Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds
Shinobu Hosono Hiromichi Takagi
math.AG


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Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds

Movable vs Monodromy Nilpotent Cones
 of Calabi–Yau Manifolds[This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.htmlhttp://www.emis.de/journals/SIGMA/modular-forms.html]

Shinobu HOSONO and Hiromichi TAKAGI

S. Hosono and H. Takagi

Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan
mailto:hosono@math.gakushuin.ac.jphosono@math.gakushuin.ac.jp, mailto:hiromici@math.gakushuin.ac.jphiromici@math.gakushuin.ac.jp

Received September 11, 2017, in final form April 23, 2018; Published online May 02, 2018

We study mirror symmetry of complete intersection Calabi–Yau manifolds which have birational automorphisms of infinite order. We observe
that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.

Calabi–Yau manifolds; mirror symmetry; birational geometry; Hodge theory

14E05; 14E07; 14J33; 14N33






§ INTRODUCTION


A smooth projective variety X of dimension n is called a Calabi–Yau n-fold if the canonical bundle K_X is trivial and H^i(X,𝒪_X)=0, 1≤ i≤ n-1. In the 90's, the idea of mirror symmetry was discovered in theoretical physics and has long been a source of many mathematical ideas related to Calabi–Yau manifolds. After more than 20 years since its discovery, we have now several approaches <cit.> toward mathematical understanding of the symmetry.

In this paper, we will focus on “classical” mirror symmetry of Calabi–Yau threefolds, i.e., we compare two different moduli spaces associated to Calabi–Yau threefolds, the Kähler moduli and the complex structure moduli spaces, considering Calabi–Yau threefolds which have several birational models. According to birational geometry of higher dimensional manifolds, if a Calabi–Yau threefold X has birational models, then the Kähler cone of X can be extended to the movable cone Mov(X) <cit.>. On the mirror side, corresponding to each birational model, there appears a special boundary point called large complex structure limit, which is characterized by unipotent monodromy <cit.>. Using this unipotent property, the so-called monodromy nilpotent cone is defined for each boundary point. We will find that, as a result of monodromy relations, the monodromy nilpotent cones glue together to define a larger cone which can be identified with the movable cone Mov(X) under mirror symmetry.

Studying birational geometry in mirror symmetry (or string theory) goes back to papers by Morrison and his collaborators in the 90's <cit.>. The birational geometry discussed in the 90's was mostly for Calabi–Yau hypersurfaces in toric varieties, and it comes from the different resolutions of ambient toric varieties. In this paper, we will study two specific examples of complete intersection Calabi–Yau threefolds for which we have birational models in slightly different form, and also have birational automorphisms of infinite order.

The construction of this paper is as follows: In Section <ref>, we will first recall some background material on mirror symmetry as formulated in the 90's. Restricting our attentions to three dimensional Calabi–Yau manifolds, we will summarize the basic properties of Calabi–Yau manifolds called A- and B-structures. In Section <ref>, we will introduce a specific Calabi–Yau threefold given by a complete intersection in ^4×^4, whose birational geometry and mirror symmetry were studied in detail in previous works <cit.>. We will describe its movable cone by studying the geometry of birational models. In Section <ref>, we will report some results of monodromy calculations, and describe the details of how the monodromy nilpotent cones glue together by monodromy relations. In Section <ref>, we will present another complete intersection given in ^3×^3. Although there do not appear other birational models to this Calabi–Yau threefold than itself, we will observe interesting gluing property of monodromy nilpotent cones which corresponds to the structure of the movable cone observed in <cit.>. Summary and discussions will be presented in Section <ref>. There we will also describe the corresponding calculations for a K3 surface in ^3×^3 which has a parallel description to the complete intersection in ^4×^4.



§ CLASSICAL MIRROR SYMMETRY
 



 §.§ Mirror symmetry of Calabi–Yau threefolds


Let us consider Calabi–Yau threefolds X and X^* which will be taken to be mirror to each other. For each of these, we have two different structures, called A-structure and B-structure.



  §.§.§ A-structure of X
 Let 𝒦_X be the Kähler cone of X and κ_1,…,κ_r∈ H^1,1(X,)=H^1,1(X)∩ H^2(X,) be ge­nerators of the Kähler cone, where for simplicity, we assume that the Kähler cone is a simplicial cone in H^2(X,ℝ). Let κ be the Kähler class which corresponds to the polarization of X and write κ by

    κ=t_1κ_1+⋯+t_rκ_r,

with t_i>0. The Lefschetz operator L_κ(-):=κ∧(-) defines a nilpotent linear action on the even cohomology H^ even(X):=⊕_pH^p,p(X). In fact, this is a part of the Lefschetz 𝔰𝔩(2,ℂ) action, and defines the following decomposition:

    H^0,0
    
    
    H^1,1
    
    
    H^2,2
    
    
    H^3,3    =    ∙
    ↓
    ∙   ∙      ∙     
    ↓   ↓   ⋯   ↓ L_κ.
    ∙   ∙      ∙     
    ↓
    ∙


From the viewpoint of homological mirror symmetry, it is natural to replace H^ even(X) with the Grothendieck group K(X) (modulo torsion) which is an abelian group equipped the symplectic form

    χ(-,-) K(X)× K(X)→ℤ

with χ defined by χ(ℰ,ℱ):=∑(-1)^i H^i(X,ℰ^*⊗ℱ) for vector bundles. Based on this integral and symplectic structure on K(X), we can introduce the corresponding structure on H^ even(X,ℚ). A-structure of X is the nilpotent action L_κ on H^ even(X,ℚ) with this integral and symplectic structure.



  §.§.§ B-Structure of X^*
 Let X^*=X_b_0^* and consider
a smooth deformation family π𝔛^*→ B of X_b_0^*(b_0∈ B) over some open parameter space B. We denote by X_b^*=π^-1(b) the fiber over b∈ B. Then we have Kodaira–Spencer map ρ_b T_bB→ H^1(X_b,𝒯X_b^*)
which we assume to be an isomorphism. Associated to this family, we naturally have the local system R^3π_*ℂ_𝔛^*
on B. In the 90's, mirror symmetry was recognized by finding some local family 𝔛|_Δ_r^*→Δ_r^* with special
properties over the product of punctured disc Δ_r^*=(Δ^*)^r where Δ^*={ z∈ℂ | 0<|z|<1} and
B=r. The required properties for the local family are described by the monodromy representation of the fundamental group π_1(Δ_r^*)≃ℤ^r for the local system R^3π_*ℂ_𝔛^* restricted to over Δ_r^*. Let T_i represent the monodromy matrix corresponding to the i-th generator of π_1(Δ_r^*) with fixing a base point b_0∈Δ_r^*. Assuming that all T_i are unipotent, we have nilpotent matrices N_i=log T_i=∑_k≥1(-1)^k-1/k(T_i- id)^k. The set

    Σ={∑λ_iN_i | λ_i∈ℝ_>0}

is called monodromy nilpotent cone consisting of nilpotent matrices on H^3(X_b_0,ℚ). It is known that each element of Σ defines the same monodromy weight filtration on H^3(X_b_0,ℚ) (see <cit.>). The following definition is due to Morrison <cit.>.

 The degeneration of the local family 𝔛|_Δ_r^*→Δ_r^* at the origin is called a large complex structure limit (LCSL) if the following hold:
=0pt

  (1) All T_i, i=1,…,r, are unipotent.


  (2) Let N_λ=∑_iλ_iN_i λ_i>0. This induces the monodromy weight filtration,

    W_0=W_1   ⊂    W_2=W_3   ⊂    W_4=W_5   ⊂    W_6    =H^3(X_b_0,ℚ)
    ∙   ←   ∙   ←   ∙   ←   ∙
          ∙   ←   ∙
          ⋮      ⋮
          ∙   ←   ∙

with W_0=1 and W_2=1+r.


  (3) Let W_0=ℚw_0 and introduce a bi-linear form on W_0 by ⟨ w_0,w_0⟩=1. This defines m_jk:=⟨ w_0,N_jw_k⟩ for a ℚ-basis [w_1],…,[w_r] of W_2/W_0. Then the r× r matrix (m_jk)_1≤ j,k≤ r is an invertible ℚ-matrix.



We note that there is a natural integral symplectic structure on H^3(X_b_0^*,ℤ), and the mono­dromy matrices T_i are given by integral and symplectic matrices if we fix a symplectic basis of H^3(X_b_0^*,ℤ). B-structure of X^* at LCSL is defined to be such an integral and symplectic basis of H^3(X_b_0^*,ℤ) with the monodromy matrices T_i which are compatible with the filtration (<ref>).



  §.§.§ Mirror symmetry
 In classical mirror symmetry, X is called a mirror to X^* if the A-structure of X is isomorphic to the
B-structure of X^*, i.e., the two nilpotent actions L_κ and N_λ are identified together with their integral and symplectic structures. To be more explicit, suppose we have a B-structure at a LCSL. Since N_λ^4=0 and N_λ^3W_6⊂ W_0, we have

    N_iN_jN_k=C_ijk𝙽_0

with a fixed rank one nilpotent matrix 𝙽_0 satisfying N_i𝙽_0=0. Corresponding to this products of nilpotent matrices, we have, in the A-structure, the cup-product

    κ_i∪κ_j∪κ_k=K_ijk𝚅_0,      K_ijk∈ℤ,

where 𝚅_0∈ H^3,3(X) normalized by ∫_X𝚅_0=1. After fixing a normalization of the matrix 𝙽_0, we have C_ijk=K_ijk if X and X^* are mirror to each other, in particular we have C_ijk∈ℤ_≥0. In fact, C_ijk is the leading coefficient of the so-called Griffiths–Yukawa coupling, and K_ijk is the leading term of the quantum product. Mirror symmetry implies the equality between the two in full orders under the so-called mirror map.



 §.§ Birational geometry and mirror symmetry


Calabi–Yau threefolds often come with birational models. Mirror symmetry in such cases has been studied in <cit.> and is known as topology change in physics <cit.>. The purpose of this paper is to elaborate such cases in more details comparing the A-structure of X and the B-structure of X^*. In the 90's, Morrison considered the movable cone of X in the context of mirror symmetry and also the topology change. We will push this perspective further by finding the corresponding cone structure in terms of the monodromy nilpotent cones in the B-structure of X^*.



  §.§.§ Movable cones of X
 As above, let us assume that Calabi–Yau threefold X=:X_1 comes with several other Calabi–Yau threefolds X_i, i=2,…,s, which are birational to each other. Let 𝒦_i⊂ H^2(X_i,ℝ) be the Kähler cone of X_i. Using the birational maps φ_i X X_i, these Kähler cones of X_i can be transformed to the corresponding cones in H^2(X,ℝ). The convex hull of the union of these cones is the movable cone Mov(X) of X. It is shown in <cit.> that the union of the transformed Kähler cones defines a chamber structure to the movable cone Mov(X) (see also <cit.>). To work with the classical mirror symmetry, in fact, we have to consider the movable cone in H^2(X,ℝ)⊗ℂ using the complexified Kähler cones 𝒦_i+√(-1)H^2(X_i,ℝ). However, in this paper, we will mostly focus on the structures in the real part of the complexified Kähler moduli.



  §.§.§ Compactification of the moduli space ℳ_X^*^ cpx


Suppose that X^* is mirror to X, i.e., we have a mirror family 𝔛^*→ B:=_X^*^ cpx over a parameter space ℳ_X^*^ cpx on which we find a local (smooth) family 𝔛|_Δ_r^*→Δ_r^*⊂ℳ_X^*^ cpx to describe the B-structure which is mirror to the A-structure of X. In the classical mirror symmetry of Calabi–Yau complete intersections in toric varieties, there is a natural (toric) compactification ℳ_X^*^ cpx <cit.> of the moduli space ℳ_X^*^ cpx, and the geometry Δ_r^*⊂ℳ_X^*^ cpx is characterized by the corresponding normal crossing boundary divisors at the origin o∈Δ_r=ℂ^r.

The following properties can be observed for an abundance of examples of complete intersection Calabi–Yau manifolds:

 
Assume X and X^* are Calabi–Yau threefolds which are mirror to each other. If Calabi–Yau threefold X=:X_1 has birational models X_i, i=2,… ,s, then there appear the corresponding boundary points o=:o_1 and o_i, i=2,… ,s, given by normal crossing divisors in ℳ_X^*^ cpx such that
=0pt

  (1) o_i are LCSLs, and

  (2) the A-structures of X_i are isomorphic to the B-structures
arising from o_i.





Let X and X^* be as above. Corresponding to the birational map φ_ji X_i X_j, there is a path connecting o_i to o_j and the connection matrix M_ji of the B-structures such that

=0pt

  (1) it preserves the monodromy weight filtrations, and

  (2) it is integral and also compatible with the symplectic structures at each o_i, i.e., ^tM_jiΣ_jM_ji =Σ_i for the symplectic matrices Σ_i representing the symplectic forms on H^3(X_b_o_i^*,ℤ).



Recently Calabi–Yau manifolds which are derived equivalent but not birational each other have been attracting attention (see, e.g., <cit.> and references therein). These are called Fourier–Mukai partners after the original work by Mukai for K3 surfaces <cit.>. As shown in examples <cit.>, if a Calabi–Yau threefold X has such Fourier–Mukai partners, then corresponding boundary points exist in ℳ_X^*^ cpx with the property (2) in Observation <ref>, but losing the property (1). If we have both the property (1) and (2), then we can see that the so-called prepotential for quantum cohomology is invariant up to quadratic terms under analytic continuations (see Proposition <ref> below), and hence the quantum cohomologies of birational Calabi–Yau threefolds are essentially the same (see <cit.> for example). However, as we see in <cit.>, quantum cohomologies of Fourier–Mukai partners are quite different to each other.


