,iφ(H_X^i)|| .

We have to find an upper bound of || (f,g)^∙,iα_i||__e-i(X/Y)_ℝ for each i=1,2.
Applying Siu's inequality (Corollary <ref>) to a = π_2^* u_i and b = π_2^* H_X and then composing with _X/Y∘π_1_* ∘ψ_Γ_f gives

    _X/Y (f^∙,i(u_i)) ⩽ C_3 || u_i||_^i(X)_ℝ(H_X^n)×_X/Y(f^∙,i(H_X^i)),

 where C_3 is a positive constant which depends only on the choice of big nef divisors.
This implies by intersecting with H_X^e-i the inequality:

    || ( (f,g)^∙,i (α_i)||__e-i(X/Y)_ℝ⩽ C_3  || u_i||_^i(X)_ℝ(H_X^n) || (f,g)^∙,i( φ(H_X^i))||__e-i(X/Y)_ℝ .

In particular we have shown that:

    1 ⩽ ||(f,g)^∙,iα ||__e-i(X/Y)_ℝ|| (f,g)^∙,iφ(H_X^i)||__e-i(X/Y)_ℝ⩽ 2 C_1 C_2 C_3(H_X^n),

which concludes the proof.










§ SEMI-CONJUGATION BY DOMINANT RATIONAL MAPS
 

In this section, we consider a more general situation than in the previous section. We still suppose that the varieties X_i and Y_i are of dimension n and l respectively such that the relative dimension is e= n-l, but we suppose the maps q_i : X_i  Y_i merely rational and dominant: they may exhibit indeterminacy points. 
Recall also that H_X_i and H_Y_i are again big and nef Cartier divisors on X_i and Y_i respectively.

 Let f : X_1  X_2, g : Y_1  Y_2, q_1: X_1  Y_1 and q_2 : X_2  Y_2 be four dominant rational maps such that q_2∘ f = g ∘ q_1. 
We define the i-th relative dynamical degree of f (still denoted _i(f)) as the relative degree _i(f̃) with respect to the rational map Γ_q_1/ Y_1 f̃gΓ_q_2 / Y_2 where Γ_q_i are the  normalization of the graphs of q_i in X_i × Y_i for each integer i∈{ 1,2} respectively and f̃: Γ_q_1Γ_q_2 is the rational map induced by f.  



 


  (i) Consider now the following commutative diagram:

    X_1 @–>[r]^f_1@–>[d]^q_1    X_2 @–>[r]^f_2@–>[d]^q_2    X_3  @–>[d]^q_3
    
    Y_1 @–>[r]^g_1    Y_2 @–>[r]^g_2    Y_3

 where f_i : X_i  X_i+1, g_i : Y_i  Y_i+1, q_1 : X_1  Y_1, q_2 : X_2  Y_2, q_3 : X_3  Y_3 are dominant rational maps for any integer j∈{1,2,3} such that q_j+1∘ f_j = g_j ∘ q_j for any integer j∈{1,2 }. Then there exists a constant C >0 which depends only e,i and the choice of big nef Cartier divisors such that:

    _i(f_2 ∘ f_1) ⩽ C _i(f_2) _i(f_1).
 

  (ii) Consider now the following commutative diagram:

    X_1' @–>[lld]_φ_1@–>[rrr]^f̃@–>[dd]          X_2' @–>[lld]_φ_2@–>[dd]
    
    X_1 @–>[rrr]^f@–>[dd]^q_1          X_2  @–>[dd]_<<<<q_2      
           Y_1' @–>[rrr]^g̃@–>[lld]_ϕ_1          Y_2' @–>[lld]^ϕ_2
    
     Y_1 @–>[rrr]^g          Y_2      ,

where f:X_1  X_2, g : Y_1  Y_2, q_1: X_1  Y_1, q_2 : X_2  Y_2 are four dominant rational maps such that q_2∘ f= g ∘ q_1. 
We consider some birational maps φ_i : X_i'  X_i and ϕ_i : Y_i'  Y_i for i = 1,2 such that  f̃  = φ_2^-1∘ f ∘φ_1 and g̃  = ϕ_2^-1∘ g ∘ϕ_1. 
Then for any integer 0 ⩽ i ⩽ e, there exists a constant C> 0 which depends on e, i, on the choice of big nef Cartier divisors and on the rational maps φ_1 and φ_2 such that:

    1C_i(f) ⩽_i(f̃) ⩽ C _i(f).
 

 




(i) 
Recall that the normalization of the graph of q_j in X_j × Y_j is birational to X_j for j∈{ 1,2}, hence one can define f̃_j : Γ_q_jΓ_q_j+1  the rational maps induced by f_j on the graph Γ_q_j of q_j for j ∈{ 1,2} respectively. Then (i) results directly from Theorem <ref> applied to the composition Γ_q_1/Y_1 f̃_̃1̃g_1Γ_q_2/Y_2 f̃_̃2̃g_2Γ_q_3/Y_3.

(ii) Let us suppose first that the maps q_j : X_j → Y_j and q_j': X_j' → Y_j' are all regular for j=1,2. 
Let us apply successively Theorem <ref> to the composition X_1'/_q_1' Y_1' φ_1ϕ_1 X_1/_q_1 Y_1 fg X_2/_q_2 Y_2 φ_2^-1ϕ_2^-1 X_2' /_q_2' Y_2'.
We obtain : 

    _i ( φ_2^-1∘ f ∘φ_1) ⩽ C_2 _i(f ∘φ_1) _i(φ_2^-1) ⩽ C_1 C_2 _i(f) _i(φ_1) _i(φ_2^-1),

where C_1 = (e-i+1)^i / (H_X_1^e · q_1^*H_Y_1^l) and C_2 = (e-i+1)^i / (H_X_2^e · q_2^* H_Y_2^l).
This proves that:

    _i(φ_2^-1∘ f ∘φ_1) ⩽ C _i(f),
 
where 

    C = (e-i+1)^2i_i(φ_1) _i(φ_2^-1) (H_X_1^e· q_1^* H_Y_1^l) (H_X_2^e · q_2^* H_Y_2^l).

The proof follows easily from the regular case since the maps Γ_q_1'Γ_q_1 and Γ_q_2'Γ_q_2 are birational where Γ_q_i' are the graphs of q_i' in X_i' × Y_i' for i=1,2. 


Proof of Theorem <ref>: 
(i) We apply Theorem <ref> to Y_1 = Y_2 = Y_3 = ( ), X_1 = X_2 = X_3 = X and H_X_1= H_X_2 = H_X_3 = H_X, we get thus the desired conclusion:

    _i (g ∘ f) ⩽(n-i+1)^i(H_X^n)_i(f) _i(g).

