Calabi--Yau 3-folds from projective joins of del Pezzo manifolds
Daisuke Inoue


In this paper, we will construct new examples of derived equivalent Calabi–Yau 3-folds with Picard number greater than one. 
We also study their mirror Calabi–Yau manifolds and find that they are given by Schoen's fiber products of suitable rational elliptic modular surfaces. 


Induction, Coinduction, and Fixed Points:

Intuitions and Tutorial

Using examples from set theory, number theory, and real
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    Moez A. AbdelGawad

moez@cs.rice.edu
    December 30, 2023
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§ INTRODUCTION

The derived equivalence between Grassmannian and Pfaffian Calabi–Yau 3-folds <cit.> is one of the interesting phenomena discovered in the study of mirror symmetry of Calabi–Yau 3-folds. 
These Calabi–Yau 3-folds share the same mirror family where the derived equivalence is indicated by the appearance of the corresponding (maximal) degenerations of the family. 
In this paper, we will construct some more examples of derived equivalent Calabi–Yau 3-folds with Picard number greater than one. 
Our construction is an application of homological projective duality (HPD) of projective joins proved in a recent paper by Kuznetsov–Perry <cit.>. 



For a Calabi–Yau 3-fold X, let us recall the so-called Fourier–Mukai partners of X defined by 

    FM(X) = {Y | D^b(X) ≅ D^b(Y) } / ∼_bir

where ∼_bir is a birational equivalence. 
The numbers of equivalent classes |FM(X)| are of considerable interest from both birational geometry as well as mirror symmetry. 
The derived equivalence between Grassmannian and Pfaffian Calabi–Yau 3-folds was the first non-trivial example of |FM(X)| > 1 with Picard number ρ(X) = 1 <cit.>. 
When ρ(X) = 1, some other interesting examples of |FM(X)| > 1 have been discovered; 
for example, the derived equivalence related to the Reye congruence and double quintic symmetroid Calabi–Yau 3-folds by <cit.>, intersections of two Grassmannians by <cit.>, G_2-Grassmann Calabi–Yau 3-folds by <cit.>, and Kapustka–Rampazzo's Calabi–Yau 3-folds by <cit.>. 

When ρ(X) > 1, several possibilities can happen. 
For example, X may have birational models which are not deformation equivalent to X. 
Also, X may have a fibration π : X  S whose general fiber is a lower dimensional Calabi–Yau manifold. 
For such a fibration, Bridgeland–Maciocia <cit.> considered a relative moduli space of stable sheaves ℳ^s(X/S) and proved derived equivalence between X and some component of ℳ^s(X/S). 
Based on the result in <cit.>, Căldăraru <cit.> gave a sufficient condition for X with elliptic fibration to be |FM(X)| > 1. 
Similarly, Schnell <cit.> has given an example X with |FM(X)| > 1 by studying a certain Calabi–Yau 3-fold with abelian surface fibration due to Gross–Popescu <cit.>. 



In this paper, we will study Calabi–Yau 3-folds with |FM(X)| > 1 and ρ(X) > 1 from the viewpoint of projective join and its categorical version <cit.>. 



We first construct Calabi–Yau 3-folds as linear sections of projective joins of two projective varieties. 
Here we recall the following well-known fact (cf. <cit.>): 


Let E_1 ↪(V_1) and E_2 ↪(V_2) be projectively normal elliptic curves where V_i are n_i-dimensional vector spaces for some n_i (i=1,2).  
Let (E_1, E_2) be the projective join of E_1, E_2 (see (<ref>) for the definition). 
Then (E_1, E_2) is normal and its dualizing sheaf is trivial. 


Based on this fact, we will construct our Calabi–Yau 3-folds by smoothings of (E_1, E_2), where E_1 is an elliptic curve of degree 5 in  and E_2 is one of three possible choices of elliptic curves given as linear sections of suitable del Pezzo manifolds (see Section <ref>). 
Combining this construction with the homological projective duality for categorical joins in <cit.>, we will naturally come to a pair of Calabi–Yau 3-folds which are derived equivalent. 
By studying birational geometry of these Calabi–Yau 3-folds, we obtain the following result: 


For each (E_1, E_2) as above, there exists a pair of Calabi–Yau 3-folds which are derived equivalent, but not birationally equivalent. Both Calabi–Yau 3-folds are realised as complete intersections in some projective bundles by relative hyperplane classes. 




We will also construct Calabi–Yau 3-folds which are mirror dual to those constructed in Theorem <ref>. 

