On the monodromy of elliptic surfaces
Genival da Silva Jr






Department of Mathematics Imperial College LondonLondon, SW7 2AZ, UK
jrriemann@gmail.com




There have been several constructions of family of varieties with exceptional monodromy group <cit.>,<cit.>. In most cases, these constructions give Hodge structures with high weight(Hodge numbers spread out). N. Katz was the first to obtain Hodge structures with low weight(Hodge numbers equal to (2,3,2)) and geometric monodromy group G_2. In this article I will give an explicit proof of Katz's result by finding all the monodromies in a given basis.


On the monodromy of elliptic surfaces
    Genival Da Silva Jr.
    
=====================================





§ INTRODUCTION

In <cit.>, Nicholas Katz studies the appearance of G_2 as the monodromy group of a family of elliptic surfaces. Starting with an elliptic curve E/k and a `seven point sheaf' on E (geometrically irreducible lisse sheaf ℱ of rank 2 on a dense open set j:U⊂ E), he wonders what are the groups G_geom,N,G_arith,N, where N=j_*ℱ(1/2)[1].

One way to obtain such ℱ is to consider E as a double covering x:E→ℙ^1 of ℙ^1. Then our ℱ is obtained by pulling back a `four point sheaf' 𝒢 on ℙ^1, and we can get such 𝒢 by taking R^1π_*ℚ_l(1/2) for an elliptic surface π: ℰ→ℙ^1.

Over ℂ, up to isogeny, there are only 4 elliptic surfaces π: ℰ→ℙ^1, they are:

    y^2 = -x(x-1)(x-λ^2), λ≠ 0,± 1, ∞
        y^2 = 4x^3 + ((λ + 2)x + λ)^2,  λ≠ 0,1,-8,∞
        y^2 = 4x^3 + (λ^2+6λ -11)x^2 + (10 - 10λ)x + 4λ -3,  λ≠ 0,∞ and λ^2+11λ-1 ≠ 0
        y^2 = 4x^3 + (3λ x + 1)^2,   λ≠∞,λ^3 ≠ 1


For each one these 4 families, we can associate a monic cubic polynomial f, whose roots are the `bad' values of λ described above. Moreover:

<cit.>
For each one of the four families above, there is an explicit nonzero integer polynomial P[T]∈ℤ[T] with the following property. For each finite field k in which l is invertible, and for each t∈ k at which P(t)≠ 0 in k, the equation 
    E_t : y^2=tf(x) + t^2
 defines an elliptic curve over k, and the N gotten by pulling back 𝒢(x) has 
    G_geom,N=G_arith,N=G_2
.


The proof of the theorem above involves the analysis of the sheaf 
    ℋ:= R^1ρ_*(𝒢(1/2)⊗ℒ_χ_2(tf(x) + t^2))
 where ρ:ℙ^1→ S, S a punctured affine t line, ℒ_χ_2 is the Kummer sheaf attached to the character χ_2 of k^×. Katz<cit.> then proves that for the first three (but not the fourth) families above the sheaf ℋ has G_geom=G_arith=G_2. In the rest of this paper I will give an alternative proof of the latter fact by explicitly finding the monodromies matrices.


The geometric monodromy group of the first three families considered above is G_2. 




 §.§ Acknowledgements
 I thank my PhD advisor Matt Kerr for sharing his ideas with me, and also acknowledge the travel support from NSF FRG Grant 1361147, the support form CNPq program Science without borders and the support from ERC Grant 682603.



§ THE EXPLICIT CONSTRUCTION OF THE FAMILY

For the sake of simplicity, I will work in this section with the first of the 3 families with monodromy group G_2, but the exact same approach applies to the remaining ones. We start off, by constructing the family of surfaces mentioned above.