(ii) Applying Theorem <ref>.(ii) to the varieties X_1'= X_2'=X_1 = X_2 = X,  Y_1'= Y_2' =Y_1 = Y_2 = (), to the choice of big nef divisors H_X_1' = H_X_2' = H_X', H_Y_1' = H_Y_2' = H_Y', H_X_1 = H_X_2 = H_X and to the rational maps φ_1 = φ_2 = _X, ϕ_1= ϕ_2 = g = _(), f : X X yields the desired result. 
 





§ MIXED DEGREE FORMULA


Let us consider three dominant rational maps f : X  X, q : X  Y, g : Y  Y such that q ∘ f = g ∘ q. Theorem <ref>.(i) implies that for any integer i⩽ e the sequence _i(f^n) is submultiplicative. Define i-th relative dynamical degree as follows. 

    λ_i(f, X/Y) := lim_p → + ∞ (_i(f^p))^1/p.

When Y is reduced to a point, then we simply write λ_i(f) := λ_i( f , X/ {pt}).

 Since _i(f^p) ∈ℕ is an integer, one has that λ_i(f, X/Y) ⩾ 1.


 Theorem <ref>.(ii) implies that λ_i(f, X/Y) is invariant by birational conjugacy, i.e λ_i(f,X/Y) does not depend on the choice of big nef Cartier divisors and on any choice of varieties X' and Y' which are birational to X and Y respectively.

Our aim in this section is to prove Theorem <ref>.
To that end, we follow the approach from <cit.>. The main ingredient (Corollary <ref>) is an inequality relating basepoint free classes which generalizes to arbitrary fields (see <cit.> and <cit.>).
This inequality is a direct consequence of Theorem <ref> which estimates the positivity of the diagonal in a quite general setting.
After this, we prove in Theorem <ref> the submultiplicativity formula for the mixed degrees.
Once the submultiplicativity of these mixed degrees holds, the proof follows from a linear algebra argument.





 §.§ Positivity estimate of the diagonal


In this section, we prove the following theorem.

 Let q: X → Y be a surjective morphism such that Y = l and such that q is of relative dimension e. There exists a constant C>0 such that for any surjective generically finite morphism π : X' → X and any class γ∈^l+e(X'× X'):
 
    (γ [Δ_X']) ⩽ C × ( γ· (π×π)^* (H_X^e · H_Y^l)) ,

 where p_1 and p_2 are the projections from X× X to the first and the second factor respectively, H_X = p_1^* H_X + p_2^* H_X and H_Y = p_1^* q^* H_Y + p_2^* q^* H_Y, and where Δ_X' (resp. Δ_X) is the diagonal of X' (resp. of X) in X'× X' (resp. in X× X).


 The fact that the constant C>0 does not depend on π but only on H_X, H_Y is crucial in the applications. 
Theorem <ref> implies that the difference (π×π)^* (H_X^e · H_Y^l) - [Δ_X'] belongs to the dual cone of the cone _e+l(X'× X')_ℝ with respect to the intersection product, however we conjecture that this class should be pseudo-effective:


    [Δ_X'] ⩽ C ψ_X'× X'((π×π)^* (H_X^e · H_Y^l)) ∈_l+e(X'× X')_ℝ.







We shall use several times the following lemma which is proved at the end of this section.
  
Let X_1/_q_1 Y_1 fg X_2/_q_2 Y_2 be two dominant rational maps where Y_1 =  Y_2 =l and X_1 =  X_2 = e+l and where q_1,q_2 are regular surjective morphisms.
We denote by Γ_f and Γ_g the normalizations of the graph of f and g in X_1× X_2 and Y_1× Y_2 respectively, π_1,π_2, π_1', π_2' are the projections from Γ_f and Γ_g on the first and the second factor respectively.
Then there exists a constant C>0 such that for any surjective generically finite morphism π: X' →Γ_f, any integer 0 ⩽ j ⩽ l and any class β∈^e+l-j(X'), one has:

    ( β·π ^*π_2^* q_2^* H_Y_2^j)  ⩽ C _j(g)(H_Y_1^l)× (β·π^* π_1^* q_1^* H_Y_1^j),

where _j(g) is the j-th degree of the rational map g with respect to the divisors H_Y_1 and H_Y_2.








Proof of Theorem <ref>.
By Siu's inequality, we can suppose that both the classes H_X and H_Y are ample in X and Y respectively.
We proceed in three steps. Fix π : X' → X.

Step 1: We suppose first that X = ^l ×^e, Y = ^l and q is the projection onto the first factor.
Since X× X is smooth, the pullback (π×π)^* is well-defined in _l+e(X× X)_ℝ because the morphism ψ_X× X : ^l+e(X× X)_ℝ→_l+e(X× X)_ℝ is an isomorphism. 
Our objective is to prove that there exists a constant C_1>0 such that 

    [Δ_X'] ⩽ C_1  ψ_X'× X'((π×π)^* (H_X^e · H_Y^l)) ∈_l+e(X'× X')_ℝ.

 As X× X is homogeneous, we apply the following lemma analogous to <cit.> which we prove at the end of the section.
   Let X be a homogeneous projective variety of dimension n and let π : X' → X be a surjective generically finite morphism. Then one has that :
  
    [Δ_X'] ⩽ (π×π)^* [Δ_X] ∈_n(X'× X')_ℝ.

 

We denote by p_1', p_2' (resp. p_1”, p_2”) the projections from Y× Y (resp. from X× X) onto the first and the second factor respectively. 
Since the basepoint free cone has a non-empty interior by Theorem <ref>.(i) and since the class p_1'^*H_Y+ p_2'^* H_Y is ample on Y × Y, there exists a constant C_2>0 such that the class - [Δ_Y] + C_2 (p_1'^* H_Y + p_2'^* H_Y)^l∈^l(Y× Y)_ℝ is basepoint free.
Since Δ_X = Δ_Y ×Δ_^e and by intersection and by pullback, we have that the class:

    -[Δ_X] + C_2   H_Y^l · p^*[Δ_^e] ∈^e+l(X× X)

is basepoint free where p denotes the projection from X× X to ^e×^e.
By the same argument, there exists a constant C_3>0 such that the class -p^*[Δ_^e] + C_3 H_X^e ∈^e(X× X)_ℝ is basepoint free. 
We have proved that the class:

    - [Δ_X] + C_2 C_3 H_Y^l · H_X^e ∈^e+l(X× X)_ℝ

is basepoint free.
Since the basepoint free cone is stable by pullback, we have thus:

    [Δ_X'] ⩽ (π×π)^* [Δ_X] ⩽ C_1   ψ_X'× X'( (π×π)^* (H_Y^l · H_X^e)) ∈_l+e(X'× X')_ℝ,

where C_1 = C_2 × C_3 as required.