We remark that the standard methods to have mirror dual do not apply to Calabi–Yau 3-folds constructed in Theorem <ref> since they are not complete intersections in Gorenstein Fano toric varieties. 
However they are given as complete intersections by linear sections of some projective bundles (Section <ref>). 
We note that the so-called I-functions of Gromov–Witten theory are known for such Calabi–Yau manifolds <cit.>. 
Since it is known that these I-functions coincide with the period integrals of the expected mirror families of Calabi–Yau 3-folds, we can try and error to find mirror duals of Calabi–Yau manifolds in Theorem <ref>. 
In fact, in the talk <cit.>, Galkin remarked that the I-functions of Calabi–Yau 3-folds in projective join of del Pezzo manifolds are given by Hadamard products of I-functions of elliptic curves which are linear sections of del Pezzo manifolds. 
We note that Hadamard products are used to construct interesting 4-th order Fuchsian differential equations (e.g. <cit.>). 
Also, Hadamard products of period integrals of families of elliptic curves are interpreted as period integrals of certain families of Schoen's Calabi–Yau 3-folds (see <cit.>). 
Combining these, we will propose in general the following: 


Mirror Calabi–Yau 3-folds of linear section Calabi–Yau 3-folds in projective join of del Pezzo manifolds are given by Schoen's Calabi–Yau 3-folds with certain choices of rational elliptic surfaces with section. 


In this paper, we will provide supporting evidences of this conjecture by constructing two parameter families of Schoen's Calabi–Yau 3-folds as the mirror manifolds of Calabi–Yau 3-folds in Theorem <ref>. 
In our construction, it turns out that these family of Calabi–Yau 3-folds have several large complex structure limit points. 
These large complex structure limit points can be interpreted which as non-trivial Fourier–Mukai partners, i.e., |FM(X)| > 1, as well as birational models of X. 

Recently, Hosono and Takagi studied a relation between mirror symmetry and birational geometry <cit.>. 
Our examples of Calabi–Yau 3-folds should also be interesting from their point of view. 



The organization of this paper is as follows. 
In Section <ref>, we will summarize projective joins. 
We will also introduce examples of pairs of varieties which are homological projective dual to each other. 
In Section <ref>, we will show the main result (=Theorem <ref>) in this paper. 
We will construct several pairs of Calabi–Yau 3-folds from mutually orthogonal linear sections of suitable projective bundles. 
We will show that each pair of Calabi–Yau 3-folds is not birational. 
In Section <ref>, we will construct Calabi–Yau 3-folds which are topological mirror of Calabi–Yau 3-folds constructed in Section <ref>. 
In Section <ref>, we will construct two dimensional families of mirror Calabi–Yau 3-folds. 
We will see that this family of Calabi–Yau 3-folds have several large complex structure limit points. 


The author is deeply grateful to Shinobu Hosono for many suggestions and helpful comments. 
The author is also grateful to Hiromichi Takagi for valuable comments to improve this paper. 
The author would like to thank Atsushi Kanazawa for encouragement during the postdoctoral period. 





§ JOINS AND HOMOLOGICAL PROJECTIVE DUALITY FOR PROJECTIVE BUNDLES

In this section, following <cit.>, we introduce projective joins and resolved joins of two projective varieties. 
We also introduce pairs of projective varieties which are homological projective dual to each other. 



 §.§ Projective joins

Let M_1 ⊂(V_1) and M_2 ⊂(V_2) be smooth projective varieties. 
The projective join of M_1 and M_2 is defined by 

    (M_1, M_2) := ⋃_x_1 ∈ M_1, x_2 ∈ M_2⟨ [x_1,0], [0,x_2] ⟩

where ⟨ [x_1,0], [0,x_2] ⟩ is the line spanned by the points [x_1,0] and [0,x_2] in (V_1  V_2). 
By the natural embedding, we can regard M_i as a subvariety of (M_1, M_2). 
In general, (M_1, M_2) is singular along the disjoint union M_1 ⊔ M_2. 
There exists a resolution of (M_1, M_2), so-called resolved join, which is given by 

    _M_1 × M_2(𝒪(-H_1) 𝒪(-H_2))

where H_i is the hyperplane class on M_i for i=1,2. 
For a projective variety M, we denote by M^∗ the projective dual variety of M. 
The following formula is well-known: 


    (M_1, M_2)^∗ = (M_1^∗, M_2^∗).

In a recent paper, Kuznetsov and Perry defined a certain category associated to the projective join (<ref>), and introduced categorical join as a categorical extension of the classical duality (<ref>) in the setting of homological projective duality. 



 §.§ Examples of homological projective dual pairs

By taking a linear section of projective joins (<ref>), we will construct Calabi–Yau 3-folds which are related to (E_1, E_2) for elliptic curves E_i in (V_i). 
To apply the homological projective duality for categorical joins to our Calabi–Yau 3-folds, we need to find homological projective dual varieties X and X^♮ together with morphisms X (V) and X^♮(V^∨) such that mutually orthogonal linear sections 

    X ×_(V)(W) and X^♮×_(V^∨)(W^⊥)

are elliptic curves where W ⊂ V is a general linear subspace with an appropriate dimension, V^∨ is the dual projective space of V, and W^⊥ is the orthogonal linear subspace to W. 