Let ℰ→ℙ^1:y^2=x(x-1)(x-z^2) be a rational elliptic surface with singular fibers at z=-1,0,1,∞. For t≠ 0,±2/3√(3) , ∞, take a base change by:

    E_t→ℙ^1:w^2=tz(z-1)(z+1)+t^2

The result is a family of elliptic surfaces X_t→ E_t with 7 singular fibers on each surface, as described below:

    [     𝒳     ↩   X_t;    ↓π        ↓π_t;     ℰ     ↩   E_t;     ↓           ↓;   ℙ^1     ↩ { t } ]



For each X_t we have dim(H^2_tr(X_t)) ≤ 7.

 
Set X := X_t , E := E_t, π := X → E. We have that X has 4 singular fibers of type I_2, 2 of type I_4 and one of type I_8. We use known formulas for the hodge numbers of Elliptic Surfaces, see <cit.>. We have: 
    b_2(X) = 2p_g(X) + 10χ(X) + 2g(E)
 where p_g(X)= H^2,0(X) is the geometric genus of X, χ(X) is the Euler characteristic and g(E) is the genus of E. In our particular case we have:

    g(E)=1, χ(X)=2; p_2(X)=χ(X)-1+g(E)=2

which amounts to b_2(X)=26. Since the components of the singular fibers give algebraic cycles, and we have 2+4+6+7=19 of them, the dimension of transcendental cycles can be at most 7, and it is 7 if we can find 7 linearly independent transcendental  cycles.
Each fiber I_n contributes n/12 to the degree of the Hodge bundle. Therefore, π_* Ω^1_X/E has degree 2,i.e π_* Ω^1_X/E = 𝒪(p+q). Now: 
    Ω^2_X ≅π^*π_* Ω^1_X/E⊗π^* Ω^1_E≅π^* Ω^1_E(p+q)

Which gives h_X^2,0 = 2, and a similar analysis gives h_X^1,0 = 1. Finally, Noether's formula gives:
    h^1,1_X = 10 - 8h_X^1,0 + 10h_X^2,0 - K_X^2 = 22

We have 19 algebraic classes coming from the singular components of the singular fibers I_n, hence the transcendental part is at most 26-19 = 7.


In fact, dim(H^2_tr(X_t)) = 7 as we shall see.

We now describe a particular choice of 7-dimensional basis of 2-cycles that we will use henceforward. First, consider the 1-cycles α,β,γ_-1,γ_0,γ_1 over each E_t, as described in figure <ref>. Denote by δ_1,δ_2 the basis for the local system over each point of E_t, with δ_1δ_2 = 1. Let's see how they behave under the action of the monodromy, but first we analyze the situation over ℙ^1(with z^2-coordinate) before the double cover, i.e on y^2=x(x-1)(x-z), so that we can predict how the cycles change after the double cover. 



The degeneration in this case is a nodal degeneration on 0,1, the monodromy matrices are then given by the Picard-Lefschetz formula:

    T_0 = ([ 1 2; 0 1 ])
        T_1 = ([  1  0; -2  1 ])

Now consider the ℙ^1 which has z coordinate. In order to make it simply connected, we draw some cuts over it. If we go through paths around -1,0,1, as described in figure <ref>, we can look at the image of those cycles under the double cover and see what the monodromy is. For example, as go around -1 on the z-plane, the image goes to once around zero, then once around one and one more time around zero again on the z^2-plane, hence we can deduce that the local monodromy around -1 is T_0*T_1*T_0^-1. Applying the same reasoning to 0 and 1, we get the resulting monodromies:

    T̃_̃-̃1̃ = ([ -3  8; -2  5 ])
       T̃_̃0̃ = ([ 1 4; 0 1 ])
       T̃_̃1̃ = ([  1  0; -2  1 ])


The vanishing cycles are: 2δ_1 + δ_2 at -1, δ_1 at 0, δ_2 at 1. Set η_1 = δ_2 and η_2 = 2δ_1 + δ_2, so η_1η_2 = -2 and the vanishing cycle at 0 is precisely δ_1=1/2(η_2-η_1).



Note that these local monodromies are quasi-unipotent, more generally we have:

(Monodromy Theorem) <cit.>
Let p:Δ^*→Γ\ D be a variation of weight w Hodge structures over the puncture disk, with monodromy generated by T. Then T is quasi-unipotent and has index of unipontency w+1.