Step 2: We now suppose that X = Y ×^e.
Since Y is projective, there exists a dominant rational map ϕ : Y ^l (ϕ can be chosen as the composition of an embedding in ^N with a linear projection on a linear hypersurface). 
Let Y' be the normalization of the graph of ϕ in X ×^e ×^l and we denote by ϕ_1 and φ_1 the projections from Y' onto the first and the second factor respectively. 
Let φ_2: Y' ×^e →^l ×^e (resp. ϕ_2 : Y'×^e → X) the map induced by φ_1 (resp. ϕ_1).
Let X” be the fibred product of X' with Y'×^e so that ϕ_3, π' are the projections from X” onto X' and Y'×^e respectively.
We obtain the following commutative diagram:

    X' [d]^π   [l]^ϕ_3 X”[d]^π'
    
    Y ×^e [dd]^q   [l]^ϕ_2 Y' ×^e [rd]^φ_2[dd]^p_Y'   
          ^l ×^e [dd]^p_^l
    
    Y    [l]^ϕ_1 Y' [rd]^φ_1
          ^l
 
where p_Y' and p_^l are the projections from Y' ×^e and ^l ×^e onto Y' and ^l respectively and where the horizontal arrows are birational maps.
Let us prove that there exists a constant C_4>0 which does not depend on the morphism π: X' → X such that for any basepoint free class γ' ∈^e+l(X”× X”), one has:

    (γ'  [Δ_X”] ) ⩽ C_4 (γ' · (ϕ_3 ×ϕ_3)^* (π×π)^* (H_X^e · H_Y^l)).
 
Fix a class γ' ∈^e+l(X”× X”).
We apply the conclusion of the first step to the surjective generically finite morphism π”:=φ_2 ∘π': X”→^l ×^e. 
There exists a constant C_1>0 such that 

    [Δ_X”] ⩽ C_1  ψ_X”× X”((π”×π”)^* (H_^l^l · H_^l ×^e^e)) ∈_l+e(X”× X”)_ℝ,

where H_^l×^e is an ample Cartier divisor in (^l ×^e )^2 and H_^l is the pullback by p_^l× p_^l of an ample Cartier divisor in ^l ×^l.
Let us apply Theorem <ref> to the class (π”×π”)^* H_^l ×^e^e and to the class (π'×π')^* (ϕ_2×ϕ_2)^* H_X, there exists a constant C_5>0 such that:

    ( π”×π”)^* H_^l ×^e^e ⩽ C_5  ((π'×π')^* ((ϕ_2×ϕ_2)^* H_X^2l+e· (φ_2×φ_2)^* H_^l ×^e^e ))((π'×π')^*(ϕ_2×ϕ_2)^* H_X^2(l+e))
    × (π'×π')^* (ϕ_2×ϕ_2)^* H_X^e ∈^e(X”× X”)_ℝ.

Since ((π'×π')^* α) = ( π') (α) for any class α∈^2l+2e((Y'×^e)^2)_ℝ, we have thus:

    (π”×π”) H_^l ×^e^e ⩽ C_6 (π'×π')^* (ϕ_2×ϕ_2)^* H_X^e ∈^e(X”× X”)_ℝ,

where C_6 = C_5 ((ϕ_2×ϕ_2)^* H_X^2l+e· (φ_2×φ_2)^* H_^l×^e^e )/ ((ϕ_2×ϕ_2)^* H_X^2l+2e)>0 does not depend on π : X' → X.
Using (<ref>) and (<ref>), we obtain:

    [Δ_X”] ⩽ C_7 ψ_X”× X”((π”×π”)^* H_^l^l ·  (ϕ_3×ϕ_3)^*(π×π)^* H_X^e) ∈_l+e(X”× X”)_ℝ,

where C_7 = C_6× C_1.
Since the basepoint free cone is contained in the nef cone by Theorem <ref>.(v), we have thus:

    (γ'  [Δ_X”]) ⩽ C_7 (γ' ·  (ϕ_3 ×ϕ_3)^* (π×π)^*H_X^e · (π”×π”)^* H_^l^l ).

Let us denote by X_1 = (Y ×^e)^2, X_2 = (^l ×^e)^2, Y_1 = Y × Y, Y_2 = ^l ×^l and let f := (φ_2 ∘ϕ_2^-1×φ_2∘ϕ_2^-1) : X_1  X_2 and g:=(φ_1 ∘ϕ_1^-1×φ_1∘ϕ_1^-1) : Y_1  Y_2 be the corresponding dominant rational maps. 
Let us apply Lemma <ref> to the class (π' ×π')^* (φ_2×φ_2)^* H_^l^l and to the class (π'×π')^* (ϕ_2 ×ϕ_2)^* H_Y^l, there exists a constant C_8>0 which is independent of the morphism π'×π' : X”× X”→ (Y'×^e)^2 such that for any class β∈^2e+l(X”× X”):

    (β·  (π' ×π')^* (φ_2×φ_2)^* H_^l^l) ⩽ C_8 _l(g)(H_Y^2l) (β· (ϕ_3×ϕ_3)^*(π×π)^*  H_Y^l).
 
Using (<ref>) and (<ref>) to the class β = γ' · (ϕ_3×ϕ_3)^* (π×π)^*H_X^e ∈^l+2e(X”× X”), we obtain:

    (γ'  [Δ_X”]) ⩽ C_4 (γ' · (ϕ_3×ϕ_3)^*(π×π)^* (H_X^e · H_Y^l )),

where C_4= C_4 × C_8 (_l(g))/ (H_Y^2l)>0 does not depend on π.  
The conclusion of the theorem follows from the projection formula and from the fact that ϕ_3 ×ϕ_3 is a birational map.
Indeed, we apply the previous inequality to γ' = (ϕ_3 ×ϕ_3)^* γ where γ∈^l+e(X'× X'), we obtain

    (γ [Δ_X']) = ( (ϕ_3 ×ϕ_3)^* γ [Δ_X”]) ⩽ C_4 ( γ· (π×π)^* (H_X^e · H_Y^l))
 
as required.


Step 3: We prove the theorem in the general case.
Suppose q : X → Y is a surjective morphism of relative dimension e and fix a class β∈^l+e(X'× X').
Since X is projective over Y, there exists a closed immersion 
i:X → Y×^N such that q = p'_Y∘ i where p'_Y is the projection of Y ×^N onto Y.
 Let us choose a projection Y ×^N  Y ×^e so that the composition with i gives a dominant rational map f : X  Y ×^e. 
Let us denote by Γ_f the normalization of the graph of f in X × Y ×^e and π_1, π_2 the projections of Γ_f onto the first and the second factor respectively.
We set X” the fibred product of X' with Γ_f and we denote by π' and ϕ the projection of X” to Γ_f and X' respectively. 
We get the following commutative diagram:

    X' [d]^π   [l]^ϕ[d]^π' X”
    
    X [d]^q@–>[rd]^f   Γ_f [d]^π_2[l]_π_1
    
    Y    [l]^p_Y Y ×^e

where p_Y is the projection of Y×^e onto Y.
We apply the result of Step 2 to the class (ϕ×ϕ)^* β∈^l+e(X”× X”) and to the diagonal of X”. 
There exists a constant C_4>0 which does not depend on π such that:

    ( (ϕ×ϕ)^* β [Δ_X”]) ⩽ C_4 ((ϕ×ϕ)^*(β· (π×π)^* H_Y^l) · ((π_2 ∘π')× (π_2 ∘π'))^* H_Y×^e^e).
 