In this paper we will focus on Calabi–Yau threefolds given by complete intersections in toric varieties. Showing two examples which exhibit interesting birational geometry, we will make Observation <ref> more explicit, e.g., we will give precise descriptions about the path connecting the boundary points. Also finding some monodromy relations, we will come to the following observation:

 Assume a Calabi–Yau threefold X and its mirror manifold X^* have the properties described in Observation <ref>. Then there are natural choices of path connecting o_i and o_j such that the monodromy nilpotent cones defined for each o_i in ℳ_X^*^ cpx are glued together. We identify the resulting structure as the mirror counter part of the movable cone obtained by gluing Kähler cones by birational maps.


The gluing will be achieved by finding monodromy relations coming from boundary divisors which have multiple tangency with some component of the discriminant (see Section <ref>). When writing the monodromy relations, we find a certain monodromy action of a distinguished form, which we call “Picard–Lefschetz formula of flopping curves” based on the mirror correspondence (cf. the same forms are known in physics literatures, <cit.> for example, as strong coupling limits associated to certain contractions of curves).



§ COMPLETE INTERSECTION CALABI–YAU SPACES
 FROM GORENSTEIN CONES


In this section, we describe mirror symmetry of a Calabi–Yau complete intersection of the form

    X:=(ℙ^4|11111
    ℙ^4|11111
    )^2,52,

i.e., a complete intersection of five general (1,1) divisors in ℙ^4×ℙ^4 which has Hodge numbers (h^1,1,h^2,1)=(2,52). In this section, we will study the A-structure of X.



 §.§ Cones for complete intersections and Calabi–Yau manifolds


To describe the complete intersection X, let us note that we can write X=s^-1(0) with a generic choice of a section of the bundle 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4. We describe this starting with the affine cone over the generalized Segre embedding s_1,1,1(ℙ^4×ℙ^4×ℙ^4), which we write by

    U_0:=Specℂ[λ_iz_jw_k | 1≤ i,j,k≤5]

with the homogeneous coordinates λ_i, z_j, w_k of ℙ^4's. Let U→ U_0 be the blow-up of the cone at the origin. It is easy to see that the exceptional divisor E is isomorphic to ℙ^4×ℙ^4×ℙ^4. In fact, U is isomorphic to the total space of the line bundle 𝒪(-1,-1,-1)→ℙ^4×ℙ^4×ℙ^4. Contracting one of the ℙ^4's (m-th factor of _λ^4×_z^4×_w^4), we have three possible contractions of U which fit in the following diagram:

    [dl]_π_1 U_1
    @<–>[d] 
        U_0        [l]_    π_2   U_2   @<–>[d]    [ul] [l] [dl]     U.     
       [ul]^π_3 U_3

Again, it is easy to see that U_α→ U_0, α=1,2,3, are small resolutions, and the geometries of U_α are of the form 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4 that are birational to each other. It is worthwhile noting that if we start with the cone over s_1,1(ℙ^1×ℙ^1) in the above construction, the resulting geometry is the standard Atiyah flop for the small resolutions of the form 𝒪(-1)⊕𝒪(-1)→ℙ^1.

Consider the potential function on U_0,

    W=∑_i,j,ka_ijkλ_iz_jw_k

with a_ijk∈ℂ being chosen generically. Let W_α:=π_α^#W be the potential functions on U_α. We denote the critical locus of W_α in each U_α by

    X_α:=Crit(W_α,U_α),     α=1,2,3.



The critical locus X_α is a Calabi–Yau complete intersection of the form (<ref>).

 By symmetry, we only consider the case X_1. To write the conditions for the criticality, it is helpful to use the homogeneous coordinate for the small resolution U_1, which is the total space (-1,-1)^⊕5→_z^4×_w^4. Let z_i,w_j denote the homogeneous coordinates of _z^4×_w^4 and λ_i be the fiber coordinate. Then the potential function is simply given by W_1=∑_i,j,ka_ijkλ_iz_jw_k, which gives the conditions for the criticality ∂ W_1/∂λ_i=∂ W_1/∂ z_j=∂ W_1/∂ w_k=0. If we denote ∂ W_1/∂λ_i=∑_j,ka_ijkz_jw_k=:f_i(z,w), the conditions ∂ W_1/∂ z_j=∂ W_1/∂ w_k=0 may be arranged into a matrix form

    (∇_zf_1   ⋯   ∇_zf_5
    ∇_wf_1   ⋯   ∇_wf_5)([ λ_1;   ⋮; λ_5 ])=0.

The last equation gives the zero section {λ_1=⋯=λ_5=0}≃_z^4×_w^4
and the conditions f_1(z,w)=… =f_5(z,w)=0 give a smooth complete intersection in the zero section if we choose a_ijk sufficiently
general. 

[<cit.>]  X_α and X_β, α≠β, are birational. The birational maps φ_βα X_α X_β are given by the Atiyah flops associated to the contractions of 50 ℙ^1s, which we summarize in the following diagram:

    (-26,0)*++X_1="Xi",
    (0,0)*++X_2="Xii",
    (26,0)*++X_3="Xiii",
    (52,0)*++X_1,="XiR",
    (-13,-13)*+Z_2="Zii",
    (13,-13)*+Z_3="Ziii",
    (39,-13)*+Z_1="ZiR",
    _π_21 "Xi";"Zii"
    ^π_22 "Xii";"Zii"
    _π_32 "Xii";"Ziii"
    ^π_33 "Xiii";"Ziii"
    _π_13 "Xiii";"ZiR"
    ^π_11 "XiR";"ZiR"
    @<–> "Xi";"Xii"
    @<–> "Xii";"Xiii"
    @<–> "Xiii";"XiR"

where Z_1⊂ℙ_z^4, Z_2⊂ℙ_w^4 and Z_3⊂ℙ_λ^4 are determinantal quintics defined by the 5×5 matrices (∑ z_ja_ijk), (∑ w_ka_ijk) and (∑λ_ia_ijk), respectively.

We refer the references <cit.> for the proof of the above proposition.
  In the above proposition, we naturally come to birational Calabi–Yau complete intersections. Some remarks related to this are in order:
=0pt

  * U_α's are birational to each other since they are all toric varieties with the same algebraic torus contained as a dense subset. In fact, they all have the form 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4. However, when defining X_α as the critical locus of the potential function, the zero section of 𝒪(-1,-1)^⊕5→ℙ^4×ℙ^4 is specified by the criticality condition. Hence, that U_α's are birational does not imply that X_α's are birational. The fact that X_α's are birational comes from different reasons as described in the above proposition.


  * The affine cone construction here is an example of more general method in toric geometry due to Batyrev and Borisov <cit.>. There, the affine cone is replaced by the so-called Gorenstein cones, and actually a pair of reflexive Gorenstein cone (C_∇,C_Δ) to describe mirror symmetry. The birational geometry we observed in the above proposition has been described by the property of so-called nef-partitions of ∇ by Batyrev and Nil <cit.>. They have found that the two different (but isomorphic) nef-partitions

    ∇=∇_1+∇_2+⋯+∇_s=∇_1'+∇_2'+⋯+∇_s'

sometimes results in dual nef-partitions

    Δ=Δ_1+Δ_2+⋯+Δ_s,    Δ'=Δ_1'+Δ_2'+⋯+Δ_k'

with Δ and Δ' having completely different shapes to each other. We can describe our birational Calabi–Yau threefolds in this general setting. See references <cit.> for recent works which shed light on this general phenomenon from the derived categories of Calabi–Yau threefolds.





 §.§ Movable cone of X:=X_1
 

Let us note that the Kähler cone of X(=X_1) is given by 𝒦_X=ℝ_>0H_1+ℝ_>0H_2 with the pull-backs H_1=π_11^*H_Z_1 and H_2=π_21^*H_Z_2 of the hyperplane classes H_Z_i of Z_i, where π_ji X_i→ Z_j is the projection in the diagram (<ref>).
  =0pt


  (1) Let 𝒦_X_2=ℝ_>0L_Z_2+ℝ_>0L_Z_3
be the Kähler cone with the generators L_Z_2=π_22^*H_Z_2
and L_Z_3=π_32^*H_Z_3. By the birational map φ_21 X_1 X_2,
the Kähler cone is transformed to

    φ_21^*(𝒦_X_2)=_>0H_2+ℝ_>0(4H_2-H_1).


  (2) Similarly, let 𝒦_X_3=ℝ_>0M_Z_3+ℝ_>0M_Z_1
be the Kähler cone of X_3 generated by M_Z_3=π_33^*H_Z_3
and M_Z_1=π_13^*H_Z_1, then we have

    φ_31^*(𝒦_X_3)=ℝ_>0(4H_1-H_2)+_>0H_1

for the birational map φ_31 X_1 X_3.



See Appendix <ref>. 

With the divisors L_Z_2, L_Z_3 and M_Z_3, M_Z_1 defined as above, we have

    φ_32^*(M_Z_1)=4L_Z_3-L_Z_2,    φ_32^*(M_Z_3)=L_Z_3

for the birational map φ_32 X_2 X_3.
 The second relation holds by definition. For the first relation, see Appendix <ref>.


Now, we define the following composite of the birational maps:

    ρ:=φ_13∘φ_32∘φ_21

with the convention φ_ij=φ_ji^-1 X_j X_i (see the diagram (<ref>)).
 The birational map ρ is not an automorphism of X. It is of infinite order.

We show that

    ρ^*H_1=-4H_1+15H_2,    ρ^*H_2=-15H_1+56H_2

for ρ^*=φ_21^*∘φ_32^*∘φ_13^*. Since φ_13^*=(φ_31^-1)^*=(φ_31)_* and using the relations φ_31^*(M_Z_3)=4H_1-H_2,φ_31^*(M_Z_1)=H_1 in Lemma <ref>(2), we have

    M_Z_3=4M_Z_1-φ_13^*(H_2),     M_Z_1=φ_13^*(H_1).

Then, using Lemmas <ref> and <ref>, it is straightforward to evaluate ρ^*(H_i), e.g., ρ^*(H_1)=φ_21^*∘φ_32^*(M_Z_1)=φ_21^*(4L_Z_3-L_Z_2)=4(4H_2-H_1)-H_2. From these actions of ρ^*, we see that ρ^*(_X)≠_X and hence ρ∉Aut(X). Also, expressing the linear
action (<ref>) by a matrix ([  -4 -15;  15  56 ]), we see that ρ has an infinite order. 

Suppose X_i≄X_j, i≠j, then the groups of birational maps of X_i are given by

    Bir(X_i)=Aut(X_i)·⟨φ_i1∘ρ∘φ_1i⟩.


 Since arguments are similar to <cit.>, here we only give a rough sketch. Also, we only describe the case i=1, φ_11=id_X. Take a birational map τ X X. We denote by 𝙴(τ) the locus where τ is not defined or non-isomorphic. Consider an ample divisor D and its transform D'=(τ^-1)_*D. Under this setting, we consider the two cases: (i) If D' is nef, then using <cit.> we have D|_𝙴(τ^-1)≡0, i.e., numerically equivalent to zero. Since D is ample, this implies 𝙴(τ^-1)=∅, i.e., τ∈Aut(X). (ii) If D' is not nef, the restriction D'|_𝙴(τ) is not nef, too. This is because if D'|_𝙴(τ) were nef, then D'=(τ^-1)_*D must be nef because D is ample. Therefore D'|_𝙴(τ) is not nef and there exists a curve C⊂𝙴(τ) such that D'· C<0. Now, since K_X|_𝙴(τ)≡0, we know that K_X+ε D', 0<ε≪1, is not nef and (X,ε D') is klt. From the theory of minimal models, we know that there exists an extremal ray of NE(X) and its associated contraction, which must be either X→ Z_1 or X→ Z_2 up to automorphisms. Now, corresponding to these two possibilities, we make the following diagrams:

    (-50,0)*++X="Xl",
    (-30,0)*++X_2="Xii",
    (-10,0)*++X="Xll",
    (-40,-13)*++Z_1="Zi",
    
    (10,0)*++X="Xr",
    (30, 0)*++X_3="Xiii",
    (50,0)*++X.="Xll",
    (20,-13)*++Z_2="Zii",
    (0,-5)*or
    @/^3mm/@–>^τ (-46,3);(-13,2)
    @–>^φ_21 "Xl";"Xii"
     "Xl";"Zi"
     "Xii";"Zi"
    
    @/^3mm/@–>^τ (14,3);(47,2)
    @–>^φ_31 "Xr";"Xiii"
     "Xr";"Zii"
     "Xiii";"Zii"

Depending on the two cases, we set D”=(φ_21τ^-1)_*D or D”=(φ_31τ^-1)_*D and consider inductively the above two cases (i) and (ii) again. Due to <cit.>, this process terminates arriving at the case (i) in the end. We can deduce that there are only two possibilities under the assumption X≄X_i, i=2,3:

    (0,0)*+X=X_1_φ_21
    X_2_φ_32
    X_3_φ_13
    X_1_φ_21
    X_2⋯_φ_13
    X_1
    X_1=X,,
    (0,-12)*+X=X_1_φ_31
    X_3_φ_23
    X_2_φ_12
    X_1_φ_31
    X_3⋯_φ_12
    X_1
    X_1=X.,
    
    
    @/_2.5mm/@–>^ρ (-35,-3);(-7,-3)
    @/_2.5mm/@–>^ρ^-1 (-35,-15);(-7,-15)

Corresponding to these two, we have the decomposition τ=φ_Lρ^n or τ=φ_R(ρ^-1)^m with φ_L,R∈Aut(X).

 We use the assumption X_i≄X_j at the very end of the above proof. If X_1≃ X_i, then it is easy to deduce that we only have to include φ_i1 in the generators of Bir(X_1). Similar modification in Bir(X_i) is required if X_i≃ X_j, i≠j. These do not affect the form of the movable cone determined below. The assumption in the above proposition has been made just for simplicity.


Let us denote by Mov(X_i) be the movable cones generated
by movable divisors on X_i. Since the transforms of movable divisors
by flops are movable, we have

    Mov(X)=Mov(X_1)=φ_21^*Mov(X_2)=φ_31Mov(X_3).

 The following result is known by <cit.>. For completeness of our arguments, we present it here with a general proof.