The most important example of X and X^♮ comes from G(2, V_5), the Grassmannian of 2-dimensional subspaces of V_5 ≅^5. 
Let G(2, V_5) (∧^2 V_5) be the Plücker embedding of G(2, V_5). 
It is known <cit.> that the homological projective dual of the G(2, V_5) with G(2, V_5) (∧^2 V_5) is the dual Grassmannian G(2, V_5^∨) with G(2, V_5^∨) (∧^2 V_5^∨). 
Let W ⊂∧^2 V_5 be a general 5-dimensional linear subspace and W^⊥⊂∧^2 V_5^∨ be the orthogonal linear subspace to W. 
Then, both orthogonal linear sections 

    G(2, V_5) ×_(∧^2 V_5)(W) and G(2, V_5^∨) ×_(∧^2 V_5^∨)(W^⊥)

are elliptic curves of degree 5. 



Other examples of X and X^♮ may be given by the linear duality <cit.>, which is the homological projective duality valid for projective bundles. 
The following construction gives Calabi–Yau n-folds as linear sections of projective bundles over an (n+1)-dimensional Fano manifold Z. 
This is a special case of <cit.> when we take = _Z^⊕ i in the notation there. 






Let Z be an (n+1)-dimensional Fano manifold. 
Let  be a locally free sheaf on Z of rank r. 
Let () be the projective bundle over Z associated to . 
We denote by π the projection of () to Z. 
We denote by L the relative hyperplane class of (). 
Assume the following conditions: 


  *  the dual ^∨ is globally generated. 

  *  φ_|L|(()) ≥ r where φ_|L| is the morphism defined by |L|. 

  *  c_1() = K_Z. 

Let X be a general complete intersection of r relative hyperplanes in (). 
Then X is a Calabi–Yau n-fold, i.e., X is an n-dimensional smooth projective variety satisfying K_X ∼ 0 and h^i(_X) = 0 if 0 < i < n and h^i(_X) = 1 if i=0, n. 


Since the linear system |L| is base point free by the assumption <ref>, by the Bertini theorem, a general complete intersection of r divisors of |L| is smooth. 
By the assumption <ref>, we have 

    X = () - r = n.

The canonical divisor of () is 

    K_() = -r L -  + K_Z = -r L.

Hence, we see that K_X ∼ 0 by the adjunction formula. 
To compute of dimensions h^i(_X)   (0 ≤ i ≤ n), we use the Koszul resolution 

    0 _()(-r L) ⋯_()(-2L)^⊕r2_()(-L)^⊕ r_()_X  0

and the spectral sequence 

    E_1^-p, q = H^q((), _()(-p L)^⊕rp) ⇒ H^q-p(X, _X).

The E_1-term of (<ref>) can be computed by the spectral sequence 


    E_1^p,q = H^q(Z, R^p π_∗_()(-i L)) ⇒ H^p+q((), _()(-i L))

for i=0, …, r. 
We can determine the higher direct images R^p π_∗_()(-i L) by the Bott–Borel–Weil theorem. 
It turns that the following are the only cases where R^p π_∗_()(-i L) are non-trivial: 

    R^0 π_∗_() = _Z,  R^r-1π_∗_()(-r L) = _Z(K_Z).

Combining the spectral sequences (<ref>), (<ref>) and the Kodaira vanishing on Z, we obtain the claimed results for h^i(_X). 


For a locally free sheaf  as in Proposition <ref>, we consider the orthogonal vector bundle of  which is defined by the kernel of the natural surjective homomorphism 

    0 ^⊥ H^0(Z, ^∨) ⊗_Z^∨ 0.

From the exact sequence (<ref>), we see that (^⊥)^∨ is globally generated and c_1(^⊥) = K_Z. 
We denote by r' the rank of ^⊥. 
We denote by L' the relative hyperplane class of (^⊥). 
For simplicity, we now make the further assumptions that 



  *  φ_|L'|((^⊥)) ≥ r'. 

  *  the natural homomorphism H^0(Z, ^∨)^∨ H^0(Z, (^⊥)^∨) is an isomorphism. In other words, H^1(Z, ) = 0. 

Applying Proposition <ref> to (^⊥), we have a Calabi–Yau n-fold as a general complete intersection of r' relative hyperplanes in (^⊥). 

Let us define V = H^0(Z, ^∨)^∨. 
The homological projective dual of the projective bundle () with φ_|L| : () (V) is the projective bundle (^⊥) with φ_|L'| : (^⊥) (V^∨). 
This result is called linear HPD proved by Kuznetsov. 
As a consequence of HPD, we obtain the following result. 