Let us apply Theorem <ref> to the class ((π_2 ∘π')× (π_2 ∘π'))^* H_Y×^e^e and to the class (ϕ×ϕ)^* (π×π) H_X.
There exists a constant C_9>0 such that:

    ((π_2 ∘π')× (π_2 ∘π'))^* H_Y×^e^e ⩽ C_9 ( (π'×π')^* (  (π_2×π_2)^* H_Y×^e^e · (π_1×π_1)^* H_X^2l+e ))  ((π'×π')^*(π_1×π_1)^* H_X^2l+2e)
    
    (ϕ×ϕ)^* (π×π) H_X^e ∈^e(X”× X”)_ℝ.

Since ( (π'×π')^* (  (π_2×π_2)^* H_Y×^e^e · (π_1×π_1)^* H_X^2l+e ))  /((π'×π')^*(π_1×π_1)^* H_X^2l+2e) = _e(f× f)/ (H_X^2l+2e)and using (<ref>), we obtain:

    ((ϕ×ϕ)^* β [Δ_X”] )⩽ C ( (ϕ×ϕ)^* ( β· (π×π)^* (H_X^e · H_Y^l)) ),

where C=C_4 C_9 _e(f× f) / (H_X^2l+2e). 
Since the morphism π_1: Γ_f → X is birational, the map ϕ: X”→ X' is also birational and we conclude using the projection formula and since (ϕ×ϕ)_* [Δ_X”] = [Δ_X']:

    (β [Δ_X'] ) ⩽ C (β· (π×π)^* (H_X^e · H_Y^l)).



Recall that X,Y are normal projective varieties and H_X,H_Y are ample divisors on X and Y respectively.
  Let q: X → Y be a surjective morphism of relative dimension e where Y = l. Then there exists a constant C>0 such that for any surjective generically finite morphism π : X' → X such that for any class α∈^i(X') and any class β∈^l+e-i(X'), one has:

    (β·α)  ⩽  C ∑_max(0, i-l) ⩽ j ⩽min(i,e) U_j(π_* ψ_X'(α))×  (β·π^* (q^* H_Y^i-j· H_X^j)),
 
where U_j(π_* ψ_X'(α)) = ( H_X^e-j· q^* H_Y^l-i+jπ_* ψ_X'(α)).



 Note that when i⩽ e, then the inequality is already a consequence of Siu's inequality (Theorem <ref>). 
Indeed, the term on the right hand side of (<ref>) with j = i corresponds exactly to the term C (π^*H_X^n-i·α) ×π^* H_X^i. 


Equation (<ref>) proves that the class

    -ψ_X'(α) + C  ∑_max(0, i-l) ⩽ j ⩽min(i,e) U_j(π_* ψ_X'(α))  ψ_X'(π^* (q^* H_Y^i-j· H_X^j)) ∈_n-i(X')_ℝ

is in the dual of the basepoint free cone ^n-i(X'). 
Moreover, if (<ref>) is satisfied, then this class is pseudo-effective.



We apply Theorem <ref> to the class γ = p_1^* β· p_2^* α∈^n(X'× X').
There exists a constant C_1>0 such that for any surjective generically finite morphism π: X'→ X and any class γ∈^n(X'× X'), one has:

    (γ [Δ_X']) ⩽ C_1 (γ· (π×π)^* (H_X^e · H_Y^l)).

We denote by p_1 and p_2 the projections of X'× X' onto the first and the second factors respectively.
Fix α∈^i(X') and β∈^n-i(X'). 
Let us apply the previous inequality to γ = p_1^* β· p_2^* α∈^n(X'× X'). 
We obtain:

    (β·α) = (p_1^* β· p_2^* α [Δ_X']) ⩽ C_1( p_1^*β· p_2^* α· (π×π)^* (H_X^e · H_Y^l)).

Since ( p_1^*π^*( H_X^m · q^*H_Y^j) · p_2^*(π^*(q^* H_Y^l-m· H_X^e-j) ·γ)) = 0 when m + j ≠ i, we obtain :

    (β·α ) ⩽   C ∑_max( 0 ,i-l) ⩽ j ⩽min(e,i) (π^*(q^* H_Y^l-i+j· H_X^e-j ) ·α)  (  π^*(q^* H_Y^i-j· H_X^j) ·β ) .

where C = C_1  ( 1 + max (  ( [ e; j ] )  ( [   l; i-j ] )  )  ).
Hence by the projection formula, we have proved the required inequality:

    ( β·α) ⩽ C ∑_max( 0 ,i-l) ⩽ j ⩽min(e,i) U_j(π_* ψ_X' (α) ) × ( β·π^* (q^* H_Y^i-j· H_X^j))) .



Proof of Lemma <ref>: (see <cit.>)
Since X is homogeneous, it is smooth.
Let G be the automorphism group of X × X, we denote by · the (transitive) action of G on X× X. 
By generic flatness (see <cit.>), there exists a non empty open subset V ⊂ X × X such that the restriction of π×π to U := (π×π)^-1(V) is flat over V.
Recall that two subvarieties F ⊂ X× X and W ⊂ X× X intersect properly in X × X if (F ∩ W) =  F +  W - 2n. 
Since G acts transitively on X × X, there exists by <cit.> a Zariski dense open subset O ⊂ G such that for any point g ∈ O,
 the cycle g · [Δ_X] intersects properly every component of X× X ∖ V. 
 In particular, there exists a one parameter subgroup τ : 𝔾_m → G such that τ(1) = ∈ G and such that τ maps the generic point of 𝔾_m to a point in O.
 Let S be the closure in X'× X'×^1 of the set { (x',t) ∈ U ×𝔾_m  |  (π×π)(x') ∈τ(t) ·Δ_X }.
Let p: X'× X' ×^1 → X'× X' be the projection onto X'× X' and let f : S →^1 be the morphism induced by the projection of  X'× X' ×^1 onto ^1.
 As in <cit.>, we denote by S_t:= p_*[f^-1(t)] ∈ Z_n(X'× X') for any t ∈𝔾_m. 
 By construction the cycle S_1 ∈ Z_n(X'× X') is effective and its support contains the diagonal Δ_X' in X'× X', hence:
 
    [Δ_X'] ⩽ S_1 ∈_n(X'× X')_ℝ.