The closure of the movable cone Mov(X) is given by

    Mov(X)=ℝ_≥0(-H_1+(2+√(3))H_2) +ℝ_≥0(H_1+(-2+√(3))H_2).


By Lemmas <ref> and <ref>, it is easy to see that the closure of the set φ_21^*(𝒦_X_2)∪𝒦_X_1∪φ_31^*(𝒦_X_3) is given by

    C_123:=ℝ_≥0(4H_2-H_1)+ℝ_≥0(4H_1-H_2).

We define

    M:=⋃_n∈ℤ(ρ^*)^nC_123:=⟨ρ^*⟩·C_123.

Then, from a linear algebra, it is straightforward to see that the r.h.s. of (<ref>) coincides with the closure M. Since any automorphism of X_i preserves the generators of _X_i or exchanges them, using Proposition <ref>, we have ∪_iφ_i1^*(Bir(X_i)^*_X_i)=M. Hence we have M⊂Mov(X).

To show the other inclusion, take a rational point d∈Mov(X). There exist m≫1 and an effective movable divisor D such that
md=[D]. If D is nef, then d∈_X and hence d∈ M. If D is not nef, we do the same inductive process as in the proof of Proposition <ref> and find a birational map τ X X_i, τ∈⟨ρ,φ_21, φ_31⟩, such that D'=τ_*D is a nef divisor on X_i, i.e., D'∈_X_i. Namely, we have D=τ^*D'∈𝒦_X and 𝒦_X⊂ M, which imply Mov(X_i)()⊂ M. Hence we have Mov(X)⊂M.






 §.§ Mirror symmetry of X
 

For the complete intersection Calabi–Yau threefolds X (=X_1), the mirror family can be obtained by a straightforward application of the Batyrev–Borisov toric mirror construction. However, the construction involves complications in combinatorics for toric geometry. In our case, we can avoid these complications and find the mirror family of X by the so-called orbifold mirror construction starting with a special family <cit.>.

Define the following special family of X_1:

    X_ sp:= { z_iw_i+a z_iw_i+1+b z_i+1w_i=0,   i=1,…,5 }⊂ℙ_z^4×ℙ_w^4,

where the indices of z_i, w_j should be considered modulo 5.


For general values of a, b, we have the following properties:
=0pt


  (1) X_ sp is singular along 20 lines of singularity of A_1 type.


  (2) There exists a crepant resolution X^*→ X_ sp with X^* being a Calabi–Yau threefold with h^1,1(X^*)=52, h^2,1(X^*)=2.


  (3) The resolution X^* parametrized by (a,b)∈ℂ^2 defines a family 𝔛^*→ℳ_X^*^ cpx∖ Dis with ℳ_X^*^ cpx=ℙ^2 and Dis=D_1∪ D_2∪ D_3∪ Dis_0 where D_i are the coordinate lines of ℙ^2 and Dis_0 is an irreducible (singular) curve of degree 5. The fiber over [a^5,b^5,1]∉ Dis is given by the resolution X^* with (a,b).


Proofs of these properties are given in <cit.>.


We can verify that all the properties in Observation <ref> hold for the family 𝔛^*→ℙ^2∖ Dis.
We set

    o_1=D_1∩ D_2=[0,0,1],     o_2=D_2∩ D_3=[1,0,0],     o_3=D_3∩ D_1=[0,1,0].

All these boundary points o_1, o_2, o_3 are LCSLs whose B-structures are identified with the A-structures of the birational models X_1, X_2 and X_3, respectively. See Fig. <ref> in the next section.


The above proposition has been derived by introducing integral and symplectic structures at each o_i and calculating the monodromies around the divisors D_i, see <cit.> for details. Our focus in what follows will be gluing the monodromy cones (<ref>) which are defined for each boundary point o_i.



§ GLUING MONODROMY NILPOTENT CONES I
 

For the example in the preceding section, we will find a path o_i→ o_j which we can identify with the birational map φ_ji X_i X_j as described in Observation <ref>. We will find that the monodromy nilpotent cones (<ref>) at each boundary point are naturally glued together by the monodromy relations coming from the path. Also, in the next section (Section <ref>) we will study another interesting example, which has no other birational models other than itself but has a birational automorphism of infinite order.



 §.§ B-structures of X^*
 

Associated to the family π𝔛^*→ℳ_X^*^ cpx∖ Dis, we have the local system R^3π_*_𝔛^* which introduces the Gauss–Manin system on the moduli space, or equivalently the Picard–Fuchs differential equation for the period integrals of holomorphic three form. This Picard–Fuchs equation has been studied in our previous work <cit.>, where we have described the B-structure for the boundary points o_i, i.e., the integral and symplectic basis for the local solutions as well as integral monodromy matrices using the central charge formula given in <cit.> which goes back to the study of GKZ system <cit.> in the 90's (see <cit.> for details). Here we briefly recall the integral and symplectic basis referring to <cit.> for its explicit form, and define the monodromy nilpotent cones for each o_j from the monodromy matrices calculated there.



  §.§.§ B-structure at o_1
 
Let [-x,-y,1]∈ℙ^2 be the affine coordinate with the origin o_1 (where the minus signs are required to have the canonical integral and symplectic structure based on the central charge formula). The canonical, integral and symplectic structure appears from a unique power series solution w_0(x,y) of the Picard–Fuchs differential equation around the origin o_1. Including the logarithmic solutions, the result can be arranged as follows:

    Π(x,y) =^t(w_0(x,y),w_1^(1)(x,y),
    w_2^(1)(x,y),w_2^(2)(x,y),w_1^(2)(x,y),w^(3)(x,y))
     = ^t(∫_A_0Ω_x,∫_A_1Ω_x,
     ∫_A_2Ω_x,∫_B_2Ω_x,∫_B_1Ω_x,
     ∫_B_0Ω_x),

where { A_0,A_1,A_2,B_2,B_1,B_0}⊂ H_3(X_b_o^*,ℤ) is a symplectic basis satisfying A_i∩ B_j=δ_ij, A_i∩ A_j=B_i∩ B_j=0 representing the integral and symplectic solutions of the Picard–Fuchs equation <cit.>. The monodromy matrix 𝚃_x of Π(x,y) for a small loop around x=0 and similarly 𝚃_y for y=0 have been determined as follows:

    𝚃_x=([  1  0  0  0  0  0;  1  1  0  0  0  0;  0  0  1  0  0  0;  5 10 10  1  0  0;  2  5 10  0  1  0; -5 -3 -5  0 -1  1 ]),    𝚃_y=([  1  0  0  0  0  0;  1  1  0  0  0  0;  1  0  1  0  0  0;  2 10  5  1  0  0;  5 10 10  0  1  0; -5 -5 -3 -1  0  1 ]).

We define

    ℬ_1:= {α_0,α_1,α_2,β_2,β_1,β_0}⊂ H^3(X_b_o^*,ℤ)

to be the dual basis satisfying ∫_A_iα^j=δ_i^ j=∫_B_iβ^j and ∫_A_iβ^j=∫_B_iα^j=0. Since the monodromy actions on the period integrals, i.e., on H_3(X_b_o^*,ℤ), are translated into the dual space via the transpose and inverse, we define the linear action N_λ=∑λ_iN_i on H^3(X_b_o^*,ℤ) by

    N_1:=- ^t(log𝚃_x),     N_2:=- ^t(log𝚃_y).

Then we define the monodromy nilpotent cone at o_1 by

    Σ_o_1:={∑λ_iN_i | λ_i>0}⊂End(H^3(X_b_0,ℚ)).

For general values of λ_i>0, it is easy to see that the nilpotent matrix N_λ induces the monodromy weight filtration W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ)
given by

    W_0=⟨α_0⟩,           W_2=⟨α_0,α_1,α_2⟩,   

    W_4=⟨α_0,α_1,α_2,β_2,β_1⟩,           W_6=⟨α_0,α_1,α_2,β_2,β_1,β_0⟩.    


Using the matrices N_1, N_2, it is easy to see the following property:
We have

    N_iN_jN_k=C_ijk𝙽_0

with totally symmetric C_ijk given by C_111=C_222=5, C_112=C_122=10 and C_ijk=0 for other cases, and 𝙽_0=([   0   1; O_5   0 ]) where O_5 is the zero matrix of size 5×5.
 As we see above, the monodromy matrices of the period integrals act on H_3(X_b_o^*,ℤ) while the monodromy weight filtration is defined in the dual space H^3(X_b_o^*,ℤ). Hence, we translate any monodromy matrix 𝙰 obtained from the analytic continuations of the period integ­ral Π(x,y) to the corresponding matrix A in the dual space by A= ^t𝙰^-1.




  §.§.§ B-structures at o_2, o_3
  In a similar way to the last paragraph, we determine the B-structure from the boundary points o_2 and o_3, which are given by the origins of the affine charts [1,-y',-x']∈ℙ^2 and [-x”,1,-y”]∈ℙ^2. As described in detail in <cit.>, we have the canonical integral and symplectic basis

    Π'(x',y')=x'Π(x',y')    and    Π”(x”,y”)=y”Π(x”,y”)

in terms of the same Π(x,y) as (<ref>) for o_2 and o_3, respectively. Since both of (<ref>) have essentially the same form as Π(x,y), we have

    𝚃_x'^'=𝚃_x”^''=𝚃_x    and    𝚃_y'^'=𝚃_y”^''=𝚃_y

for the monodromy matrices with the base points b_o' and b_o” near the origins. Hence for o_2 and o_3 we have isomorphic
B-structures with

    Ñ_1'=log T_x'',    Ñ_2'=log T_y''    and    Ñ_1”=log T_x””,    Ñ_2”=log T_y””,

where T_x''=( ^t𝚃_x'')^-1, T_y''=( ^t𝚃_y'')^-1 and similarly for T_x””, T_y””. These nilpotent matrices determine the respective monodromy weight filtrations in H^3(X_b_o'^*,ℚ) and H^3(X_b_o”^*,ℚ) with the basis

    {α_0',α_1',α_2',β_2',β_1',β_0'}    and    {α_0”,α_1”,α_2”,β_2”,β_1”,β_0”} ,

as described above. We denote the monodromy nilpotent cones at o_2 and o_3 by

    Σ_o_2'={∑λ_iÑ_i' | λ_i>0}⊂End(H^3(X_b_o'^*,ℚ)),
    Σ_o_3”={∑λ_iÑ_i” | λ_i>0}⊂End(H^3(X_b_o”^*,ℚ)).

These are the B-structures which we identify with the A-structures of the birational models X_2 and X_3, respectively, in <cit.>.



 §.§ Gluing the monodromy nilpotent cones
 

The monodromy matrices are transformed by conjugation when the base point is changed along a path. We can transform the monodromy nilpotent cones (<ref>) into H^3(X_b_o^*,ℚ) once we fix paths p_b_0'← b_o and p_b_0”← b_o. Let us denote by φ_b_o'b_o the resulting isomorphism φ_b_o'b_o H^3(X_b_o^*,ℚ)≃ H^3(X_b_o'^*,ℚ) and similarly for φ_b_o”b_o. We define the transforms of the nilpotent cones (<ref>) by these isomorphisms by

    Σ_o_2:=(φ_b_o'b_o)^-1Σ_o_2' φ_b_o'b_0 ,    Σ_o_3:=(φ_b_o”b_o)^-1Σ_o_3”φ_b_o”b_o.

Then the cones Σ_o_2 and Σ_o_3 are generated by

    N_i':=(φ_b_o'b_o)^-1Ñ_i' φ_b_o'b_0,     N_i”:=(φ_b_o”b_o)^-1Ñ_i”φ_b_o”b_0,      i=1,2,
 respectively. Note that Σ_o_1, Σ_o_2, Σ_o_3 are cones in End(H^3(X_b_o^*,ℚ)).



  §.§.§ Path p_o_2←o_1


The transform Σ_o_2 of the nilpotent cone obviously depends on the choice of the path. Looking the moduli space ℳ_X^*^ cpx closely, we find that there is a natural choice of the path by which the cone Σ_o_2 is glued with Σ_o_1 along a common face (boundary ray) of them.

The moduli space ℳ_X^*^ cpx has been studied in detail in <cit.>. Here we recall the structure of the discriminant Dis= Dis_0∪ D_x∪ D_y∪ D_z. As we schematically reproduce the results in Fig. <ref>, the irreducible component Dis_0 of the discriminant touches the divisor D_y={ y=0} at (x,y)=(1,0) with fifth-order tangency as we can see in the expression

    Dis_0={ (1-x-y)^5-5^4xy(1-x-y)^2+5^5xy(xy-x-y)=0} .

We introduce the affine chart ℂ_(1,0)^2 with the origin (1,0). After blowing-up at the origin five times, we can remove the tangential intersection of the proper transform Dis_0 of Dis_0 with the exceptional divisors (see Fig. <ref>). We denote the exceptional divisors by E_1,…,E_5.

Let q_12 be a point near the intersection E_1∩ D_y, and b_o, b_o' be points near the origins o_1 and o_2, respectively. We define a path p_b_o'← b_0 to be the composite path p_b_o'← q_12∘ p_q_12← b_o of the following straight lines:

    p_q_12← b_o={(1-t)b_o+tq_12 | 0≤ t≤1},
    
    p_b_o'← q_12={(1-t)q_12+tb_o' | 0≤ t≤1}.






  §.§.§ The isomorphisms φ_b_o'b_0, φ_b_o”b_o' and φ_b_ob_o”


We first calculate the connection matrix of the local solution Π(x,y) along the path p_b_o'← b_o.



With respect to the basis (<ref>) and (<ref>), the isomorphism φ_b_o'b_o H^3(X_b_o^*,ℚ)≃ H^3(X_b_o'^*,ℚ) along the path p_b_o'← b_o is given by

    φ_b_o'b_o=(-1     0     0     0     0     0
    
    0     1     -4     2     25     0
    
    0     0     -1     0     -2     0
    
    0     0     0     -1     -4     0
    
    0     0     0     0     1     0
    
    0     0     0     0     0     -1
    ).