Let  be a locally free sheaf of rank r on Z satisfying the assumptions <ref>–<ref>. 
Let W be a general codimension r linear subspace of V. 
Let W^⊥ be an orthogonal linear subspace to W. 
Let X = () ×_(V)(W) and Y = (^⊥) ×_(V^∨)(W^⊥) be linear sections of projective bundles. 
Then X and Y are n-dimensional Calabi–Yau manifolds. 
Moreover, there exists an equivalence 

    D^b(X) ≅ D^b(Y)

of the derived categories of these manifolds. 



Let D be the image of X under the projection π : ()  Z. 
This is equal to the image of Y under the projection π' : (^⊥)  Z, so we have the following diagram: 

    [thick,->] (1.9,1.5) – (0.2,1);
    	[thick,->] (1.9,0.5) – (0.2,0);
    	[thick,->] (-1.9,1.5) – (-0.2,1);
    	[thick,->] (-1.9,0.5) – (-0.2,0);
    	[above] at (2,1.5) (^⊥);
    	[above] at (2,1) ∪;
    	[above] at (2,0.5) Y;
    	[below] at (0,1) Z;
    	[below] at (0,0.5) ∪;
    	[below] at (0,0) D;
    	[above] at (-2,1.5) ();
    	[above] at (-2,1) ∪;
    	[above] at (-2,0.5) X;
 
If s_1, …, s_r ∈ H^0(Z, ^∨) are defining sections of X, then the image D is the anti-canonical hypersurface defined by s_1 ∧⋯∧ s_r ∈ H^0(Z, ∧^r ^∨) = H^0(Z, _Z(-K_Z)). 
Therefore we have a birational map X  Y. 



In the next section, we will use the construction of a Calabi–Yau n-fold in Proposition <ref> in the case that S is a del Pezzo surface. 




§ CALABI–YAU 3-FOLDS FROM PROJECTIVE JOINS

In this section, we will construct three examples of Calabi–Yau 3-folds, X_i   (i=1,2,3), as linear sections of resolved joins (<ref>). 
For each of these Calabi–Yau 3-folds, we obtain a Calabi–Yau 3-fold Y_i which is derived equivalent to X_i. 
By studying birational geometry of these Calabi–Yau 3-folds, we will find that X_i and Y_i are not birational, hence non-trivial Fourier–Mukai partners to each other. 



 §.§ Joins of G(2, V_5) and projective bundles over del Pezzo surfaces

Let M_1 = G(2, V_5) and M_2 be a projective bundle _S() over a del Pezzo surface S satisfying the assumptions <ref>–<ref>. 
We denote by Σ_1 the image of M_1 under the Plücker embedding. 
We also denote by Σ_2 the image of M_2 under the map corresponding to the relative hyperplane class. 


Let _M_1, M_2 = _M_1 × M_2((-H_1) (-H_2)) be the resolved join of M_1 and M_2, where H_1 is the Schubert divisor class of M_1 and H_2 is the relative hyperplane class of M_2. 
Let L be the relative hyperplane class of _M_1, M_2. 
Then a general complete intersection of r+5 divisors in |L| is a Calabi–Yau 3-fold. 



We will write k=r+5 here and hereafter in this section. 
Let us denote by X a general complete intersection of k divisors of |L| in _M_1, M_2. 

Since H^0(_M_1, M_2, (L)) = H^0(M_1, (H_1))  H^0(M_2, (H_2)) and H_i   (i=1,2) is base point free by the assumption <ref>, the relative hyperplane class L is also base point free. 
Then a general complete intersection X is smooth by Bertini theorem. 

The image of _M_1, M_2 under the map φ_|L| is (Σ_1, Σ_2), which is the projective join of Σ_1 and Σ_2. 
By the assumption <ref>, we have 

    (Σ_1, Σ_2) = Σ_1 + Σ_2 + 1 ≥ k.

Then we obtain X = _M_1, M_2 - k = 3. 

Similar to the proof of Proposition <ref>, we have 

    K_X = ((r + 3) L - 4H_1 - (r-1) H_2)|_X.

Let D_i be the divisor in _M_1, M_2 corresponding to the injection (-H_i) ↪(-H_1) (-H_2) for i=1,2. 
The image of D_i under the map φ_|L| is Σ_i ⊂(Σ_1, Σ_2). 
Since Σ_2 < k, a general X does not intersect with D_2. 
Similarly if r ≥ 2, then a general X does not intersect with D_1. 
These results imply that 

    (L - H_i)|_X ∼ 0  (i=1,2).

By (<ref>), we have 

    K_X = 4(L - H_1)|_X + (r-1)(L - H_2)|_X ∼ 0

when the rank of  is greater than or equal to two. 
Similar result holds for the case r=1. 