 Let t ∈𝔾_m such that τ(t) ∈ O.
 Since S_1 = S_t ∈ A_n(X'× X') for any t ∈^1, we have thus:
 
    [Δ_X'] ⩽ S_t ∈_n(X'× X')_ℝ.

Since the cycle τ(t) · [Δ_X] intersects properly every component of X× X∖ V and since the restriction of π×π to U = (π×π)^-1(V) is flat over V, <cit.> asserts that the pullback of (π×π) ^* τ(t) · [Δ_X] is rationnally equivalent to the cycle [ (π×π)_|U^-1 (τ(t) ·Δ_X) ]. We have thus:

  
    S_t = [ (π×π)_|U^-1 (τ(t) ·Δ_X) ] = (π×π)^* [Δ_X] ∈ A_n(X'× X').

Hence:

    [Δ_X'] ⩽ (π×π)^* [Δ_X] ∈_n(X'× X')_ℝ.






Proof of Lemma <ref>.
Observe that one has the following commutative diagram:

    X' [d]^π   
       Γ_f [ld]_π_1[rd]^π_2   
    
    X_1 @–>[rr]^f[d]^q_1       X_2[d]^q_2
    
    Y_1  @–>[rr]^g       Y_2
       Γ_g [ul]^π_1'[ur]_π_2'

Fix a class β∈^e+l-j(X').  
By linearity and by Proposition <ref>, we can suppose that the class β is induced by a product of nef divisors D_1 ·…· D_e_1 + e+ l -j where D_i are nef divisors on X_1' where p: X_1' → X' is a flat morphism of relative dimension e_1.
The intersection (β·π^* π_2^* q_2^* H_Y_2^j) is thus given by the formula:

    (β·π^* π_2^* q_2^* H_Y_2^j )= (D_1 ·…· D_e_1 + e + l-j· p^*  π^* π_2^* q_2^* H_Y_2^j).

Take A an ample Cartier divisor on X_1' and set α_ϵ = (D_1 + ϵ A) ·… (D_e_1 + e + ϵ A) ∈^e_1+e(X_1')_ℝ for any ϵ>0. 
Since the class α_ϵ is a complete intersection and since the morphisms q_i are surjective, there exists a cycle V_ϵ∈ Z_l(X'_1)_ℝ such that ψ_X'_1 (α_ϵ) = { V_ϵ}∈_l(X'_1)_ℝ and such that the restrictions of the morphisms π_1 ∘π∘ p and π_2 ∘π∘ p to the support of V_ϵ are surjective and generically finite onto Y_1 and Y_2 respectively.
 We apply Theorem <ref> to the class (p^*π^* π_2^* q_2^* H_Y_2^j)_|V_ϵ and to (p^*π^* π_1^* q_1^* H_Y_1)_|V_ϵ, we get:
 
    p^* π^* π_2^* q_2^* H_Y_2^j ·α_ϵ⩽ C  (p^*π^*( π_2^* q_2^* H_Y_2^j ·π_1^* H_Y_1^l-j ) {V_ϵ} )(p^* π^*π_1^* q_1^* H_Y_1^l {V_ϵ})× p^*π^* π_1^* q_1^* H_Y_1^j ·α_ϵ∈^j+e_1 + e(X'_1)_ℝ.

 By the projection formula applied to the morphism π∘ p, we have that 
    (p^*π^*( π_2^* q_2^* H_Y_2^j ·π_1^* H_Y_1^l-j ) {V_ϵ} )/ ( p^*π^*π_1^* q_1^* H_Y_1^l {V_ϵ}) = _j(g) / (H_Y_1^l),
 hence:
 
    p^*π^* π_2^* q_2^* H_Y_2^j ·α_ϵ⩽ C _j(g)(H_Y_1^l) p^*π^* π_1^* q_1^*  H_Y_1^j ·α_ϵ∈^j+e_1 + e(X'_1)_ℝ.

 We intersect with the class (D_e_1 + e + 1·…· D_e_1 + e + l -j) ∈^l-j(X_1')_ℝ and take the limit as ϵ tends to zero. We obtain:
 
    (β·π^* π_2^* q_2^* H_Y_2^j )= (D_1 ·…· D_e_1 + e + l-j· p^*  π^* π_2^* q_2^* H_Y_2^j)  ⩽ C _j(g)(H_Y_1^l)(β·π^* π_1^* q_1^* H_Y_1^j ),

 as required.

















 §.§ Submultiplicativity of mixed degrees



 
Let X_1/_q_1 Y_1 fg X_2/_q_2Y_2 be rational maps where e =  X_i -  Y_i and l =  Y_i for i=1,2. We fix some ample divisors H_X_i and H_Y_i on each variety respectively. We define for any integer 0 ⩽ i ⩽ n:

    a_i,j (f) :=  {[ ((H_Y_1^l-j· H_X_1^e+j-i)  f^∙,i (H_X_2^i))                      if max(0,i-e) ⩽ j ⩽ l,;                                           0                                  otherwise. ].



 For j = 0, it is the i-th relative degree a_i,0 (f) = _i(f) and
when j= l, it corresponds to the i-th degree of f, a_i,l(f) = _i(f).



  Let q_1: X_1 → Y_1, q_2 : X_2 → Y_2, q_3: X_3 → Y_3 be three surjective morphisms such that X_i = e+ l and Y_i = l for all i ∈{ 1,2,3}.
Then there exists a constant C>0 such that for any rational maps X_1/_q_1 Y_1 f_1g_1 X_2/_q_2Y_2, X_2/_q_2 Y_2 f_2g_2 X_3/_q_3Y_3 and for all integers 0 ⩽ j_0 ⩽ l:

    a_i,j_0(f_2 ∘ f_1) ⩽ C ∑_max(0,i-l ) ⩽ j ⩽min (e , i)_i-j(g_1) a_i, i-j(f_2) a_j, j + j_0 -i(f_1).



 
Since we are in the same situation as Theorem <ref>, we can consider the diagram (<ref>) and we keep the same notations. 
We denote by n=e+l the dimension of X_i.