This isomorphism preserves the monodromy weight filtrations and also the symplectic structures described in Section <ref>. 

 To determine the matrix form of φ_b_o'b_o, we do first the analytic continuation of the period integral Π(x,y) along
the path p_q_12← b_o by making local solutions around q_12=E_1∩ D_y in terms of the blow-up coordinates s_1=x-1, s_2=y/(1-x)^5 which represent q_12 by s_1=s_2=0. There are two local solutions which are given by regular powerseries, and others contain logarithmic singularities given by log s_1 and log s_2,…,(log s_2)^3. For a fixed value of y, |y|≪1, we analytically continue these solutions to Π(x,y) as functions of s_1=x-1. Note that, under the analytic continuation, the powers of log y are unchanged. Hence the connection matrix follows from the analytic continuation of the period integrals Π(x,0) where we set log y=0 and y=0. In our actural calculation, we set s_2=0 and log s_2=-5log(1-x) for the local solutions around (s1,s2)=(0,0), and relate these solutions numerically to Π(x,0) using powerseries expansions with sufficiently high degrees. In a similar way, we can calculate the connection matrix for the latter half p_b_o'← q_12 of the path p_b_o'← b_o. Actually, we can avoid the above numerical calculations finding an analytic formula for Π(x,0). However, since the details are technical, we will report them elsewhere. It is clear that the connection matrix φ_b_o'b_o preserves the filtrations since it is block diagonal with respect to the basis compatible with the filtrations W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ) and W_0'⊂ W_2'⊂ W_4'⊂ W_6'=H^3(X_b_o'^*,ℚ). Moreover, we can verify directly that it preserves the symplectic structure given by (<ref>).


From the forms of period integrals given in (<ref>), it is easy to deduce that we have the isomorphisms

    φ_b_o”b_o' H^3(X_b_o'^*,ℤ)≃ H^3(X_b_o”^*,ℤ)    and    φ_b_ob_o” H^3(X_b_o”^*,ℤ)≃ H^3(X_b_o^*,ℤ)

by simply exchanging the bases α_1↔α_2 and β_1↔β_2 suitably, i.e., φ_b_o”b_o'=φ_b_o'b_op_23p_45 and φ_b_ob_o”=p_23p_45φ_b_o'b_op_23p_45 with the permutation matrices p_ij for the transposition (i,j). Explicitly, they are given by

    φ_b_o”b_o'=([ -1  0  0  0  0  0;  0 -4  1 25  2  0;  0 -1  0 -2  0  0;  0  0  0 -4 -1  0;  0  0  0  1  0  0;  0  0  0  0  0 -1 ]),    φ_b_ob_o”=([  -1   0   0   0   0   0;   0  -1   0  -2   0   0;   0  -4   1 -25   2   0;   0   0   0   1   0   0;   0   0   0  -4  -1   0;   0   0   0   0   0  -1 ]).

Here we note that these isomorphisms preserve the monodromy weight filtrations and also the symplectic structures described in Sections <ref> and <ref>. Also it should be noted that we have verified Observation <ref> in Section <ref> in the present case.

Let us introduce the following notation:

    _21:=φ_b_o'b_o,    _32:=φ_b_o”b_o',    _13:=φ_b_ob_o”

and also set _ij:=_ji^-1. As this notation indicates, we expect certain correspondence of these _ij to the birational maps φ_ij X_j X_i under mirror symmetry. In order to make this more explicit, we note the groupoid structure associated to the isomorphisms _ij.
We denote by G_{1,2,3} the groupoid generated by _21, _32, _13.

Let G_ij be the subset of G_{1,2,3} consisting of elements _ii_1_i_1i_2⋯_i_kj, k≥0. It is easy to see that

    G_11={^n |  n∈ℤ} ,     G_21={_21^n |  n∈ℤ} ,     G_31={_31^n |  n∈ℤ} ,

where set :=_13_32_21.



  §.§.§ Groupoid actions on the nilpotent cones
 
We define the following conjugates of the nilpotent cones (<ref>) and (<ref>):

    Σ_o_1^(n):=(^-1)^nΣ_o_1^n,
    Σ_o_2^(n):=(^-1)^n_21^-1Σ_o_2'_21^n =(^-1)^nΣ_o_2^n,
    Σ_o_3^(n):=(^-1)^n_31^-1Σ_o_3”_31^n =(^-1)^nΣ_o_3^n.

These are cones in End(H^3(X_b_o^*,ℝ)) and generalize the nilpotent cones Σ_o_k=Σ_σ_k^(0), k=1,2,3, introduced in the beginning of this subsection. It is easy to see that these cones are generated by

    N_i(n):=(^-1)^n N_i^n,     N_i'(n):=(^-1)^n N_i' ^n,     N_i”(n):=(^-1)^n N_i”^n,

respectively, where we set N_i':=_21^-1Ñ_i'_21 and N_i”:=_31^-1Ñ_i”_31, i=1,2.



  §.§.§ Monodromy relations

To see how the (closure of the) cone Σ_o_2=Σ_o_2^(0) is connected to (that of) Σ_o_1=Σ_o_1^(0), we calculate the generators N_i' in End(H^3(X_b_o^*,ℤ)). By the definition of N_i', it suffices to calculate

    T_x':=_21^-1 T_x'^'_21,     T_y':=_21^-1 T_y'^'_21,

since we can use T_x''=T_x, T_y''=T_y for the local monodromy matrices as we remarked in (<ref>). Similarly, using the connection matrix along the path p_q_12← b_o, we can express the local monodromy around the exceptional divisor E_1 as a linear (integral and symplectic) action on H^3(X_b_o^*,ℤ) which we denote by a matrix T_E_1 using the basis ℬ_1 in (<ref>).
[`Picard–Lefschetz formula' for flopping curves] Using the basis ℬ_1 in (<ref>), we have

    T_E_1=([  1;     1       50;        1;           1;              1;                 1 ]),     i.e.,     α_1→α_1+50β_1,
    β_1→β_1,
    α_i=α_i,  β_i=β_i,   i≠1.


As sketched briefly in the proof of Proposition <ref>, we make the local solutions of the Picard–Fuchs equation around the point of the blow-up q_12=E_1∩ D_y, and calculate the local monodromy around the divisor E_1. The claimed monodromy
follows from the analytic continuation of the local solutions in the period integral Π(x,y) near the origin o_1. In our actual calculations, we only have powerseries expressions for the local solutions around q_12 and evaluate them numerically for the analytic continuation. However, as in Proposition <ref>, we can attain sufficient precision having an analytic formula for Π(x,0). 

  The `Picard–Lefschetz formula' above is written using the symplectic basis {α_i,β_j} of H^3(X_b_o^*,ℤ). When we translate this into the dual basis { A_i,B_j} of H_3(X_b_o^*,ℤ), we have

    A_1→ A_1,     B_1→ B_1-50A_1

with the rest of the basis left invariant. This should be contrasted to the genuine Picard–Lefschetz monodromy

    A_0→ A_0+B_0,     B_0→ B_0,

which we can see for the monodromy transformation around the proper transform Dis_0 of the discriminant. In the latter case, we see the topology of the cycles as A_0≈ T^3, B_0≈ S^3, where S^3 is a vanishing cycle and T^3 is its dual torus cycle. Recently, the construction of the A_k-cycles (k≠0) has been discussed in general in <cit.>. It is interesting to see how the dual B_k-cycles are constructed, and how the above `Picard–Lefschetz formula' are explained by the geometry of these cycles.


We have the following monodromy relations:

    T_x'=T_E_1^-1T_x^-1T_y^4,     T_y'=T_y.


Recall that we have the relations T_x=( ^t𝚃_x)^-1, T_y=( ^t𝚃_y)^-1 (see Remark <ref>). Then both the relations can be verified directly using the explicit forms of T_x, T_y given in Section <ref> and T_x', T_y', T_E_1 above. The second relation also follows from the fact that the divisor { y=0} ={y'=0} intersects normally with the exceptional divisor E_1 of the blowing-up.


We have arrived at (<ref>) by explicit monodromy calculations. It is natural to expect to have a conceptual derivation of (<ref>) by studying mirror symmetry of conifold transitions, but we have to this to future investigations. Instead, in the rest of this section, we will interpret the monodromy relation (<ref>).


The following properties hold:
=0pt

  (1) Generators N_i' are expressed as

    N_1'=4N_2-N_1+Δ_1,0',     N_2'=N_2,

where Δ_1,0' is a non-zero element of End(H^3(X_b_o^*,ℝ)) which annihilates the subspace W_2, i.e., Δ_1,0'|_W_2=0.


  (2) The monodromy nilpotent cones Σ_o_2=ℝ_>0N_1'+ℝ_>0N_2' and Σ_o_1 glue together along N_2'=N_2. They are not in a two dimensional plane in End(H^3(X_b_o^*,ℝ)).

 The properties in (1) are based on explicit calculations using (<ref>). The second relation N_2'=log T_y'=log T_y=N_2 is clear. For the first relation, by evaluating the matrix logarithms, we have

    Δ_1,0'=N_1'-(4N_2-N_1)=([     0     0     0     0    25 -25/3;     0     0     0     0   -50    25;     0     0     0     0     0     0;     0     0     0     0     0     0;     0     0     0     0     0     0;     0     0     0     0     0     0 ]).

From this triangular form, we see the claimed property of Δ_1,0' (see also (<ref>)). The claims in (2) are clear from (1) and also from the fact that the cone Σ_o_1 is generated by N_1=log T_x and N_2=log T_y.


(1) It should be observed that, under the identification

    L_Z_3↔ N_1',     L_Z_2↔ N_2'    and     H_1↔ N_1,     H_2↔ N_2,

Proposition <ref> above is the mirror counter part for the gluing of Kähler cones described in Lemma <ref>.

(2) If the first monodromy relation of (<ref>) were T_x'=T_x^-1T_y^4, then we would have

    N_1'=4N_2-N_1,     N_2'=N_2,

since T_x and T_y are commutative. These relations are exactly the same as those we have seen in Lemma <ref>. However the presence of T_E_1 prevents this exact correspondence. We will see that T_E_1 represents the first order quantum correction coming from the 50 flopping curves of the contraction X_1 Z_1. Thus the gluing relation found in Proposition <ref>(1) naturally encodes the first order quantum corrections.




  §.§.§ Gluing nilpotent cones
 Before going into general descriptions, it will be helpful to see that the cone Σ_o_1 is glued with Σ_o_3 along N_1 in a similar way as above. Let
us define

    T_x”:=_31^-1T_x””_31,     T_y”:=_31^-1T_y””_31,

and also T_E_1” for the monodromy matrix around the exceptional divisor E_1”. Observing the symmetry in Fig. <ref> and (<ref>), it is easy to deduce the following monodromy relations

    T_x”=T_x,     T_y”=T_E_1”^-1T_y^-1T_x^4

with T_E_1”=p_23p_45T_E_1p_23p_45 in End(H^3(X_b_o^*,ℤ)), where p_ij are the permutation matrices. Since the generators of the cone Σ_o_3 are given by N_1”=log T_y” and N_2”=log T_x”, we can evaluate these as

    N_1”=N_1,     N_2”=4N_1-N_2+Δ_2,0”,

where Δ_2,0” is given by Δ_2,0”=p_23p_45Δ_1,0'p_23p_45 with the vanishing property Δ_2,0”|_W_2=0. As before, Δ_2,0” is a non-vanishing element. Hence, the nilpotent cones Σ_o_1 and Σ_o_3 glue together along the common half line ℝ_≥0N_1 but do not lie on the same plane. Now we generalize these properties in the following proposition.

  =0pt

  (1) The matrix  preserves the monodromy weight filtration

    W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ).


  (2) The (closures of the) monodromy nilpotent cones Σ_o_1^(n), Σ_o_2^(n), Σ_o_3^(n) glue sequentially as in

    …,Σ_o_2^(1), Σ_o_1^(1), Σ_o_3^(1), Σ_o_2, Σ_o_1, Σ_o_3, Σ_o_2^(-1), Σ_o_1^(-1), Σ_o_3^(-1),… .



  (3) The generators of the cones satisfy

    (i)         (N_1(n),N_2(n))=(N_1,N_2)(0     -1

1     4 )^3n+(Δ_1,n,Δ_2,n),   

    (ii)        (N_1'(n),N_2'(n))=(N_1',N_2')(0     -1

1     4 )^-3n+(Δ_1,n',Δ_2,n'),    

    (iii)           (N_1”(n),N_2”(n))=(N_1”,N_2”)(0     -1

1     4 )^-3n+(Δ_1,n”,Δ_2,n”),   

where Δ_i,n,Δ_i,n',Δ_i,n”∈End(H^3(X_b_o^*,ℚ)) and satisfy Δ_i,n|_W_2=Δ_i,n'|_W_2=Δ_i,n”|_W_2=0.


  (4) The following relations glue the nilpotent cones in (2) (see Fig. <ref>):

    N_1(n)=N_1”(n),     N_2(n)=N_2'(n),     N_2”(n)=N_1'(n-1).



 (1) Recall that  is defined by =_13_32_21. Each isomorphism _ij preserves the monodromy weight filtrations defined for each boundary point o_k (see Proposition <ref>). Hence, H^3(X_b_o,ℚ)→ H^3(X_b_o,ℚ) preserves the monodromy weight filtration as claimed.

(2) We have introduced the generators of the nilpotent cones Σ_o_k^(n) by N_i(n),N_i'(n) and N_i”(n) for k=1,2,3, respectively, in Section <ref>. Then the claim follows from the properties (3) and (4) (see also Fig. <ref>).