To compute of dimensions h^i(_X)   (0 ≤ i ≤ 3), we use the spectral sequence

    E_1^-p,q = H^q(_M_1, M_2, __M_1, M_2(-pL))^⊕kp⇒ H^q-p(X, _X)

associated with the Koszul resolution similar to the proof of Proposition <ref>. 
We note the following results which are obtained easily (e.g. Bott–Borel–Weil theorem): 

    H^i(M_1, (-j H_1)) = 
       (if  (i,j) = (0,0)  or  (6,5))
    
    0    (if  0 ≤ j ≤ 5  and  (i,j) ≠ (0,0)  or  (6,5)), 
    
       H^i(M_2, (-j H_2)) = 
       (if  (i,j) = (0,0)  or  (r+1,r)) 
    
    0    (if  0 ≤ j ≤ r  and  (i,j) ≠ (0,0)  or  (r+1,r)).

We also note that the higher direct images R^p π_∗__M_1, M_2(-iL)   (i=0, …, k) are nonzero only for R^0 π_∗__M_1, M_2 = _M_1 × M_2 and 

    R^1 π_∗__M_1, M_2(-i L) = ⊕_a=0^i-2_M_1 × M_2(-(a+1)H_1+(a+1-i)H_2)

for 2 ≤ i ≤ k. 
As in the proof of Proposition <ref>, these results (<ref>)–(<ref>) give h^i(_X) = 1 if i=0,3 and h^i(_X) = 0 if i=1,2, i.e., X is a Calabi–Yau 3-fold. 



We remark the relation between Fact <ref> and Proposition <ref>. 
We denote by V_M_1 = H^0(M_1, (H_1))^∨ = ∧^2 V_5 and V_M_2 = H^0(M_2, (H_2))^∨. 
Let E_1 = G(2, V_5) ×_(∧^2 V_5)(W_1) be a general linear section of G(2, V_5) by W_1, where codimension of W_1 ⊂∧^2 V_5 is five. 
Also let E_2 = M_2 ×_(V_M_2)(W_2) be a general linear section of M_2 by W_2, where codimension of W_2 ⊂ V_M_2 is r. 

Assume that the restriction of the map M_2 Σ_2 to E_2 induces an isomorphism onto the image for a general choice of W_2. 
By Proposition <ref>, for a general codimension r + 5 linear subspace W ⊂ V_M_1 V_M_2, the linear section X = _M_1 × M_2((-H_1) (-H_2)) ×_(V_M_1 V_M_2)(W) is a smooth Calabi–Yau 3-fold. 
We define X̅ = φ_|L|(X), which is a linear section of (Σ_1, Σ_2). 
If we consider a special choice W = W_1  W_2 where W_i ⊂ V_M_i  (i=1,2) are given as above, then X̅ is (E_1, E_2), which is nothing but a projective join of elliptic curves. 




 §.§ The cases where _S() is a del Pezzo manifold

We will restrict our attention to the cases where a polarized manifold (_S(), L) is a del Pezzo manifold. 
From the classification of del Pezzo manifolds by Fujita and Iskovskikh (cf. <cit.>), we consider the following three cases: 

    (1)   ^2 ×^2 = _^2(_^2(-1)^ 3), 
    
    (2)   Bl_pt^3 = _^2(_^2(-2) _^2(-1)), 
    
    (3)   ^1 ×^1 ×^1 = _^1 ×^1(_^1 ×^1(-1, -1)^ 2).

We will denote by N_i = _S_i(_i)   (i=1,2,3) the del Pezzo manifold of (1)–(3). 
Note that _i satisfies the conditions <ref>–<ref>. 
We will denote by r_i the rank of _i. 
We define V_N_i = H^0(N_i, (L_i))^∨ where L_i is the relative hyperplane class on N_i. 
For each of these, we consider the following projective bundles associated to the orthogonal vector bundles 

    (1)'   _^2(_1^ 3), 
    
    (2)'   _^2(_2 _1), 
    
    (3)'   _^1 ×^1(_1,1^ 2),

by introducing locally free sheaves _i   (i=1,2), and _1,1 defined by the following exact sequences: 

    0 _i  H^0(^2, _^2(i)) ⊗_^2_^2(i)  0   (i=1,2), 
       0 _1,1 H^0(^1 ×^1, _^1 ×^1(1,1)) ⊗_^1 ×^1_^1 ×^1(1,1)  0.

We will denote by N'_i = _S_i(_i^⊥)   (i=1,2,3) the projective bundle of (1)'–(3)'. 
We will denote by r_i' the rank of _i^⊥. 