By this theorem, the local monodromies(<ref>) have to have index of unipontency 2, which is true, as one quickly verifies. Therefore, the log monodromy is just T-I, where T is the local monodromy. Recall the formula for the local monodromy in a nodal degeneration:
(Picard-Lefschetz formula)
In a nodal degeneration with vanishing cycles δ_i, the local monodromy is: 
    T(x) = x ±∑_i ⟨ x,δ_i ⟩δ_i



In a nodal degeneration of weight 1 Hodge structures, we have up to a sign: 
    N(x):= log T(x) =∑_i ⟨ x,δ_i ⟩δ_i


This is equivalent of saying that the image of the log monodromy in a nodal degeneration of weight one Hodge structures is generated by the vanishing cycles.



§ CONSTRUCTION OF THE 2-CYCLES

We use henceforward the notation a× b to denote the 2-cycle on X_t obtained by taking the 1-cycle a on a fiber of π_t and continuing it along the 1-cycle b on E_t. Note that the cycle a has to be monodromy invariant when one goes over b, otherwise this definition doesn't make sense.
Now that our notation is established we proceed with the definition of a 7-dimensional subspace of H^2_tr(X_t):

    A_1 = η_1 ×α    C_-1   = η_2 ×γ_-1
        A_2 = η_2 ×α    C_0   = 1/2(η_2 -η_1) ×γ_0
        B_1 = η_1 ×β    C_1   = η_1 ×γ_1
        B_2 = η_2 ×β

Note that, A_1,A_2,B_1,B_2 are trivially transcendental, the same is not true for the C_i. The reason is that the C_i may–in fact they do–contain algebraic cycles resulting from classes of singular fibers. To overcome this, we have to “add" enough cycles in order to make all C_i transcendental.

Let's take a closer look at the C_-1, for example. As we can see from figure <ref>, we can pick a cycle equivalent to C_-1 but with minimal intersection, in other words:

    C_-1 D_- = -1
        C_-1 D_+ = 1
        C_-1 E_- = -1
        C_-1 E_+ = 1



Now, let's try to eliminate the intersections of C_-1 with algebraic classes. Start by setting:

    C_-1 := C_-1 + aD_- + bD_+ + cE_- + dE_+

If σ is the class of the zero section, then the transcendental condition reduces to the following system of linear equations:

    C_-1 D_- = 0
       C_-1 D_+ = 0
       C_-1σ = 0
       C_-1 E_- = 0
       C_-1 E_+ = 0


Without loss of generality we may assume a=d=0. Solving the system we get that:

    C_-1 = C_-1 + 1/2D_+  -1/2E_-


By following the exact same reasoning, we deduce that:


    C_1 = C_1 + 1/2G_+  -1/2H_-

where G_- and H_- are the components of the singular fibers of the endpoints.

Now we address C_0, consider the figure <ref>. Following the idea above, we set:

    C_0 = C_0 + aL_1 + bL_2 + cL_3 -dF_1 -eF_2 -fF_3



We again solve the system of equations required for transcendency:

    C_0 L_1 = 0
       C_0 L_2 = 0
       C_0 L_3 = 0
       C_0 F_1 = 0
       C_0 F_2 = 0
       C_0 F_3 = 0
       C_0σ = 0


The resulting cycle is:

    C_0 = C_0 + 3/4L_1 + 1/2L_2 + 1/4L_3 -3/4F_1 -1/2F_2 -1/4F_3




§ COMPUTATION OF THE MONODROMIES

Denote by V the space generated by the transcendental cycles  (A_1,A_2, B_1,B_2,C_-1,C_0,C_1). The intersection matrix is:
    Q = [    0    0    0    2    0    0    0;    0    0   -2    0    0    0    0;    0   -2    0    0    0    0    0;    2    0    0    0    0    0    0;    0    0    0    0   -1    1    2;    0    0    0    0    1 -1/2   -1;    0    0    0    0    2   -1   -1 ]

Notice that since det(Q) ≠ 0, we have dim(V)=7. Since V⊂ H^2_tr(X_t), Proposition <ref> implies that dim H^2_tr(X_t) = 7.