(3) By the definition of N_i(n), it is straightforward to calculate N_i(1) as

    N_i(1)=^-1N_i=
    -4N_1+15N_2+Δ_1,1,     i=1,
    
    -15N_1+56N_2+Δ_2,1,     i=2,

where Δ_i,1 satisfy Δ_1,1|_W_2=Δ_2,1|_W_2=0 on the subspace W_2⊂ H^3(X_b_o^*,ℚ). We note the relation ([  0 -1;  1  4 ])^3=([  -4 -15;  15  56 ]) and arrange the above relation into the claimed matrix form for n=1. Then we can obtain the claimed formula (i) for general n (in the first line) by evaluating (^-nN_1^n,^-nN_2^n) inductively. In the evaluation, we should note that ^-1Δ_i,n-1|_W_2=0 if Δ_i,n-1|_W_2=0 since  preserves the monodromy weight filtration. For the second formula (ii), we note the relation

    (N_1',N_2')=(N_1,N_2)(-1     0
    
    4     1
    )+(Δ_1,0',0)

obtained in Proposition <ref>(1). Taking the conjugations ^-n(-)^n on the both sides of this relation, and using the first formula (i) for ^-n(N_1,N_2)^n, we have the claimed formula. In the derivation, we use the relation

    (0     -1
    
    1     4
    )^3n(-1     0
    
    4     1
    )=(-1     0
    
    4     1
    )(0     -1
    
    1     4
    )^-3n

and also the property ^-1Δ'_i,n-1|_W_2=0 if Δ'_i,n-1|_W_2=0. For the third relation (iii),
calculations are similar but we need to use the relation ([  0 -1;  1  4 ])^3n([  1  4;  0 -1 ])=([  1  4;  0 -1 ])([  0 -1;  1  4 ])^-3n.

(4) Since N_i(n), N_i'(n), N_i”(n) are defined by the conjugation of N_i(n-1), N_i'(n-1) and N_i”(n-1) by , it is sufficient to show the equalities

    N_1=N_1”,     N_2=N_2',     N_2”(1)=N_1'.

The first two relations are verified already in Proposition <ref> and (<ref>). For the last relation, we evaluate N_2”(1)=^-1N_2” directly verifying its equality to N_1'.




 Consider the left ideal ℐ_2:={ X∈End(H^3(X_b_0^*,ℝ)) |  X|_W_2=0}
of End(H^3(X_b_0^*,ℝ)), and πEnd(H^3(X_b_0^*,ℝ))→End(H^3(X_b_0^*,ℝ))/ℐ_2
be the natural projection as a vector space. Then, taking the closure in End(H^3(X_b_0^*,ℝ))/ℐ_2, we have

    ⋃_nπ(^-n(Σ_o_2∪Σ_o_1∪Σ_o_3)^n)=ℝ_>0c̅_1+ℝ_>0c̅_2,

where c̅_1=-N̅_1+(2+√(3))N̅_2 and c̅_2=N̅_1-(2-√(3))N̅_2 with N̅_i=π(N_i), i=1,2. 

From Proposition <ref>, we have

    π(Σ_o_1∪Σ_o_2∪Σ_o_3) =ℝ_≥0π(N_1')+ℝ_≥0π(N_2”)
     =ℝ_≥0π(4N_2-N_1)+ℝ_≥0π(4N_1-N_2).

Evaluating the matrix power ([  0 -1;  1  4 ])^3n, it is easy to see that

    lim_n→∞ℝ_≥0π(N_1(n))=lim_n→∞ℝ_≥0π(N_2(n))=ℝ_≥0c̅_1

and

    lim_n→-∞ℝ_≥0π(N_1(n))=lim_n→-∞ℝ_≥0π(N_2(n))=ℝ_≥0c̅_2.

Then the claim follows from the gluing property (1) of Proposition <ref>.




 §.§ Flopping curves and T_E_1
 

The matrix T_E_1 arises from the tangential intersection of the relevant components of the discriminant Dis in the moduli space ℳ_X^*^ cpx. As noted in the remark above, T_E_1 may be identified with the first order correction from the quantum cohomology of X_1. To see this, let us introduce

    N_1^𝚏:=log(T_x^-1T_y^4)=4N_2-N_1

and N_2^𝚏=N_2'=N_2. Here, we should note the difference in N_1^𝚏 from the definition N_1'=log(T_E_1^-1T_x^-1T_y^4).
Define C_ijk' and C_ijk^𝚏 by N_i'N_j'N_k'=C_ijk'𝙽_0 and N_i^𝚏N_j^𝚏N_k^𝚏=C_ijk^𝚏𝙽_0 with 𝙽_0 as given in Proposition <ref>. Non-vanishing (totally symmetric) C_ijk' and C_ijk^𝚏
are given by

    (C_111',C_112',C_122',C_222')=(5,10,10,5),
    
    (C_111^𝚏,C_112^𝚏,C_122^𝚏,C_222^𝚏)=(-45,10,10,5).


We derive these numbers by direct calculations of matrix products.


The nilpotent matrices N_1', N_2' follow from the B-structure at o_2, which has been identified with the A-structure of X_2. Hence the first equality in (<ref>) is a consequence from mirror symmetry. To see more details of the equality, let us recall the so-called mirror map which are defined by

    t_i'=∫_A_i'Ω_x'/∫_A_0'Ω_x',     t_i=∫_A_iΩ_x/∫_A_0Ω_x

for each boundary point o_2 and o_1, respectively. If we relate these local definitions by the isomorphism ( ^tφ_b_o'b_o)^-1 H_3(X_b_o^*,ℤ)→ H_3(X_b_o'^*,ℤ) along the path p_b_o'← b_o (cf. Proposition <ref>), we have

    t_1'=-t_1,     t_2'=4t_1+t_2.


Let C_ijk be as defined in Proposition <ref>. Also set q_1':=e^t_1' and q_1=e^t_1. Then we have the following relations

    C_ijk^𝚏=∑_l,m,nC_lmndt_l/dt_i'dt_m/dt_j'dt_n/dt_k'

and

    C_111'+50q_1'/1-q_1'=C_111^𝚏+50q_1/1-q_1(dt_1/dt_1')^3.


It is easy to verify these. For the second relation, we note that 50q_1/1-q_1(dt_1/dt_1')^3
=50+50q_1'/1-q_1' for q_1=1/q_1'.


The equality (<ref>) is a consequence of the flop invariance of the quantum cohomology (see, e.g., <cit.>). As mentioned in Remark <ref>, the number 50 represents the flopping curves. Comparing this with Proposition <ref>, we see that the monodromy T_E_1 encodes the data of the flopping curves which is in the first order of the quantum cohomology of X_1.



 §.§ Prepotentials


The flop invariance expressed in (<ref>) is known more precisely as the invariance of quantum cohomology under analytic continuations, where all higher order quantum corrections are taken into account. Here we rephrase this property as a property of the so-called prepotentials.

For the B-structure at each boundary point, we can define the prepotential. For example for the B-structure at o_1 and o_2, respectively,
they are given by

    ℱ=1/2∑_i=0^3∫_A_iΩ_x∫_B_iΩ_x,    ℱ'=1/2∑_i=0^3∫_A_i'Ω_x'∫_B_i'Ω_x'

with the symplectic integral bases for period integrals in Π(x,y) and Π'(x',y').
By the isomorphism ( ^tφ_b_o'b_o)^-1 H_3(X_b_o^*,ℤ)→ H_3(X_b_o'^*,ℤ) along the path p_b_o'← b_o chosen as in Proposition <ref>, ℱ and ℱ' are related by

    ℱ'=ℱ+1/2∑_i,j=1^2Q_ij∫_A_iΩ_x∫_A_jΩ_x,

where (Q_ij)=([ -25  -2;   2   0 ]).
Using the basis { A_i,B_j}, { A_i',B_j'}, the connection matrix has the form

    (^tφ_b_o'b_0)^-1=([  -1   0   0   0   0   0;   0   1   0   0   0   0;   0  -4  -1   0   0   0;   0   2   0  -1   0   0;   0 -25  -2  -4   1   0;   0   0   0   0   0  -1 ]),

which gives the analytic continuation by Π'(x',y')=( ^tφ_b_o'b_0)^-1Π(x,y).
From this, we read A_0'=-A_0, B_0'=-B_0 and

    ([ A_1'; A_2' ])=R([ A_1; A_2 ]),    ([ B_1'; B_2' ])=( ^tR)^-1([ B_1; B_2 ])+([ -25  -2;   2   0 ])([ A_1; A_2 ]),

where we set R=([  1  0; -4 -1 ]). The claimed formula is immediate from these.


As we can deduce in the above proof, the prepotentials are invariant only up to quadratic terms of the A_i-periods under the analytic continuations even if they are symplectic and also preserve the monodromy weight filtrations. However the so-called Yukawa couplings are invariant since they are given by the third derivatives of the prepotentials with respect to the coordinates t_i (see (<ref>)).



§ GLUING MONODROMY NILPOTENT CONES II


We will study the following Calabi–Yau threefold of complete intersections:

    X=([ ℙ^3| 2 1 1; ℙ^3| 2 1 1 ])^2,66.

We assume the defining equations of X are chosen general unless otherwise mentioned. For such X, there is no other birational model than X. However X has an interesting birational automorphism of infinite order <cit.>, and also has a non-trivial movable cone similar to the one in the preceding section.



 §.§ Birational automorphisms of infinite order


Let π_i X→ℙ^3 be the projections to the first and second factor of ℙ^3×ℙ^3 for i=1 and 2, respectively. It is easy to see that the projection π_i is surjective and generically 2:1. We consider the Stein factorization X→ W_i→ℙ^3 of the morphism π_i X→ℙ^3 and denote the morphism by ϕ_i X→ W_i for i=1,2.
 For i=1,2, the morphism W_i→ℙ^3 is a double cover of ℙ^3 branched along an octic, and W_i is a (smooth) Calabi–Yau threefold. 
 We omit proofs since they are standard (see, e.g., <cit.>).


Let τ̃_i W_i≃ W_i^+ be the deck transformation of the covering W_i→ℙ^3. Then we have the map τ_i which covers τ̃_i as in the following diagram:

    (0,0)*++X="cX",
    (-15,0)*++X="lX",
    ( 15,0)*++X="rX",
    (-10,-20)*++ℙ^3="lP",
    ( 10,-20)*++ℙ^3="rP",
    (-10,-10)*+W_2="lW",
    (-26,-10)*+W_2^+="lWp",
    ( 10,-10)*+W_1="rW",
    ( 26,-10)*+W_1^+.="rWp",
    ( 17.5,-8)*++ ^τ̃_1,
    (-17.5,-8)*++ ^τ̃_2,
    ( 17.5,-10)*++≃,
    (-17.5,-10)*++≃,
    ( 7,1)*++∼,
    ( 7,3)*++ _τ_1,
    (-7,1)*++∼,
    (-7,3)*++ _τ_2,
    ( 7,-4)*++ _ϕ_1,
    (-7,-4)*++ _ϕ_2,
    @–>_ "cX";"lX"
    @–>^ "cX";"rX"
    ^π_2 "cX";"lP"
    _π_1 "cX";"rP"
     "cX";"lW"
    _ "lW";"lP" 
     "lX";"lWp"
    _2:1 "lWp";"lP"
     "cX";"rW"
     "rW";"rP"
    ^2:1 "rWp";"rP" 
     "rX";"rWp"


The following hold:
=0pt

  (i) The map τ_i X X is birational but not bi-holomorphic.

  (ii)The morphism ϕ_i X→ W_i contracts 80 lines and 4 conics to points, and the birational map τ_i is an Atyah's flop of these curves.


(i), (ii) See the reference <cit.>. 

 (1) Bir(X)=Aut(X)·⟨τ_1,τ_2⟩. (2) τ_i^2= id for i=1,2. Also τ_1τ_2 has infinite order. 

See <cit.>.




 §.§ Mirror family of X


We can describe the mirror family 𝔛^*→ℳ_X^*^ cpx of X by writing X in terms of a Gorenstein cone following Batyrev–Borisov. The parameter space of the defining equations up to isomorphisms naturally gives the moduli space ℳ_X^*^ cpx, which turns out to be compactified to ℙ^2 as before. Here we will not go into the details of the mirror family, but we only write the form of the Picard–Fuchs differential operator in the affine coordinate [1,x,y]∈ℳ_X^*^ cpx=ℙ^2.
 Picard–Fuchs equations of the family on the affine coordinate [1,x,y] are given by 𝒟_1w(x,y)=𝒟_2w(x,y)=0 with

    𝒟_1= (3θ_x^2-4θ_xθ_y+3θ_y^2)-(θ_x+θ_y)(2θ_x+2θ_y-1)(10x+6y)
     +4θ_x(2θ_x+2θ_y-1)(x-y),
    𝒟_2= (θ_x^3-θ_x^2θ_y+θ_xθ_y^2-θ_y^3)-2(θ_x+θ_y)^2(2θ_x+2θ_y-1)(x-y),

where θ_x=x∂ /∂ x,θ_y=y∂ /∂ y. The discriminant locus of this system is given by Dis=D_1∪ D_2∪ D_3∪ Dis_0 with

    Dis_0={ (1-4x-4y)^4-128xy(17+56(x+y)+16(x^2+y^2))=0} ,

and the coordinate lines D_i of ℙ^2. 
The differential operators 𝒟_1 and 𝒟_2 arise from the Gel'fand–Kapranov–Zelevinski system after finding suitable factorizations of differential operators. See <cit.> for more details. Once 𝒟_1 and 𝒟_2 are determined,
it is straightforward to determine the discriminant locus (singular locus) of the system from the equations of the characteristic variety.


From the forms of 𝒟_1and 𝒟_2, the origin x=y=0 is expected to be a LCSL. In fact, we can verify all the properties for the LCSL in Definition <ref>. We also verify that there is no other LCSL point in ℳ_X^*^ cpx=ℙ^2. In Fig. <ref>, we schematically describe the structure of the moduli space ℳ_X^*^ cpx. There, as in the preceding example, we see that the component Dis_0 intersects tangentially with the divisors D_1={ x=0} and D_2={ y=0}. This time, we blow-up at these two intersection points successively four times to make the intersections normal crossing (see Fig. <ref> right).
 As in the previous example, we should be able to arrive at the mirror family 𝔛^*→ℳ_X^*^ cpx starting with a special family { X_ sp} _a,b. But we leave this task for other occasions, since we have the mirror family in any case as above.