Let 𝒥_i = _M_1 × M_2((-H_1) (-H_2)) be the resolved join associated with M_1 = G(2, V_5) and M_2 = N_i. 
Note that the maps φ_|L| : 𝒥_i  J_i are, respectively, the resolutions of the following projective joins: 

    J_1    = (G(2, V_5), ^2 ×^2), 
    
    J_2    = (G(2, V_5), Bl_pt^3), 
    
    J_3    = (G(2, V_5), ^1 ×^1 ×^1) .


Let X_i be a general complete intersection of k_i:=r_i + 5 divisors of |L| in 𝒥_i. 
In other words, we define X_i by 𝒥_i ×_(∧^2 V_5  V_N_i)(W) for a general codimension k_i subspace W in ∧^2 V_5  V_N_i. 
By Proposition <ref>, X_i is a Calabi–Yau 3-fold for a general W. 


The restriction of φ_|L| to X_i induces an isomorphism X_i to its image φ_|L|(X_i) which is a linear section of the projective join of the corresponding del Pezzo manifolds. 
This construction of Calabi–Yau 3-folds is due to S. Galkin <cit.>. 


Let 𝒥_i' = _G(2, V_5^∨) × N_i'((-H_1') (-H_2')) be the resolved join associated with G(2, V_5^∨) and N_i' where H_1' is the Schubert divisor class of G(2, V_5^∨) and H_2' is the relative hyperplane class of N_i'. 
We denote by L' the relative hyperplane class of _i'. 
We denote by Σ'_i the image under the map φ_|H'_i| for i=1,2. 
Since H^0(𝒥_i', (L')) is 

    H^0(G(2, V_5^∨), (H_1'))  H^0(N_i', (H_2')) = ∧^2 V_5  V_N_i

by the assumption <ref>, a linear subspace W in ∧^2 V_5  V_N_i defines a linear section of 𝒥_i'. 
In other words, we define Y_i by 

    Y_i = 𝒥_i' ×_(∧^2 V_5^∨ V_N_i^∨)(W^⊥)

where W^⊥ is the orthogonal subspace to W. 



Let us fix a general linear subspace W of codimension k_i in ∧^2 V_5  V_N_i. 
Then the orthogonal linear section Y_i is a Calabi–Yau 3-fold. 
The Hodge numbers of these Calabi–Yau 3-folds are given as in the following table: 



By definition, the equality V_N_i = r_i + r_i' holds. 
Then the codimension of W^⊥ in ∧^2 V_5^∨ V_N_i^∨ is r_i' + 5. 
If we take a general codimension r_i + 5 linear subspace W in ∧^2 V_5 V_N_i, then W^⊥ is also a general codimension r_i' + 5 linear subspace in ∧^2 V_5^∨ V_N_i^∨. 
By Proposition <ref>, this implies that Y_i is a Calabi–Yau 3-fold. 

The calculations of the Hodge numbers of these Calabi–Yau 3-folds are similar to the proof of Proposition <ref>. 
For example, to compute the Hodge numbers of X_1, we use the conormal sequence

    0 (-L)^ k_1|_X_1Ω_𝒥_1|_X_1Ω_X_1 0.

The cohomologies of (-L)^ k_1|_X_1 and Ω_𝒥_1|_X_1 are calculated by the Koszul resolution and Bott–Borel–Weil theorem. 
Details are left to the readers. 



Applying the homological projective duality for categorical joins due to Kuznetsov and Perry, we obtain the following result. 

Assume that X_i and Y_i are given by mutually orthogonal linear sections in 𝒥_i and 𝒥_i'. 
Then these Calabi–Yau 3-folds are derived equivalent. 



In the following, we only consider the case i=1. 
The other cases are proved similarly. 

   
Let 𝒜^1 = (G(2, V_5)) and 𝒜^2 = (_^2((-1)^ 3)) be the categories of perfect complexes. 
Let (𝒜^1)^♮ and (𝒜^2)^♮ be the HPD categories of these categories. 
There are equivalences

    (𝒜^1)^♮≅(G(2,V_5^∨)), (𝒜^2)^♮≅(_^2(_1^ 3))

as Lefschetz categories over (∧^2 V_5^∨) and (V_N_1^∨) by <cit.> and <cit.> respectively. 
Let (𝒜^1, 𝒜^2) be the categorical join of 𝒜^1 and 𝒜^2 defined by <cit.>. 
By <cit.>, there exists an equivalence 

    (𝒜^1, 𝒜^2)^♮≅((𝒜^1)^♮, (𝒜^2)^♮)

of Lefschetz categories over (∧^2 V_5^∨ V_N_1^∨). 
By the main theorem of HPD for joins (see <cit.>), we obtain an equivalence 

    (𝒜^1, 𝒜^2)_(W)≅((𝒜^1)^♮, (𝒜^2)^♮)_(W^⊥)

for a general codimension k_1 linear subspace W of ∧^2 V_5  V_N_1. 