Let us denote by d the topological degree of the map f_2.
We apply Corollary <ref> to the pliant class α:=  (1/d) v^* π_4^* H_X_3^i ∈^i(Γ), to the class β := u^* π_1^* (H_X_1^e-i+j_0· q_1^* H_Y_1^l-j_0) ∈^n-i(Γ) and to the morphism π = φ∘π_3 ∘  v.  
There exists a constant C_1 >0 which depends only on the choice of divisors H_Y_2 ×^e and H_Y_2 such that:

    a_i,j_0(f_2∘ f_1) ⩽ C_1 ∑_max(0,i-l ) ⩽ j ⩽min (e , i) U_j(π_* ψ_Γ(α)) ( β·π^* (H_X_2^j · q_2^* H_Y_2^i-j)),

where U_j( γ) = (H_X_2 ^e-j· q_2^*H_Y_2^l-i+jγ) for any class γ∈_n-i(X_2)_ℝ.
We observe that U_j(π_* ψ_Γ(α))= a_i,i-j(f_2). 
We have thus:

    a_i,j_0(f_2∘ f_1) ⩽ C_1 ∑_max(0,i-l ) ⩽ j ⩽min (e , i) a_i,i-j(f_2) (u^* (π_1^* (H_X_1^e-i+j_0· q_1^* H_Y_1^l-j_0)·π_2^*(H_X_2^j · q_2^* H_Y_2^i-j))).

Applying Lemma <ref> to the class u^* π_2^* q_2^* H_Y_2^i-j∈^i-j(Γ) and to β' = β· u^* π_2^* H_X_2^j ∈^n-i+j(Γ), there exists a constant C_2 >0 such that :

    (  β' · u^* π_2^* q_2^* H_Y_2^i-j) ⩽ C_2 _i-j(g_1) ( u^* (π_1^* (H_X_1^e-i+j_0· q_1^*H_Y_1^l-j_0 + i-j) ·π_2^* H_X_2^j)).

Since the map u : Γ→Γ_f_1 is birational, we have that:

    (u^* (π_1^* (H_X_1^e-i+j_0· q_1^* H_Y_1^l-j_0)·π_2^*(H_X_2^j · q_2^* H_Y_2^i-j))) ⩽ C_2 _i-j(g_1) a_j, j_0 +j -i(f_1).
 
Finally, (<ref>) and (<ref>) imply:

    a_i,j_0(f_2∘ f_1) ⩽ C ∑_max(0,i-l ) ⩽ j ⩽min (e , i) a_i,i-j(f_2) a_j,j_0 + j-i(f_1) _i-j(g_1),

where C = C_2 C_1>0 is a constant which is independent of f_1 and f_2 as required.















 §.§ Proof of Theorem <ref>


Recall that we want to prove the following formula:

    λ_i(f) = max_j⩽ i (λ_j(f, X/Y) λ_i-j(g)).

By definition of the relative degrees, we are reduced to prove the theorem when q: X→ Y is a proper surjective morphism.
Recall that X = n and Y = l such that q: X → Y has relative dimension e= n-l.
Let us consider the following commutative diagram:

    Γ_f [rd]^π_2[ld]_π_1@/^1pc/[ddd]^ϖ   
    
    X @–>[rr]^f[d]^q       X [d]^q
    
    Y@–>[rr]^g       Y 
       Γ_g [ru]_π_2'[lu]^π_1'

where f: X  X, g : Y  Y are dominant rational maps, Γ_f, Γ_g are the normalization of the graph of f and g respectively, π_1, π_2, π_1', π_2'  are the projections from Γ_f and Γ_g onto the first and second factor respectively and ϖ: Γ_f →Γ_g is the restriction of q × q to Γ_f. 


The following lemma proves that max_j ⩽ i ( λ_j(f,X/Y) λ_i-j(g)) ⩽λ_i(f). 
  For any integer max(0, i-l) ⩽ j ⩽min( i , e), there exists a constant C>0 such that for any rational map X/_q Y fg X/_q Y,  we have  _i-j(g ) _j(f) ⩽ C _i(f).


Granting the above lemma, then we obtain the lower bound on λ_i(f) as:

    λ_i(f) ⩾λ_j(f,X/Y) λ_i-j(g).





 It suffices to consider the product (π_1^* (H_X^e-j· q^* H_Y^l-i+j) ·π_2^* (H_X^j · q^* H_Y^i-j)). Since π_i ∘ q = ϖ∘π_i' for i ∈{ 1,2}, we obtain:
 
    (π_1^* (H_X^e-j· q^* H_Y^l-i+j) ·π_2^* (H_X^j · q^* H_Y^i-j)) =( ϖ^* (π_1'^* H_Y^l-i+j·π_2'^* H_Y^i-j) ·π_1^* H_X^e-j·π_2^* H_X^j ).

Moreover, one has that π_1'^* H_Y^l-i+j·π_2'^* H_Y^i-j = (π_1'^* H_Y^l-i+j·π_2'^* H_Y^i-j) [p_0] = _i-j(g)  [p_0] where p_0 is a general point in Γ_g. 
We can hence apply Proposition <ref> to the morphism ϖ: Γ_f →Γ_g and obtain:

    (π_1^* (H_X^e-j· q^* H_Y^l-i+j) ·π_2^* (H_X^j · q^* H_Y^i-j)) = _i-j (g) ( π_1^* H_X^e-j·π_2^* H_X^j [Γ_f_p_0] ).

Since π_1' is a birational morphism, a general fiber of ϖ is equal to a general fiber of π_1' ∘ϖ. 
In other words, we have that _Γ_f/Γ_g = _Γ_f/Y and since π_1^* H_X^e-j·π_2^* H_X^j [Γ_f_p_0]  = _Γ_f / Γ_g (π_1^* H_X^e-j·π_2^* H_X^j), we obtain:

    (π_1^* (H_X^e-j· q^* H_Y^l-i+j) ·π_2^*( H_X^j · q^* H_Y^i-j)) = _i-j(g) ×_j(f).

As H_X is ample, we apply Theorem <ref> to the classes π_2^*q^* H_Y and π_2^*H_X:

    π_2^* q^*H_Y^i-j⩽ (n-i+j+1)^i-j(π_2^* q^* H_Y^i-j·π_2^* H_X^n-i+j)(π_2^* H_X^n)π_2^*H_X^i-j = C_1 π_2^* H_X^i-j∈^i-j(X)_ℝ,

where C_1 = (n-i+j+1)^i-j(q^* H_Y^i-j· H_X^n-i+j) / (H_X^n) depends only on n,i and the choice of big nef Cartier divisors. 
Intersecting with π_1^* H_X^n-i·π_2^* H_X^j, one obtains:

    _i-j(g) ·_j(f) ⩽ C_1 (π_2^* H_X^i·π_1^* (H_X^e-j· q^* H_Y^l-i+j)).

By the same argument, there exists a constant C_2>0 which depends only on H_Y, H_X and i such that:

    π_1^* q^* H_Y^l-i+j⩽ C_2 π_1^* H_X^l-i+j.

Hence, we obtain:

    _i-j(g) _j(f) ⩽ C _i(f),

where C = C_1 C_2. 