 §.§ B-structure of X at the origin o
 

As in Section <ref>, the canonical integral and symplectic structure can be introduced from the power series solution

    w_0(x,y)=∑_n,mΓ(1+2n+2m)Γ(1+n+m)^2/Γ(1+n)^4Γ(1+m)^4x^ny^m

around the origin o:=[1,0,0]. Simply replacing the necessary parameters in the general formula <cit.>, and fixing the so-called quadratic ambiguities there by C_kl=0, we obtain the canonical integral and symplectic structure in the form of period integrals Π(x,y) with the corresponding symplectic basis

    { A_0,A_1,A_2,B_2,B_1,B_0}⊂ H_3(X_b_o^*,ℤ)    with     A_i∩ B_j=δ_ij,A_i∩ A_j=B_i∩ B_j=0,

where a base point b_o is taken near the origin. We denote by 𝚃_x the matrix of the monodromy transformation of Π(x,y) for a small loop around the divisor D_1= { x=0 }, and similarly denote by 𝚃_y for a small loop around D_2= { y=0 }. Writing the local solutions explicitly, it is straightforward to have

    𝚃_x=([  1  0  0  0  0  0;  1  1  0  0  0  0;  0  0  1  0  0  0;  3  6  6  1  0  0;  1  2  6  0  1  0; -4 -1 -3  0 -1  1 ]),    𝚃_y=([  1  0  0  0  0  0;  0  1  0  0  0  0;  1  0  1  0  0  0;  1  6  2  1  0  0;  3  6  6  0  1  0; -4 -3 -1 -1  0  1 ]).

We introduce the dual basis ℬ= {α_i,β_i} of H^3(X_b_o^*,ℤ) and consider the dual actions T_x:=( ^t𝚃_x)^-1 and T_y:=( ^t𝚃_y)^-1 on H^3(X_b_o^*,ℤ) which are clearly unipotent.

We define the monodromy nilpotent cone at o by

    Σ_o={∑λ_iN_i | λ_i≥0}⊂End(H^3(X_b_o^*,ℤ))

with N_1=- ^t(log𝚃_x) and N_2=- ^t(log𝚃_y).


Using the explicit forms of these matrices, we verify the following properties:
  =0pt 
  (1) The nilpotent element N_λ:=∑_iλ_iN_i, λ_i>0, defines the weight monodromy filtration W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_0^*,ℤ) with the same form W_2i as given in (<ref>).


  (2) We have

    N_iN_jN_k=C_ijk𝙽_0

with totally symmetric C_ijk defined by C_111=C_222=2, C_122=C_112=6, C_ijk=0 otherwise, and 𝙽_0=([   0   1; O_5   0 ]) where O_5 is the zero matrix of size 5×5.



From the above proposition, we see that the origin satisfies the conditions for LCSL. Also, looking other boundary points in the moduli space
ℳ_X^*^ cpx, we see that no other LCSL exists in ℳ_X^*^ cpx.



 §.§ Gluing the monodromy nilpotent cone Σ_o


Although there is only one LCSL in the mirror family 𝔛^*→ℳ_X^*^ cpx, we can find the monodromy transformations which correspond to the birational automorphisms τ_1 and τ_2 of X. We observe that the monodromy nilpotent cone Σ_o extends to a larger cone (or cone structure) using these monodromy transformations, and we will identify the resulting cone structure with the movable cone Mov(X) of X.



  §.§.§ Path p_o← E_i← o, i=1,2


As shown in Fig. <ref>, the discriminant locus Dis has non-normal crossing intersection at three points. To make the intersections normal, we blow-up successively four times at the two points near the origin o. We denote by E_1 and E_2, respectively, the exceptional divisors introduced by the blow-ups (see Fig. <ref> right). As we see in the form of the discriminant Dis, the family over ℳ_X^*^ cpx is symmetric under x↔ y. Because of this symmetry reason, it suffices to describe the divisor D_2= { y=0} which intersects with E_1 at [1,x,y]=[1,1/4,0]∈ℙ^2. Explicitly, we introduce the blow-up coordinate at the origin q_12:=E_1∩ D_y by

    s_1=4x-1,     s_2=1/2^6y/(1-4x)^4.


 Let R_12={1/4+s_1/4e^iθ | 0≤θ≤2π} be a small loop around E_1 on D_2. We denote by p_q_12← b_o= { (1-t)b_o+tq_12 | 0≤ t≤1-ε} the straight line connecting the base point b_o near o and a point q_12 on the small loop R_12. Then we define

    p_b_o← E_1← b_0:=(p_q_12← b_o)^-1∘ R_12∘ p_q_12← b_o

to be the composite path which encircles the divisor E_1 from the base point b_o. In a similar way, we define a closed path p_b_0← E_2← b_o which encircle the divisor E_2 from b_0 (see Fig. <ref>).



  §.§.§ Monodromy around E_i
 

Let (x',y')=(1/x,y/x) be the affine coordinate with the origin [0,1,0]∈ℙ^2 and b_o' be a base point near the origin. We denote by 𝚃_x'' and 𝚃_y'' the local monodromy around x'=0 and y'=0, respectively. Conjugating 𝚃_x'', 𝚃_y'' by the connection matrix for the path p_b_o'← b_0=p_b_0'← q_12∘ p_q_12← b_o, we define the corresponding monodromy matrices 𝚃_x' and 𝚃_y' for loops with the base point b_o. We define T_x':=(^t𝚃_x')^-1 and T_y':=( ^t𝚃_y')^-1 to be the linear actions on the dual space H^3(X_b_o^*,ℤ). 
We have

    T_x'=([  -1  -1   3   6 -10   2;   0   1  -6 -12   8   2;   0   0  -1   0  12  -6;   0   0   0  -1  -6   3;   0   0   0   0   1  -1;   0   0   0   0   0  -1 ]),     T_y'=([  1  0 -1  1  3  4;  0  1  0 -6 -6 -3;  0  0  1 -2 -6 -1;  0  0  0  1  0  1;  0  0  0  0  1  0;  0  0  0  0  0  1 ]).

In particular we have T_y'=T_y. 

 These are based on explicit calculations. Here we only sketch the calculations. We first make local solutions using the coordinate (s_1,s_2) centered at q_12. Then their domain of convergence have overlap both with the local solutions around (x,y)=(0,0) and (x',y')=(0,0). Then it is straightforward to obtain the connection matrices. The local monodromy matrices 𝚃_x'' and 𝚃_y'' are easily read off from the local solutions. Then by conjugating these local monodromy matrices by the connection matrix, we have the expressions for 𝚃_x', 𝚃_y' as the linear actions on H_3(X_b_0^*,ℤ). Translating these to H^3(X_b_o^*,ℤ), we obtain T_x' and T_y'.


Similarly we define the monodromy matrix 𝚃_E_1 along the loop p_b_o← E_1← b_o and set T_E_1:=( ^t𝚃_E_1)^-1. Corresponding to Proposition <ref> we have
[`Picard–Lefschetz formula' for the flopping curves] 
  =0pt

  (1) The monodromy matrix is given by

    T_E_1=([ -1  0  0  0  0  0;  0  1 -6  0 48  0;  0  0 -1  0  0  0;  0  0  0 -1 -6  0;  0  0  0  0  1  0;  0  0  0  0  0 -1 ]).

In particular, this is quasi-unipotent.


  (2) For T_E_1^2 we have

    T_E_1^2=([  1;     1       96;        1;           1;              1;                 1 ]),    i.e.,    α_1→α_1+96β_1,
    β_1→β_1,
    α_i=α_i,   β_i=β_i,    i≠1.



  (3) By symmetry, we have similar formula for T_E_2 and T_E_2^2. In particular, T_E_2^2 is given by
α_2→α_2+96β_2, β_2→β_2, with α_i=α_i, β_i=β_i for i≠2.


These results follow from making local solutions and the analytic continuations of them. Again, calculations are straightforward since local solutions around (x,y)=(0,0) and (s_1,s_2)=(0,0) have overlap in their domains of convergence. 

 (1) As before, the monodromy action (2) in the above proposition is expressed in terms of the symplectic basis { A_i,B_j} of H_3(X_b_o^*,ℤ) as

    A_1→ A_1,     B_1→ B_1-96 A_1.


(2) We have seen in Proposition <ref> that each τ_i X X is an Atiyah's flop with respect to 80 lines and also 4 conics. We observe that 96=80+4×2^2 holds for the number in T_E_i^2. We can verify the corresponding relations also for other examples. Based on these, we conjecture the following general form:

    A_1→ A_1,     B_1→ B_1-(n_0(1)+n_0(2)×2^2) A_1

for the Atiyah's flops of n_0(1) lines and n_0(2) conics associated to the contractions to the double cover of ℙ^3.




  §.§.§ Monodromy relations

Take affine coordinates (x,y), (x',y') and (x”,y”) of ℙ^2 as shown in Fig. <ref>. Let T_x', T_y' be as defined in Proposition <ref>.
 The following monodromy relations holds

    T_x'=T_E_1^-1T_x^-1T_y^3,     T_y'=T_y,     T_E_1T_y=T_yT_E_1.


 We have the second and the third relations since all the divisors are normal crossing after the blow-ups. We can verify the first relation directly by using T_x=( ^t𝚃_x)^-1, T_y=( ^t𝚃_y)^-1 given in Section <ref> and T_x', T_E_1 in Section <ref>.



Define the following conjugations of T_x,T_y by T_E_1:

    T̃_x:=T_E_1^-1T_xT_E_1,    T̃_y:=T_E_1^-1T_yT_E_1.

Using these, we define the monodromy nilpotent cone by

    Σ̃_o:={∑λ_iÑ_i | λ_i>0}⊂End(H^3(X_b_o^*,ℝ)),

where Ñ_1:=logT̃_x and Ñ_2:=logT̃_y.


The (closures of the) monodromy nilpotent cones Σ_o and Σ̃_o glue along the ray ℝ_≥0N_2, but they are not on the same two dimensional plane. 
Using the monodromy relations in Proposition <ref>, we have T̃_y'=T_y. Hence the claim is immediate since we have Ñ_2=N_2 by definition. To see the second claim, we use again the monodromy relations to have

    T̃_x=T_E_1^-1T_xT_E_1=T_E_1^-1T_y^3T_x'^-1=T_E_1^-1T_x'^-1T_y^3,

which is reminiscent of the relation (<ref>). In fact, after some matrix calculations, we obtain

    Ñ_1=6N_2-N_1+Δ_1,    Δ_1=([     0     0     0     0    48 -44/3;     0     0     0     0  -112    48;     0     0     0     0     0     0;     0     0     0     0     0     0;     0     0     0     0     0     0;     0     0     0     0     0     0 ]),

where Δ_1 satisfies Δ_1|_W_2=0. Since the nilpotent cone Σ_o lies on the plane spanned by N_1 and N_2, and aN_1+bN_2|_W_2≠0 holds for any a, b, the basis element Ñ_1 does not lie on the same plane as Σ_o.




  §.§.§ Gluing nilpotent cones
 As the example in the previous section, the structure of the moduli space ℳ_X^*^ cpx is
symmetric under the exchange of x and y. Hence, corresponding to (<ref>), we have

    T_x”=T_x,     T_y”=T_E_2^-1T_y^-1T_x^3,     T_E_2T_x=T_xT_E_2.

When we define T̃_x':=T_E_2^-1T_xT_E_2, T̃_y':=T_E_2^-1T_yT_E_2,
we have the following relations

    Ñ_1'=N_1,    Ñ_2'=6N_1-N_2+Δ_1'

for Ñ_1':=logT̃_x, Ñ_2':=logT̃_y with Δ'_1|_W_2=0. This entails the corresponding
gluing property described in Proposition <ref>. We summarize these two actions into the following general form.

We denote by τ_E_i the conjugations by T_E_i on End(H^3(X_b_o^*,ℚ)),
which act on the nilpotent matrices N in general as

    τ_E_1(N)=T_E_1^-1NT_E_1,    τ_E_2(N)=T_E_2^-1NT_E_2.

We set G:=⟨τ_E_1,τ_E_2⟩, i.e., the group generated by τ_E_1 and τ_E_2.


  =0pt

  (1) The actions of τ_E_i^n∈ G on N_1=log T_x, N_2=log T_y are summarized as

    (τ_E_1^n(N_1),τ_E_1^n(N_2))=(N_1,N_2)(-1     0
    
    6     1
    )^n+(Δ_n,0),
    (τ_E_2^n(N_1),τ_E_2^n(N_2))=(N_1,N_2)(1     6
    
    0     -1 )^n+(0,Δ_n'),

where Δ_n, Δ_n' are elements in End(H^3(X_b_o^*,ℚ)) satisfying Δ_n|_W_2=Δ_n'|_W_2=0. In particular, we have

    τ_E_1^n(N_2)=N_2,    τ_E_2^n(N_1)=N_1.



  (2) The action of σ∈ G on Δ_n,Δ_n' preserves the vanishing properties of Δ_n, Δ_n' on
W_2, i.e., σ(Δ_n)|_W_2=σ(Δ_n')|_W_2=0.


  (3) Δ_n, Δ_n' have the following forms:

    Δ_2m=(
    O_24   [ -96m    0;    0 -96m ]
    
    O_44    O_42),    Δ_2m-1=(
    O_24   [ 96(m-1/2)     -44/3;      -122 96(m-1/2) ]
    
    O_44    O_42)

and Δ_n'=p_23p_45Δ_np_23p_45, where O_ab is the a× b zero matrix and p_ij represents the permutation matrix for the transposition (i,j).


 These properties are verified by explicit calculations using the matrix representa­tions T_x, T_y and T_E_i given previous sections. The vanishing properties follow inductively from Δ_1|_W_2=Δ_1'|_W_2=0 and the fact that both T_E_1 and T_E_2 preserve the monodromy weight filtration W_0⊂ W_2⊂ W_4⊂ W_6=H^3(X_b_o^*,ℚ).