Similar to the proof of <cit.>, the left hand side of (<ref>) is equivalent to (X_1). 
Similarly, the right hand side of (<ref>) is equivalent to (Y_1). 
Therefore we have an equivalence 

    (X_1) ≅(Y_1)

for general W. 
By the argument of <cit.>, this equivalence implies that 

    D^b(X_1) ≅ D^b(Y_1)

as desired. 




Let X_i and Y_i be general orthogonal linear sections of 𝒥_i and 𝒥'_i for 1 ≤ i ≤ 3. 
Then X_i and Y_i are not birationally equivalent. 


We prove this result by studying birational geometry of X_i and Y_i. 
We give our proof only for the case i=1. 
The other cases are similar and left to the readers. 



Let X_1 be a linear section Calabi–Yau 3-fold in _1 = _×^2 ×^2((-H_1) (-H_2)). 
We denote by π_i the composition of the projection _1 to ^2 ×^2 and the projections to the i-th factor for i=1,2. 


The restriction of π_i to X_1 induces an elliptic fibration on X_1 for i=1,2. 
The divisors giving these elliptic fibrations generate the Kähler cone of X_1. 
In particular, the birational class of X_1 consists of X_1 itself. 



The second assertion follows from the first assertion. 
In fact, X_1 has no flopping contraction because X_1 has Picard number two by Proposition <ref>. 

Let us prove the first assertion. 
Let D_i be the pull-back to X_1 of the hyperplane class of the i-th factor of ^2 ×^2. 
The restriction of π_1 to X_1 induces a map π_1 : X_1 ^2, which corresponds to the divisor D_1. 
This map gives an elliptic fibration on X_1. 
Indeed, a general fiber over a point x ∈^2 is 

    (G(2, V_5), { x }×^2) ∩(W)

where W is the linear subspace of ∧^2 V_5  V_N_1 which defines X_1. 
Since (G(2, V_5), { x }×^2) is a projective cone of G(2, V_5) with its vertex ^2, the fiber (<ref>) is isomorphic to a linear section of G(2, V_5) of codimension five, which is a degree 5 elliptic curve. 
Similarly, the restriction of π_2 induces an elliptic fibration on X_1. 


Let us consider a linear section of 𝒥'_1 = _G(2, V_5^∨) ×_^2(_1^ 3)((-H_1') (-H_2')). 
In the following, we will write the projective bundle _^2(_1^ 3) as _(V_3)(_1 ⊗ W_3^∨) or ℙ_(W_3)(V_3^∨⊗_1) where V_3 ≅^3 and W_3 ≅^3. 
Let us consider the following diagram: 

    [thick,->] (2.9,1) – (0.5,0) node[midway,below] φ_2;
    	[thick,->] (-2.9,1) – (-0.5,0) node[midway,below] φ_1;
    	[above] at (3,1) ℙ_(W_3)(V_3^∨⊗_1);
    	[below] at (0,0) (V_3^∨⊗ W_3^∨);
    	[above] at (-3,1) ℙ_(V_3)(_1 ⊗ W_3^∨);

where φ_i is the map defined by the relative hyperplane class for i=1,2. 
We use the following result in the appendix of <cit.>: 


The maps ℙ_(V_3)(_1 ⊗ W_3^∨) (V_3^∨⊗ W_3^∨) and ℙ_(W_3)(V_3^∨⊗_1) (V_3^∨⊗ W_3^∨) are flopping contractions onto the common image

    Σ'_2 = { M ∈(V_3^∨⊗ W_3^∨) |  M < 3}

where  is a rank of a linear map V_3  W_3^∨ representing M. 
Moreover, the induced birational map ℙ_(V_3)(_1 ⊗ W_3^∨) ℙ_(W_3)(V_3^∨⊗_1) is a flop. 


Note that if we identify (V_3^∨⊗ W_3^∨) with the projectivization of the space of 3 × 3 matrices, then Σ'_2 is a cubic hypersurface defined by the determinant. 

Let Y_1 be the linear section Calabi–Yau 3-fold in 𝒥'_1 = _G(2, V_5^∨) ×_(V_3)(_1 ⊗ W_3^∨)((-H_1') (-H_2')) by a linear subspace W' ⊂∧^2 V_5^∨ V_N_1^∨ of codimension 11. 
We denote by Y̅_1 the image of Y_1 under the map φ_|L'| on 𝒥_1'. 


For general W', the image Y̅_1 has 30 ODPs. 
Moreover, the restriction map φ_|L'| : Y_1 Y̅_1 is a small contraction. 