With our transcendental basis (A_1,A_2,B_1,B_2,C_-1,C_0,C_1) defined, we now compute the monodromies matrices at the singular points t=-2/3√(3),0,2/3√(3),∞. For computational purposes, we will work with figure <ref> instead of figure <ref>.

When t→±2/3√(3), we have a nodal degeneration on the base curve E_t. In figure <ref>, such degeneration can be described as when the “x” of one cut merges itself with an “x” of the other cut. 

It's straightforward to conclude that in this case, the C_i remain unchanged, while in the other cases the cycles over the vanishing cycles remain unchanged. 

Finally, the cycles that do change, do it so according to the Picard-Lefschetz formula (<ref>), since the degeneration is nodal. In conclusion, we have the following monodromies for 2/3√(3),-2/3√(3) respectively:

    M_+   = [ 1 0 1 0 0 0 0; 0 1 0 1 0 0 0; 0 0 1 0 0 0 0; 0 0 0 1 0 0 0; 0 0 0 0 1 0 0; 0 0 0 0 0 1 0; 0 0 0 0 0 0 1 ]
    
            M_-   = [  1  0  0  0  0  0  0;  0  1  0  0  0  0  0; -1  0  1  0  0  0  0;  0 -1  0  1  0  0  0;  0  0  0  0  1  0  0;  0  0  0  0  0  1  0;  0  0  0  0  0  0  1 ]


The situation when t→ 0 is much more subtle. If one looks at figure <ref>, the endpoints of the cuts behave roughly as -1-t/2,t and 1-t/2, therefore when t go through a path around 0, the endpoints will certain move, but this time not in a nice way as they did in the case above, they will instead make the γ_i cycles cross each other and also α and β. This is the crucial point which results in G_2 monodromy, as we shall verify. 


Let's start off by analyzing the resulting cycle α of the monodromy action on α. If we look at figure <ref>, we see not only α is no longer a vanishing cycle, but also that it crosses the cuts trivializing the local system. What that means basically is that η_1 and η_2 might change after monodromy; this is in fact the case, as we shall see.   

Now, consider α-α, as depicted in figure <ref>. Note that the vanishing cycle at 1 is η_2, hence any cycle which is the continuation of η_2 won't have monodromy around 1, so we can simplify α-α to encircle only 0, and vice-versa. Using the expression for the local monodromies <ref> and formula <ref>, we can compute the resulting 2-cycles for the ones that are over α, i.e A_1,A_2. Denote by M_0 the monodromy at 0, then:

    M_0(A_1) = A_1 - 2A_2 + 2B_1 - 2B_2 -4C_0
        M_0(A_2) = 2A_1 - 3A_2 + 6B_1 - 2B_2 -4C_0 -4C_1




Similarly, we can follow exact the same procedure for β. We get:

    M_0(B_1) = -2A_1 +6A_2 -3B_1 + 2B_2 -4C_-1 + 4C_0
        M_0(B_2) = -2A_1 +2A_2 -2B_1 + B_2 + 4C_0


Now, as figure <ref> suggest, the case for each γ_i is more subtle. Contrary to the α,β cases, the 2-cyle C_0, for example, is formed by continuing a 1-cycle that involves both η_1,η_2, therefore we can't ignore any of the points -1,0,1 in computing the monodromy. At this point though, we can use the computer and impose the condition that M_0 has to preserve Q and solve the system of equations. The result is the following:

    M_0(C_0) = -A_1 +3A_2 -3B_1 + B_2 -2C_-1 + C_0 + 2C_1
        M_0(C_-1) = 2A_1 -4A_2 +6B_1 -2 B_2 +C_-1 - 4C_0 - 4C_1
        M_0(C_1) = -2A_1 +6A_2 -4B_1 + 2B_2 -4C_-1 + 4C_0 + C_1


Now we can write our full monodromy M_0:

    M_0 = [  1  2 -2 -2  2 -1 -2; -2 -3  6  2 -4  3  6;  2  6 -3 -2  6 -3 -4; -2 -2  2  1 -2  1  2;  0  0 -4  0  1 -2 -4; -4 -4  4  4 -4  1  4;  0 -4  0  0 -4  2  1 ]


Since we can rearrange the loops around -1,0,1,∞ so that their product is the identity, we naturally get the expression for M_∞ as the inverse of the prodcut M_- M_0  M_+, leading to:


    M_∞ = [  0 -4  1  0 -4  2  2;  4  0  4  1 -2  2  4; -1  4 -3 -2  6 -3 -4;  0 -1  2  1 -2  1  2; -4  0 -4  0  1 -2 -4;  0  0  4  4 -4  1  4;  0 -4  0  0 -4  2  1 ]




§ THE GEOMETRIC MONODROMY GROUP


Recall that by the Monodromy theorem<ref>, all the monodromies are quasi-unipotent. Hence, all of them have a well-defined logarithm, which we will denote by N_i:= log(M_i), see chapter 1 for a brief review of this topic. 

A quick computation shows that M_0 is in fact semi-simple, so the unipotent part (M_0)_un is the identity and hence N_0=0. The remaining monodromies do have non trivial logarithms: M_+,M_- are actually unipotent and M_∞ is the only non-unipotent. We can easily check that M_∞^3 is unipotent though. 

If M_∞ = M_s M_u is the Jordan-Chevalley decomposition and I is the 7x7 identity matrix, then:

    N_+ = M_+ - I
        N_- = M_- - I
        N_∞ := log(M_u) = 1/3log(M_∞^3)


We have the following result concerning the monodromy group of the family X_t:

The log-monodromies N_+,N_-,N_∞ generate 𝔤_2.


Consider the elements:

    Y_1 = [N_-,N_+]      Y_8    = [Y_5,Y_6]
        Y_2 = [N_-,N_∞]      Y_9    = [N_∞,Y_5]
        Y_3 = [N_+,N_∞]      Y_10   = [N_∞,Y_9]
        Y_4 = [Y_1,Y_2]     Y_11   = [N_∞,Y_10]
        Y_5 = [Y_1,Y_3]     Y_12   = [N_+,Y_11]
        Y_6 = [Y_2,Y_3]     Y_13   = [N_∞,Y_12]
        Y_7 = [Y_2,Y_6]     Y_14   = [N_-,Y_13]

A quick computation leads us to:

The elements N_-,N_+,Y_1,Y_4,Y_5,Y_6,Y_7,Y_8,Y_9,Y_10,Y_11,Y_12,Y_13,Y_14 are linearly independent over ℚ.


Now define t_1 := Y_1 and t_2 := [Y_4,Y_5], a direct computation gives us that [t_1,t_2 ]=0, moreover they both are diagonalizable. Let ad(.) denotes the adjoint representation, if we act through ad(t_i), i=1,2, on 𝔤, we get 14 linearly independent (in both cases) eigenvectors with 1-dimensional eigenspaces, moreover we have:

    
  * 1 with eigenvalue -2
    
  * 4 with eigenvalue -1
    
  * 4 with eigenvalue 0
    
  * 4 with eigenvalue 1
    
  * 1 with eigenvalue 2 
 
Which are in 1-1 correspondence with the roots of 𝔤_2(see figure <ref>), therefore 𝔥:= ⟨ t_1,t_2⟩ is a Cartan subalgebra and 𝔤 = 𝔤_2.



This gives us the immediate corollary:

The geometric monodromy group of the family X_t is G_2.



The asymptotics of the period map for this family is given by the limiting mixed Hodge structure(LMHS) at each singularity of the local system.They are easily computed using the expression of the monodromies obtained above, and one hopes that using that fact, one can prove the generic global Torelli theorem for some quotient of this family.