As before, let ℐ_2:={ X∈End(H^3(X_b_0^*,ℝ)) |  X|_W_2=0} be an left ideal of End(H^3(X_b_0^*,ℝ)), and πEnd(H^3(X_b_0^*,ℝ))→End(H^3(X_b_0^*,ℝ))/ℐ_2 be the natural projection. Since T_E_i preserve the monodromy weight filtration, and by the definition of τ_E_i, it is easy to see that σ(ℐ_2)⊂ℐ_2 for all σ∈ G. Hence we have the naturally induced G action on the quotient End(H^3(X_b_0^*,ℝ))/ℐ_2 by σ(X+ℐ_2):=σ(X)+ℐ_2. Note that, if we denote by σ̅ the action of σ∈ G on the quotient space, we have στ=τ̅σ̅ (i.e., anti-homomorphism by our convention for the adjoint action) for all σ,τ∈ G.

Denote by N̅_i:=π(N_i) the basis of the cone π(Σ_o) in the quotient space. Then the following hold:
=0pt

  (1) Define τ_12:=τ_1τ_2 with τ_i=τ_E_i. We have

    (τ̅_12^n(N̅_1),τ̅_12^n(N̅_2))=(N̅_1,N̅_2)(35     6
    
    -6     -1
    )^n.



  (2) {σ∈ G | σ̅(N̅_i)=N̅_i,   i=1,2 } =⟨τ_1^2,τ_2^2⟩.


  (3) Taking the closure in End(H^3(X_b_0^*,ℝ))/ℐ_2, we have

    ⋃_σ∈ Gπ(σ(Σ_o))=ℝ_>0(-N̅_1+(3+2√(2))N̅_2)+ℝ_>0(N̅_1+(3-2√(2))N̅_̅2̅).


 The equality (1) follow from Proposition <ref>(1) and τ̅_12=τ̅_2τ̅_1. By definition, G is generated by τ_1, τ_2. Then the claim (2) follows from Proposition <ref>(1) and the above equality (1). To show the claim (3), we write by (N_1,N_2)_>0 the cone generated by N_1 and N_2. Then we first show that
the following cones successively glue together to a large cone:

    (τ_12^n(N_1),τ_12^n(N_2))_>0,    (τ_12^n(τ_1N_1),τ_12^n(τ_1N_2))_>0,      n∈ℤ.

Using the property τ_1(N_2)=N_2, τ_2(N_1)=N_1, we have

    τ_12^n(τ_1N_1)=τ_12^n+1(N_1),    τ_12^n(τ_1N_2)=τ_12^n(N_2),

by which we can arrange a sequence of cones schematically as follows:

    [                        (τ_12^n+1(τ_1N_1), τ_12^n+1(τ_1N_2))_>0     (τ_12^n(τ_1N_1),   τ_12^n(τ_1N_2))_>0                    ⋯;                                                                                                                              ;                                         ⋯      (τ_12^n+1(N_2),    τ_12^n+1(N_1))_>0        (τ_12^n(N_2),      τ_12^n(N_1))_>0 ]

Let us note that τ_2τ_1=τ_2^2τ_12^-1τ_1^2 and τ_2=τ_2^2τ_12^-1τ_1 hold. Then, using these relations, we can deduce the decomposition

    G=⟨τ_12,τ_1^2,τ_2^2⟩∪⟨τ_12,τ_1^2,τ_2^2⟩τ_1.

Since τ_1^2, τ_2^2 have trivial actions on N̅_i, i=1,2, the above sequence of the cones explain the union ⋃_σ∈ Gπ(σ(Σ_o)). After some linear algebra of the matrix power ([ 35  6; -6 -1 ])^n, we can determine the infinite union in the claimed form.






 §.§ Flopping curves and T_E_1


The monodromy T_E_1 has appeared in the moduli space from the tangential intersection of the discriminants. This is quite parallel to Section <ref>. However, T_E_1 is not unipotent but only quasi-unipotent in the present case. This prevents a parallel definition to the second equation in (<ref>), but this time we set

    N_1^𝚏:=6N_2-N_1

with Ñ_1=N_1^𝚏+Δ_1 (see (<ref>)) and also Ñ_2=N_2^𝚏=N_2. Then we have

Let Ñ_iÑ_jÑ_k=C̃_ijk𝙽_0 and N_i^𝚏N_j^𝚏N_k^𝚏=C_ijk^𝚏𝙽_0 with 𝙽_0 as given in Proposition <ref>. Non-vanishing (totally symmetric) C̃_ijk and C_ijk^𝚏 are given by

    (C̃_111,C̃_112,C̃_122,C̃_222)=(2,6,6,2), 
    (C_111^𝚏,C_112^𝚏,C_122^𝚏,C_222^𝚏)=(-110,6,6,2).



As before, the first equality of (<ref>) is explained by mirror symmetry, i.e., the isomorphism of the B-structure at o with the A-structure of X. To see this isomorphism more explicitly, we recall the mirror map

    t_i=∫_A_iΩ_x/∫_A_0Ω_x

defined by the B-structure at the LCSL o. The monodromy matrix 𝚃_E_1=( ^tT_E_1)^-1 represents the isomorphism H_3(X_b_o^*,ℤ)→ H_3(X_b_o'^*,ℤ) which follows from the analytic continuation of the period integral Π(x,y) along the path p_b_0← E_1← b_o. After the continuation, the coordinate (t_1,t_2) transformed to (t_1',t_2') with

    t_1'=-t_1,     t_2'=6t_1+t_2.

 Corresponding to Proposition <ref>, we now have
Let C_ijk be as defined in Proposition <ref>. Also set q_1':=e^t_1' and q_1=e^t_1. Then we have the following relations

    C_ijk^𝚏=∑_l,m,nC_lmndt_l/dt_i'dt_m/dt_j'dt_n/dt_k'

and

    C̃_111+80q_1'/1-q_1'+42^3q_1'^2/1-q_1'^2=C_111^𝚏+(80q_1/1-q_1+42^3q_1^2/1-q_1^2)(dt_1/dt_1')^3.



In the above equality, we see the invariance of the quantum cohomology of X under birational transformations. We note that the equality (<ref>) has a slightly more general form than the familiar form (<ref>) due to the existence of 4 conics in the flopping curves.



§ SUMMARY AND DISCUSSIONS


We have studied gluings of monodromy nilpotent cones through monodromy relations coming from boundary divisors. Under the mirror symmetry, we have identified them with the corresponding gluings along codimension-one walls of the Kähler cones in birational geometry. In this paper, we confined ourself to two specific examples by doing explicit monodromy calculations. However, it is naturally expected that the observed gluings of monodromy nilpotent cones and their interpretation in mirror symmetry hold in general.

We present below some discussions and related subjects in order. In particular, we briefly report the gluing in the case of K3 surfaces whose moduli spaces have parallel structures to the Calabi–Yau threefolds X and X^* studied in Sections <ref> and <ref>.

6.1. The gluing of monodromy nilpotent cones has been done naturally through the monodromy relations (<ref>), (<ref>) and also (<ref>), (<ref>). These relations came from boundary divisors which have tangential intersections with some component of discriminant and the blowing-ups at the intersection points. As remarked in Remarks <ref>, <ref>, these tangential singularities are related to the contractions in the birational geometry of the mirror Calabi–Yau manifolds. We expect some generality in the degenerations of the mirror families 𝔛^* when we approach to the exceptional divisors E_i of the blow-ups. We have to leave this for future investigations although we note that a categorical study of the mirror symmetry for conifold transitions has been put forward in a recent work <cit.>.

6.2. In the homological mirror symmetry due to Kontsevich <cit.>, monodromy transformations in B-structures are interpreted as the corresponding transformations in the derived category of coherent sheaves D^b(X). From this viewpoint, the gluing of nilpotent cones in End(H^3(X^*,)) suggests the corresponding gluing of Kähler cones in End(K(X)) as a homological extension of the movable cones. The resulting wall structures of the gluing in End(K(X)) should be regarded as the wall structures in the stability space <cit.> of the objects in D^b(X).

6.3. As addressed in Remark <ref>, one can expect non-trivial birational geometry also for other examples of complete intersections described by Gorenstein cones <cit.>. Among such examples, there are complete intersections whose projective geometry fits well to the so-called linear duality (see Appendix <ref>). We have for example the following complete intersection:

    X=(^4| 2 1 1 1
    ^3| 1 1 1 1
    )^2,56,

which shares many properties with (<ref>) in Section <ref>, e.g., three birational models come together when we construct the complete intersection of the form X. Although we do not have birational
automorphisms of infinite order in this example, these three birational models are explained nicely by “double linear duality”, a certain composite of two different linear dualities. We will report this elsewhere.

6.4 (Cayley model of Reye congruences). Historically the Calabi–Yau complete intersection studied in Section <ref> is a generalization of the following K3 surface:

    X=(^3 |  1 1 1 1
    ^3 |  1 1 1 1
    ),

which is called a Cayley model of Reye congruences. When we take the defining equations general, X is a smooth K3 surface of the Picard lattice isomorphic to M:=(^2,([ 4 6; 6 4 ])). This K3 surface has been studied in <cit.> as an example which has an automorphism ρ of infinite order and also positive entropy. Actually, we have the same diagram as (<ref>) with the parallel definitions of X_i (X_1:=X) and Z_i as well as ρ in Proposition <ref>. The difference is in that all X_i and Z_i are smooth K3 surfaces and hence
isomorphic to each other under the morphisms, e.g., π_ij and φ_ij. For K3 surfaces, we have the so-called counting formula <cit.> for the number of Fourier–Mukai partners. Based on it, it is easy to see that the set FM(X) of Fourier–Mukai partners consists of only X itself.

The construction of the mirror family of X is similar to Section <ref>, and there appear three LCSL o_i, i=1,2,3, on the compactified moduli space ℳ_X^*^ cpx=^2. As before, we determine the connecting matrices _ij by blowing-up at three points with (fourth) tangential intersections (cf. Fig. <ref>). Making similar canonical bases of period integrals as
in (<ref>) at each point, which represents bases of the transcendental lattice T_X^*≃ U⊕ M of the mirror K3 surface X^*, we obtain

    _21=([ -1  0  0  0;  0  1 -3  0;  0  0 -1  0;  0  0  0 -1 ]),    _32=([ -1  0  0  0;  0 -3  1  0;  0 -1  0  0;  0  0  0 -1 ]),    _13=([ -1  0  0  0;  0 -1  0  0;  0 -3  1  0;  0  0  0 -1 ]),
    :=_13_32_21=([  -1   0   0   0;   0   3  -8   0;   0   8 -21   0;   0   0   0  -1 ])

as elements in O(U⊕ M,). Here we define U= e⊕ f to be the hyperbolic lattice (^2,([ 0 1; 1 0 ])) and order the bases of U⊕ M as e⊕ M⊕ f when writing the above matrix forms.

The classical mirror symmetry summarized in Section <ref> applies to the so-called (families of) lattice polarized K3 surfaces replacing the Kähler cone with the ample cones <cit.>. In our case here, we consider a primitive lattice embedding M⊕ U⊕M̌⊂ L_K3 with a fixed decomposition M^⊥=U⊕M̌. Then X is a member of the M-polarized K3 surfaces, while the mirror X^* is a member of M̌-polarized K3 surfaces (whose transcendental lattice is M̌^⊥=U⊕ M). The classical mirror symmetry in this case may be summarized in the following isomorphism:

    V_M^++√(-1)M⊗≃Ω^+(U⊕ M)

for the period domain Ω^+(U⊕ M)={ [ω]∈((U⊕ M)⊗) | ω.ω=0,ω.ω̅>0} ^+ where we take one of the connected components, and the corresponding component of the tube domain V_M^+={ v∈ M⊗ | (v,v)_M>0} ^+.

Since there are no elements with (v,v)_M=-2 in M, the ample cones of general members of M-polarized K3 surfaces coincide with the positive cone, which is isomorphic to V_M^+. Similarly to what we described in Section <ref>, by gluing the cone _≥0H_1+_≥0H_2⊂ H^2(X,) by the morphisms φ_ij, we arrive at the positive cone V_M^+ which is an irrational cone (see <cit.> and <cit.>). This gluing exactly matches to the gluing the monodromy nilpotent cones at each boundary point o_i by the connection matrix φ̌_ij. The monodromy relations play the key roles for the gluing, and they follow from the parallel calculations to those in Section <ref>. For example, we have

    T_x'=T_x^-1T_y^3,     T_y'=T_y,     T_x”=T_x,     T_y”=T_y^-1T_x^3

corresponding to (<ref>) and (<ref>), respectively, with

    T_x=([  1 -1  0 -2;  0  1  0  4;  0  0  1  6;  0  0  0  1 ]),     T_y=([  1  0 -1 -2;  0  1  0  6;  0  0  1  4;  0  0  0  1 ]),
    
     T_x'=_21^-1T_x_21,     T_y'=_21^-1T_y_21    and     T_x”=_31^-1T_x_31,     T_y”=_31^-1T_y_31.

 As in Section <ref>, exceptional divisors E_1, E_1' and E_1” have to be introduced to determine the connection matrices _ij, but it turns out that their monodromies are trivial, i.e., T_E_1=T_E'=T_E_1”=id. Clearly, this is consistent to our interpretation of these monodromies in terms of the flopping curves (Proposition <ref>) for the case of Calabi–Yau threefolds.

As this example shows, irrational ample cones indicate infinite gluings of the nilpotent cones in the mirror side. It is natural to expect that the corresponding property holds for the mirror symmetry of Calabi–Yau threefolds in general with ample cones replaced by movable cones and the morphisms by birational maps as known in the so-called movable cone conjecture <cit.>. We have shown in this paper that, in three dimensions, the gluings of monodromy nilpotent cones encode the non-trivial monodromies T_E_i which correspond to the flopping curves.