Let us prove the converse inequality.
We fix an integer 0 ⩽ i ⩽ n. 
Let us apply Theorem <ref> to f_1=f, f_2=f^p, g_1 =  g and g_2=g^p, we can rewrite the inequality as:

    a_i,j_0 (f^p+1) ⩽ C ∑_max(0 , i-e) ⩽ j ⩽min(i,l)_j(g) a_i-j, j_0 -j(f) a_i, j(f^p).

Let us denote by U_i(f) the column vector given by:

    U_i(f) = ( a_i,j(f))_0 ⩽ j ⩽ l =  ( [  a_i,0(f);         …; a_i, l(f) ] ).

Let us also denote by M_i(f) the (l+1)× (l+1) lower-triangular matrix given by:

    M_i(f) :=( _j(g) a_i-j,m-j(f) ×χ_ [i-e, min(i,l)](j)  )_ 0 ⩽  m ⩽ l, 0 ⩽ j ⩽ l   ,

where χ_A denotes the characteristic function of the set A. 
Therefore, (<ref>) can be rewritten as:

    U_i(f^p+1) ⩽ C M_i(f) · U_i(f^p),

where · denotes the linear action on ℤ^l+1.
A simple induction proves:

    U_i(f^p) ⩽ C^p (M_i(f))^p-1· U_i(f)

Since the (l+1)-th entry of the vector U_i(f^p) corresponds to _i(f^p), we deduce that:

    _i(f^p)^1/p⩽ C ⟨ e_l , ( M_i(f) )^p·  U_i(f)⟩^1/p,

where (e_0, …, e_l) denotes the canonical basis of ℤ^l+1.
In particular, _i(f^p)^1/p is controlled up to a constant by the eigenvalues of the matrix M_i(f) which are _j(g) _i-j(f) for max(0,i-e)⩽ j⩽min(i,l) since M_i(f) is lower-triangular. 
Applying (<ref>) to f^r, we get:


    _i(f^pr)^1/(pr)⩽ C^1/r ||  U_i(f^r)||^1/prmax_max(0,i-e) ⩽ j ⩽min(i,l)  (_j(g^r) _i-j (f^r))^1/r.

We conclude by taking the lim sup as r → +∞, p→ + ∞:

    λ_i(f) ⩽max_max(0, i-l)⩽ j ⩽min(i,e)λ_i-j(g) λ_j(f,X/Y).



 Note that the previous theorem gives information only on the dynamical degrees of f. Lemma <ref> provides a lower bound on the degree of f^p. However, one cannot find an upper bound for _i(f^p) which would only depend on the relative degrees and the degree on the base. If X = E × E is a product of two elliptic curves and if f : (z, w) ∈ E × E → (z , z + w) is an automorphism of X, then the degree growth of f^p is equivalent to p^2 whereas the degree on the base and on any fiber are trivial.






§ KÄHLER CASE
 


We prove the submultiplicativity of the k-th degrees in the case where (X,ω) is a complex compact Kähler manifold. For any  closed smooth (p,q)-form α on X, we denote by {α} its class in the Dolbeault cohomology H^p,q (X)_ℝ. 
 Let (X,ω) be a compact Kähler manifold. A class α∈ H^1,1 (X)_ℝ is nef if for any ϵ >0,  the class α + ϵ{ω} is represented by a Kähler metric.




A class α of degree (i,i) is pseudo-effective if it can be represented by a closed positive current T. Moreover, one says that α is big if there exists a constant δ>0 such that T -δω is a closed positive current and we write T ⩾δω^i.


 (cf <cit.>, <cit.>)Let (X,ω) be a compact Kähler manifold of dimension n. Let k be an integer and α,β be two nef classes in H^1,1(X) such that α^i ∈ H^i,i(X) is  big and such that ∫_X α^n -  ( [ n; i ] ) ∫_X α^n-i∧β ^i >0. Then the class α^i - β^i is big.  



Recall that the degree of a meromorphic selfmap f : X  X when (X,ω) is given by:

    _i(f) := ∫_Γ_fπ_1^* ω^n-i∧π_2^* ω^i,

where Γ_f is the desingularization of the graph of f and π_j are the projections from Γ_f onto the first and the second factor respectively. 

When X is a projective variety and ω represents the class of a hyperplane section H_X, then the intersection of the form coincides with the cup-product in cohomology, hence _i(f) = _i,H_X(f).
 
 Let (X_1,ω_X_1), (X_2,ω_X_2) and (X_3,ω_X_3) be some compact Kähler manifolds of dimension n. Then there exists a constant C> 0 which depends only on the choice of the Kähler classes ω_X_j such that for any dominant meromorphic maps f_1: X_1  X_2 and f_2 : X_2  X_3, one has:

    _i(f_2 ∘ f_1) ⩽ C _i (f_1) _i(f_2).

Moreover, the constant C may be chosen to be equal to ( [ n; i ] ) / (∫_X_2ω_X_2^n).


 The previous theorem gives that for any big nef class β^i ∈ H^i,i(X), for any nef class α∈ H^1,1(X), one has:

    α^i ⩽ ( [ n; i ] ) ∫_X α^i ∧β^n-i∫_X β^n ×β^i.
 
Then, the proof is formally the same as Theorem <ref>. Indeed, one only needs to consider the diagram (<ref>) where Y_1 = Y_2 = Y_3 are reduced to a point and where Γ_f_1, Γ_f_2, Γ are the desingularizations of the graph of f_1, f_2 and π_3^-1∘  f_1 ∘π_1 respectively. 
We apply (<ref>) to α = v^* π_4^* ω_X_3 and β = v^* π_3^* ω_X_2 to obtain:

    v^* π_4^* ω_X_3^i ⩽ ( [ n; i ] ) _i(f_2)∫_X_2ω_X_2^n× v^* π_3^* ω_X_2^i.

By intersecting the previous inequality with the class u^* π_1^* ω_X_1^n-i, we finally get:

    _i(f_2 ∘ f_1) ⩽ ( [ n; i ] ) _i(f_2)  _i(f_1)∫_X_2ω_X_2^n.
















§ COMPARISON WITH FULTON'S APPROACH




In <cit.>, a cycle z ∈ Z_i(X) on a variety X is defined to be numerically trivial if (c z) for any product c =c_i_1(E_1) ·…· c_i_p(E_p) ∈ A^i(X) of Chern classes c_i_j(E_j) where E_j is a vector bundle on X and i_1 + … + i_p = i. This appendix is devoted to the proof of the following result:

  Let X be a normal projective variety of dimension n. For any z ∈ Z_i(X), the following conditions are equivalent:


  (i) For any product of Chern classes c = c_i_1(E_1) ·…· c_i_p(E_p) ∈ A^i(X)_ℝ where E_j are vector bundles on X and i_1 + … + i_p = i, we have (c  z) = 0.  