§ THE FOURTH FAMILY

Now we analyze the family X_t given by the fourth family in <ref>. It is a family of elliptic surfaces obtained by base changing the elliptic surface 
    y^2=4x^3+9z^2x^2 + 6zx +1  (z∈ℙ^1)
 by w^2 = t(z^3-1) + t^2 (t≠ 0,1,∞). Repeating the same argument as we did before, now we have 6 fibers of type I_1 and one of type I_18(at ∞). Moreover, the local monodromies around 1,r=-1+√(3)i/2 and r̅ are obtained using the Picard-Lefschetz formula (<ref>) again, since the degeneration is nodal. We have:


    T_1=( [ 1 1; 0 1 ])
    
    , T_r= ( [  1  0; -1  1 ])
    
    , T_r̅= ( [  2  1; -1  0 ])


Let δ,η be the basis for the local system with δ·η=1, then the vanishing cycles are δ at 1, η at r and δ - η at r̅. Let γ_i the path connecting the 2 points in E_t which are the pre-image of `i', and α,β the basis for the first homology of the Elliptic curve E_t, see Figure <ref>. By using the same idea of the first case, we define the 7 cycles:


    A_1 = δ×α    C_1   = δ×γ_1
        A_2 = η×α    C_r   = η×γ_r
        B_1 = δ×β    C_r̅   = (δ - η )×γ_r̅
        B_2 = η×β


Again, the cycles A_1,A_2,B_1,B_2 are trivially transcendental but the Cs are not. Let E_i,E_i' be the two nodal curves over the end of γ_i, then the cycles C_i = C_i + E_i+E_i'-σ, for i=1,r,r̅ and σ the zero section. The intersection matrix with respect to the basis (A_1,A_2,B_1,B_2,C_1,C_r,C_r̅) is:

    Q = [  0  0  0 -1  0  0  0;  0  0  1  0  0  0  0;  0  1  0  0  0  0  0; -1  0  0  0  0  0  0;  0  0  0  0 -2  1 -1;  0  0  0  0  1 -2  1;  0  0  0  0 -1  1 -2 ]

In particular, Q ≠ 0 again and the cycles above are indeed a basis. The monodromy M_1 around t=1 is really simple, the crosses × rotate counter-clockwise around each other until they meet the spot the other was. As the picture <ref> suggests, the C_i don't move, so we can ignore them. A quick computation using <cit.> leads us to:

    α→α-β,β→α

As t→ 1, the branch cuts of E_t don't cross the cuts trivializing the ℙ^1 local system, hence the cycles {δ,η} don't change and our matrix becomes:

    M_1 = [  1  0  1  0  0  0  0;  0  1  0  1  0  0  0; -1  0  0  0  0  0  0;  0 -1  0  0  0  0  0;  0  0  0  0  1  0  0;  0  0  0  0  0  1  0;  0  0  0  0  0  0  1 ]

The Jordan Block of this matrix is:

    [      1      0      0      0      0      0      0;      0      1      0      0      0      0      0;      0      0      1      0      0      0      0;      0      0      0    ζ_6      0      0      0;      0      0      0      0    ζ_6      0      0;      0      0      0      0      0 ζ̅_̅6̅      0;      0      0      0      0      0      0 ζ̅_̅6̅ ]

where ζ_6 is the 6-th root of unity. Our result matches Katz's description <cit.> of the Jordan  block structure of the monodromy around 1.


On the other hand, when t→ 0 the situation is really complicated. As before, the branch cuts cross the cuts of the
local system, the × in Figure <ref> encircle r,r̅,1 counter-clockwise once. 

According to Katz <cit.>, the Jordan Block of M_0 should be

    [ -1  0  0  0  0  0  0;  0 -1  0  0  0  0  0;  0  0 -1  0  0  0  0;  0  0  0 -1  0  0  0;  0  0  0  0  1  0  0;  0  0  0  0  0  1  0;  0  0  0  0  0  0  1 ]


Note, in particular, that M_0 has to be of finite order.

We couldn't find this monodromy, but by looking at the representation description given by Katz, we conjecture that the monodromy group is reducible and is either SL(2)× SL(2)× SL(2) or SL(2)× SL(2)× U(1).
amsplain
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