§ PROOF OF LEMMAS <REF> AND <REF>




 §.§ Proof of Lemma <ref>


Let us consider the projective spaces (V_i) with V_i≃^5, i=1,2. Here we will only present a proof of (1), but it should be clear how to modify the following setting to show (2).

We start with our discussion with the following exact sequence, which we obtain by tensoring the Euler sequence of (V_2) with V_1:

    0→ V_1⊗_(V_2)(-1)→ V_1⊗ V_2⊗_(V_2)→ V_1⊗ T_(V_2)(-1)→0.

In the following arguments, we denote this sequence by

    0→→ V_1⊗ V_2⊗_(V_2)→(^⊥)^*→0

with defining :=V_1⊗_(V_2)(-1) and ^⊥:=V_1^*⊗Ω_(V_2)(1). We also have the following diagram of a linear duality (cf. <cit.>):

    (-25,0)*++X_1  ⊂  ="Xi",
    (-15,0)*++()="PE",
    (15,0)*++(^⊥)="PEp",
    (25,0)*++  ⊃ X_2,
    (-30,-13)*++(V_1⊗ V_2)="PVV",
    ( 0,-13)*++(V_2)="PV",
    ( 30,-13)*++(V_1^*⊗ V_2^*)="PVsVs",
    ( 45,-13)*+⊃  Z_3. "PE";"PVV"
     "PE";"PV"
     "PEp";"PV"
     "PEp";"PVsVs"

Note that () is isomorphic to (V_1)×(V_2), and _()(1)≃_(V_1)×(V_2)(1,1) since it is the pull-back of _(V_1⊗ V_2)(1) by construction. Therefore X_1 is a codimension 5 complete intersection in () with respect to _()(1), and we have _()(1)|_X_1=H_1+H_2.

We see that

    (^⊥)={(w,M) |  Mw=0}⊂(V_2)×(V_1^*⊗ V_2^*),

where we consider V_1^*⊗ V_2^*≃(V_2,V_1^*) and M is a 5×5 matrix. Therefore the image  of the map (^⊥)→(V_1^*⊗ V_2^*) consists of 5×5 matrices of rank ≤4, thus  is so-called the determinantal quintic. Note that we can write the determinantal quintic Z_3⊂ℙ_λ^4 in Proposition <ref> by Z_3=∩ P_4 for a 4-dimensional linear subspace P_4⊂(V_1^*⊗ V_2^*) with identifying P_4 with ℙ_λ^4. Moreover, the pull-back of Z_3 to (^⊥) is X_2.

By a general fact on linear duality (<ref>) in Appendix <ref>, we have

    _()(1)|_X_1+_(^⊥)(1)|_X_1=^*=5H_2,

where we denote by _(^⊥)(1)|_X_1 the strict transform of _(^⊥)(1)|_X_2 and abbreviate the notation for the pull-back for ^*. In this appendix, unless stated otherwise, we will write proper transforms of a divisor by the same symbol omitting the pull-backs by birational maps. Using this convention, we have _()(1)|_X_1=H_1+H_2 and also _(^⊥)(1)|_X_1=L_Z_3. Then we have

    (H_1+H_2)+L_Z_3=5H_2,

which gives L_Z_3=4H_2-H_1. Therefore, restoring the pull-backs by birational maps, we have

    φ_21^*L_Z_3=4H_2-H_1,    φ_21^*L_Z_2=H_2

in N^1(X), which determine φ_21^*(_X_2) as claimed.



 §.§ Proof of Lemma <ref>


Basic idea is very similar to the linear duality in the previous section. We consider the following diagram:

    (-25,0)*++(V_1^*⊗Ω_(V_2)(1))="PVO",
    ( 0,-13)*++(V_1^*⊗ V_2^*).="PVsVs",
    ( 25,0)*++(Ω_(V_1)(1)⊗ V_2^*)="PVOt"
     "PVO";"PVsVs"
     "PVOt";"PVsVs"


(V_1^*⊗Ω_(V_2)(1))→(V_1^*⊗ V_2^*) and (Ω_(V_1)(1)⊗ V_2^*)→(V_1^*⊗ V_2^*) are flopping contractions onto the common image . Moreover, it is of Atiyah type outside the locus in  of corank ≥2.

 This is standard since (V_1^*⊗Ω_(V_2)(1))→(V_1^*⊗ V_2^*) and (Ω_(V_1)(1)⊗ V_2^*)→(V_1^*⊗ V_2^*) are the Springer type resolutions of the image 𝒵 (see (<ref>)).


As we have seen in the proof of Lemma <ref>, X_2 is contained in
(V_2^*⊗Ω_(V_1)(1)). Similarly, X_3
is contained in (Ω_(V_2)(1)⊗ V_1^*). Indeed,
for the 4-dimensional linear subspace P_4⊂(V_1^*⊗ V_2^*)
such that Z_3=∩ P_4, X_2 and X_3 are the pull-backs
of P_4 to (V_1^*⊗Ω_(V_2)(1)) and
(Ω_(V_1)(1)⊗ V_2^*), respectively.

Now we take the fiber product

    :=(V_1^*⊗Ω_(V_2)(1))×_(V_1^*⊗ V_2^*)(Ω_(V_1)(1)⊗ V_2^*).


It holds that =_(V_1)×(V_2)(Ω_(V_1)(1)⊠Ω_(V_2)(1)).

Note that

    ={(w,M,z) |  Mw=0,^tzM=0}⊂(V_2)×(V_1^*⊗ V_2^*)×(V_1).

Thus the fiber of →(V_1)×(V_2) over (w,z) is nothing but ((V_1/ w)^*⊗(V_2/ z)^*) and the assertion follows.


Note that the tautological divisor _(1) of (Ω_(V_1)(1)⊠Ω_(V_2)(1)) defines a map to (V_1^*⊗ V_2^*) and it is the pull-back of _(V_1^*⊗ V_2^*)(1). We will denote it by L_(V_1^*⊗ V_2^*). By the canonical bundle formula of (Ω_(V_1)(1)⊠Ω_(V_2)(1)), we have

    K_=-16L_(V_1^*⊗ V_2^*)+K_(V_1)×(V_2)+{Ω_(V_1)(1)⊠Ω_(V_2)(1)}^*,

where we omit the notation of the pull-backs for K_(V_1)×(V_2) and {Ω_(V_1)(1)⊠Ω_(V_2)(1)}^*. Since K_(V_1)×(V_2)=-5L_(V_1)-5L_(V_2), where L_(V_1) and L_(V_2) are the pull-backs of _(V_i)(1)'s of (V_i) on the left and right factors of (V_1)×(V_2), respectively, and {Ω_(V_1)(1)⊠Ω_(V_2)(1)}^* =4L_(V_1)+4L_(V_2), we have

    K_=-16L_(V_1^*⊗ V_2^*)-L_(V_1)-L_(V_2).

By the canonical bundle formula of (V_1^*⊗Ω_(V_2)(1)), we have

    -K_(V_1^*⊗Ω_(V_2)(1))=20L_(V_1^*⊗ V_2^*).

Pushing forwards (<ref>) to (V_1^*⊗Ω_(V_2)(1)), we obtain

    -K_(V_1^*⊗Ω_(V_2)(1))=16L_(V_1^*⊗ V_2^*)+L_(V_1)+L_(V_2).

Therefore we have

    L_(V_1)+L_(V_2)=4L_(V_1^*⊗ V_2^*).


Now, restricting the above construction over the linear subspace P_4⊂(V_1^*⊗ V_2^*), we have

    (0,0)*++_|P_4="topP",
    (-15,-10)*++X_2="Xl",
    ( 15,-10)*++X_3,="Xr",
    ( 0,-20)*++P_4="botP"
     "topP";"Xl"
     "topP";"Xr"
     "Xl";"botP"
     "Xr";"botP"

where we denote by _|P_4 the restriction of  over P_4. Restricting (<ref>) to X_2, we have

    φ_32^*(M_Z_1)+L_Z_2=4L_Z_3.

This is the claimed relation.
(V_1^*⊗Ω_(V_2)(1))(Ω_(V_1)(1)⊗ V_2^*) is the flop. Similarly, X_2 X_3 is the flop. 

Note that L_(V_1) and L_(V_2) are relatively ample for (Ω_(V_1)(1)⊗ V_2^*)→(V_1) and (V_1^*⊗Ω_(V_2)(1))→(V_2), respectively. Since L_(V_1^*⊗ V_2^*) is the pull-backs of a divisor on (V_1^*⊗ V_2^*), we see that -L_(V_1) is relatively ample for (V_1^*⊗Ω_(V_2)(1))→(V_2) by (<ref>). Therefore (V_1^*⊗Ω_(V_2)(1))(Ω_(V_1)(1)⊗ V_2^*) is the flop. We can show the assertion for X_2 X_3 in the same way using (<ref>).




§ LINEAR DUALITY


Having the case W=V_1⊗ V_2 and B=(V_2) in mind, we consider the exact sequence of sheaves (vector bundles) in the following general form with W=N:

    0→→ W⊗_B→(^⊥)^*→0,
    
    0→^⊥→ W^*⊗_B→^*→0.

Under this general setting, we have the following natural morphisms:

    (-15,0)*++()="PE",
    ( 15,0)*++(^⊥)="PEp",
    (-30,-13)*++(W)="PW",
    ( 0,-13)*++B="B",
    ( 30,-13)*++(W^*).="PWs",
    _f "PE";"PW"
    "PE";"B"
    "PEp";"B"
    ^g "PEp";"PWs"


 Let _b and _b^⊥ be the fibers over b∈ B of  and ^⊥, respectively. Then it holds

    (_b∩ L_r^⊥)=(_b^⊥∩ L_r)

for any r-dimensional linear subspace L_r⊂ W^* and the orthogonal linear subspace L_r^⊥ in W. 
We calculate the dimensions as follows: (_b∩ L_r^⊥)=_b+ L_r^⊥-(_b+L_r^⊥)=r+(N-r)-(_b+L_r^⊥)=(_b^⊥∩ L_r).


The complete intersections X_1, X_2 in Appendix <ref> may be described, respectively, in general terms as

    X_L_r^⊥=f^-1(L_r^⊥)∩(),     Y_L_r=g^-1(L_r)∩(^⊥)

for a fixed subspace L_r⊂ W^*, which we call orthogonal linear sections. Consider the Grassmannian =Gr(r,N) of r-spaces in W^* and define the following family of orthogonal
linear sections:

    𝒳_r:={ ([L_r],x)∈×() |  f(x)∈(L_r^⊥)} ,
    𝒴_r:={ ([L_r],y)∈×(^⊥) |  g(y)∈(L_r)} .

Also we define

    Σ_0:={ ([L_r],b)∈× B | _b∩ L_r^⊥≠0} ={ ([L_r],b)∈× B | _b^⊥∩ L_r≠0} ,

where the second equality is valid due to Lemma <ref>. Then 𝒳_r, 𝒴_r are orthogonal linear sections fibered over Σ_0 and, with natural morphisms, they
can be arranged in the following diagram:

    ( 0,13)*++𝒳_r×_Σ_0𝒴_r="XrYr",
    (-15,0)*++𝒳_r="Xr",
    ( 15,0)*++𝒴_r="Yr",
    ( 0,-13)*++Σ_0="Sigma",
    (-30,0)*++()="PE",
    (30,0)*++(^⊥).="PEp",
    ( 12,-13)*⊂ × B,
    (-6,9.5);"Xr"
    (6,9.5);"Yr"
    "Xr";"Sigma"
    "Yr";"Sigma"
    "Xr";"PE"
    "Yr";"PEp"

Let us introduce the following divisors related to the diagram:

    H_:=_()(1),     H_^⊥:=_(^⊥)(1),     H_:=_(1).


 Abbreviating the pull-back symbols by the morphisms in the diagram (<ref>), we have

    K_𝒳_r×_Σ_0𝒴_r=-(N-2)H_-H_-H_^⊥+K_B+2 ^*

and

    K_𝒳_r=-(N-1)H_+K_B+^*,
    
     K_𝒴_r=-(N-1)H_+K_B+(^⊥)^*.


We leave the proofs for readers.


It is easy to recognize that the proofs of the above proposition rely on the projective geometry behind the diagram (<ref>). We will report the proofs elsewhere with some additional properties which we can extract from the diagram (<ref>); for example, we can show that the morphisms 𝒳_r→Σ_0, 𝒴_r→Σ_0 are flopping contractions and the naturally induced birational map 𝒳_r𝒴_r in the diagram is the flop for these contractions.
Pushing forward K_𝒳_r×_Σ_0𝒴_r to 𝒳_r, and equating toK_𝒳_r, we have a relation

    H_+H_^⊥=^*+H_

on 𝒳_r. Similarly, we have a corresponding relation on 𝒴_r,

    H_+H_^⊥=(^⊥)^*+H_.



Now restricting the relation (<ref>) on 𝒳_r to 𝒳_r|_[L_r]×()=X_L_r^⊥, we obtain

    H_|_X_L_r^⊥+H_^⊥|_X_L_r^⊥= ^*,

which is the relation we used in (<ref>).



 §.§ Acknowledgements
 The results of this work have been reported by the first named author (S.H.) in several workshops; “Modular forms in string theory” at BIRS (2016), “Categorical and Analytic invariants IV” at Kavli IPMU (2016), “Workshop on mirror symmetry and related topics” at Kyoto University (2016) and “The 99th Encounter between Mathematicians and Theoretical Physicists” at IRMA, Strasbourg (2017). He would like to thank the organizers for the invitations for the workshops where he had valuable discussions with the participants. Writing this paper started when S.H. was staying at Brandeis University and Harvard University in March, 2017. He would like to thank B. Lian and S.-T. Yau for their kind hospitality and also valuable discussions during his stay. The authors would like to thank anonymous referees for valuable comments which helped them improve this paper. This work is supported in part by Grant-in Aid Scientific Research JSPS (C 16K05105, JP17H06127 S.H. and C 16K05090 H.T.).

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