  (ii) For any integer e, any flat morphism p_1 : X_1 → X of relative dimension e where X_1 is a projective scheme and any Cartier divisors D_1 , … , D_e+i on X_1, we have (D_1 ·…· D_e+i p_1^*z) =0. 

  (iii)For any integer e, any flat morphism p_1 : X_1 → X of relative dimension e between normal projective varieties and any Cartier divisors D_1 , … , D_e+i on X_1, we have (D_1 ·…· D_e+i p_1^*z) =0. 






The implication (ii) ⇒ (i) follows immediately from the definition of Chern classes. 
The implication (ii) ⇒ (iii) is also straightforward.
For the converse implications (i) ⇒ (ii) and (i) ⇒ (iii), we rely on the following proposition. 

 Let q : X → Y be a flat morphism of relative dimension e where X is a projective scheme and Y is a normal projective variety. For any Cartier divisors D_1, … , D_e+i be some ample Cartier divisors on X, there exist vector bundles E_j, and a homogeneous polynomial c=P(c_i_1(E_1),  … , c_i_p(E_p)) of degree i with respect to the weight (i_1, … , i_p), with rational coefficients such that for any cycle z ∈ Z_i(X), (c · z ) = (D_1 ·…· D_e+i· q^* z).











We take some ample Cartier divisors D_1 , … , D_e+i on X. We denote by ℒ_i the line bundle 𝒪_X(D_i).
By Grauert's Theorem (cf <cit.>), the sheaves R^i q_*(ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i) are locally free.
By <cit.>, we have that R^i q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i) = 0 for i>0 and m_i large enough since the line bundle ℒ_i are ample.
So the sheaf q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i) is locally free and we have in K_0(Y):

    q_* [ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i] = ∑ (-1)^i [R^i q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i)] =[ q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i)].


 For any j ⩽ i:


  * The function (m_1, …, m_e+i) →_j ( q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i)) ∈^j(Y)_ℝ is a polynomial of degree e+j with coefficients in ^j(Y).

  * For any cycle z ∈ Z_j(Y), the coefficient in m_1 ·…· m_e+i in (_j(q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i))   z) is ((D_1 ·…·  D_e+i)  q^* z).



 Let us set ℱ = ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i. We prove the result by induction on 0 ⩽ j ⩽ i. 


For j = 0, choosing a point y ∈ Y(), the number _0(q_* (ℱ)) is equal to h^0( X_y , ℱ_|X_y ). 
By asymptotic Riemann-Roch, for m_1, …, m_e+i large enough, it is a polynomial of degree X_y = e. Moreover, Snapper's theorem (see <cit.>) states that the coefficient in m_1 ·…· m_e+i is the number (D_1 ·…· D_e+i [X_y]).


We suppose by induction that _i( q_* (ℱ)) is a polynomial of degree e +i for any i ⩽ j where j ⩽ i-1. For any subvariety V of dimension j+1 in Y, we denote by W its scheme-theoretic preimage by q. 


For any scheme V, let us denote by τ_V the morphisms:

    τ_V : K_0(V) ⊗ℚ→ A_∙ (V) ⊗ℚ.
 
We refer to <cit.> for the construction of this morphism and its properties. 
We apply Grothendieck-Riemann-Roch's theorem for singular varieties (see <cit.>) and using (<ref>), we get in A_∙(Y)_ℚ: 

    ( q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i) ) τ_V( 𝒪_V ) =  q_* (  (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i) τ_W(𝒪_W)).

The term in A_0(Y)_ℚ in the left handside of the previous equation is equal to:

    _j+1 ( q_* (ℱ))  [V] + ∑_i ⩽ j_i ( q_*(ℱ)) τ_V,i (𝒪_V),

where τ_V,i (𝒪_V) is the term in A_i(Y) of τ_V(𝒪_V).
By the induction hypothesis, every _i(q_* ℱ) is a polynomial of degree e+i, and the right hand side of equation (<ref>) is a polynomial of degree e+j + 1, so _j+1(q_*( ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i)) is also a polynomial of degree e + j + 1. Now we identify the coefficients in m_1 ·…· m_e+i of the term in _0(Y) in equation (<ref>). 
It follows from <cit.> that τ_W(𝒪_W) = [W] + R_W where  R_W is a linear combination of cycles of dimension < e+i. Therefore, the coefficient in m_1 ·…· m_e+i of the right hand side of equation (<ref>) in _0(Y) is ((D_1 ·…· D_e+i)  [W]) if j+1 = i or 0 otherwise. 
 
We have proved that the coefficient of _j+1(q_* ( ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i))  [V] is ((D_1 ·…· D_e+i)  [W]) if V = i or 0 otherwise. Extending it by linearity, one gets the desired result.
 




We have that _i( q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i)) is by definition a polynomial in Chern classes of vector bundles on Y. Using the previous lemma, the coefficient U(D_1, … , D_e+i) in m_1 ·…· m_e+i of _i( q_* (ℒ_1^m_1⊗…⊗ℒ_e+i^m_e+i)) is equal to P(c_i_1(E_1) , … , c_i_p(E_p)) where P is a homogeneous polynomial with rational coefficients of degree i with respect to the weight (i_1, …, i_p) and E_i are vector bundles on Y. We have proven that for any cycle z ∈ Z_i(Y): 

    (P(c_i_1(E_1) , … ,c_i_p(E_p) )  z ) = ((D_1 ·…· D_e+i ) q^*z).

As any Cartier divisor can be written as a difference of ample Cartier divisors. The proposition provides a proof for the implication (i) ⇒ (ii) of Theorem <ref>. 




 In codimension 1, the intersection product (D_1 ·…· D_e+1 q^*z) is represented by Deligne's product I_X(𝒪_X(D_1), … , , 𝒪_X(D_e+1)) ∈^1(X)_ℝ (see <cit.> for a reference). Indeed, one has by <cit.> that for any cycle z ∈_1(X): 

    c_1( I_X( 𝒪_X(D_1), … , , 𝒪_X(D_e+1)) )  z =  D_1 ·…· D_e+1 q^*z.

 





This gives an answer to the question of numerical pullback formulated in <cit.>.
  Let q: X → Y be a flat morphism of relative dimension e between normal projective varieties. Then the morphism q^* : A_∙(Y)_ℚ→ A_e+ ∙(X)_ℚ induces a morphism of abelian groups q^* : _∙(Y)_ℚ→_e+∙(X)_ℚ. 
By duality, the morphism q_* : A^∙(X)_ℚ→ A^∙ - e(Y)_ℚ induces a morphism of abelian groups q_*: ^∙(X)_ℚ→^∙ - e(Y)_ℚ.










amsalpha









   Nguyen-Bac Dang

   CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex,  

   France
   nguyen-bac.dang@polytechnique.edu