The image of 𝒥'_1 under the map φ_|L'| is 

    (Σ'_1, Σ'_2) ⊂(∧^2 V_5^∨ V_3^∨⊗ W_3^∨).

By using the map 𝒥'_1 (Σ'_1, Σ'_2), we see that the singular locus of (Σ'_1, Σ'_2) is equal to 

    Σ'_1 ∪Σ'_2 ∪(Σ'_1, D')

where D' ⊂Σ'_2 is a closed subvariety consisting of matrices with rank less than two. 
Note that D' is isomorphic to ^2 ×^2. 
Since Y̅_1 is the linear section of (Σ'_1, Σ'_2) by a linear subspace of codimension 11, Y̅_1 does not intersect with the components Σ'_1 ∪Σ'_2. 
But Y̅_1 must intersect with the component (Σ'_1, D') at finitely many points. 
These points are ODPs of Y̅_1 because Σ'_2 has A_1-singularity along D'. 
Since (Σ'_1, D') = 30, Y̅_1 has 30 ODPs for general W'. 
The restriction map φ_|L'| : Y_1 Y̅_1 induces the isomorphism φ_|L'|^-1(U) ≅ U where U = Y̅_1 ∖Sing(Y̅_1). 
Therefore the map φ_|L'| : Y_1 Y̅_1 is a crepant resolution, and the last claim follows from this fact. 


From the above diagram, we have the following diagram: 

    [thick,->] (2.9,1) – (0.5,0);
    	[thick,->] (-2.9,1) – (-0.5,0);
    	[above] at (4,1) _G(2, V_5^∨) ×_(W_3)(V_3^∨⊗_1)((-H_1') (-H_2'));
    	[below] at (0,0) (Σ'_1, Σ'_2);
    	[above] at (-4,1) _G(2, V_5^∨) ×ℙ_(V_3)(_1 ⊗ W_3^∨)((-H_1') (-H_2'));
 
Combining Lemma <ref> with Lemma <ref>, we obtain the following result. 

Let Y̅_1 be a general linear section of (Σ'_1, Σ'_2). 
Let Y_1 and Y_1' be the pull-backs of Y̅_1 to _G(2, V_5^∨) ×ℙ_(V_3)(_1 ⊗ W_3^∨)((-H_1') (-H_2')) and _G(2, V_5^∨) ×_(W_3)(V_3^∨⊗_1)((-H_1') (-H_2')), respectively. 
Then both Y_1 and Y_1' are Calabi–Yau 3-folds and the maps Y_1 Y̅_1 and Y_1' Y̅_1 are small contractions. 
Moreover, the birational map Y_1  Y_1' is a flop. 


Similar to the proof of Lemma <ref>, we have the following result. 

There exist elliptic fibrations on Y_1 and Y_1'. 


The following diagram describes birational geometry of Y_1. 

    [thick,->] (3.1,1) – (5.5,0);
    	[thick,->] (2.9,2) – (0.5,1);
    	[thick,->] (2.9,1) – (0.5,0);
    	[thick,->] (-2.9,2) – (-0.5,1);
    	[thick,->] (-2.9,1) – (-0.5,0);
    	[thick,->] (-3.1,1) – (-5.5,0);
    	[above] at (4,2) _G(2, V_5^∨) ×_(W_3)(V_3^∨⊗_1)((-H_1') (-H_2'));
    	[below] at (6,0) ^2;
    	[above] at (3,1.5) ∪;
    	[above] at (3,1) Y_1';
    	[below] at (0,1) (Σ'_1, Σ'_2);
    	[below] at (0,0.5) ∪;
    	[below] at (0,0) Y̅_1;
    	[above] at (-4,2) _G(2, V_5^∨) ×ℙ_(V_3)(_1 ⊗ W_3^∨)((-H_1') (-H_2'));
    	[above] at (-3,1.5) ∪;
    	[above] at (-3,1) Y_1;
    	[below] at (-6,0) ^2;
 


Now, Theorem <ref> follows for X_1 and Y_1. 
Indeed, X_1 has no flopping type contraction, so X_1 and Y_1 cannot be birational. 
Other cases i=2,3 are proved by similar arguments. 




Both Calabi–Yau 3-folds X_1 and Y_1 (or Y_1') have elliptic fibrations over the same base ^2. 
It is interesting to see whether Y_1 (or Y_1') is a component of the relative moduli space of stable sheaves on the elliptic fibrations π_i : X_1 ^2 (i=1,2). 


In what follows, we summarize briefly the birational geometry of X_2, Y_2 and X_3, Y_3 in order. 
We first describe the birational geometry of X_2 and Y_2 in the following diagrams: 

    [thick,->] (0.1,1.5) – (1.4,0) node[midway,right] f;
    	[thick,->] (-0.1,1.5) – (-1.4,0) node[midway,left] π;
    	[below] at (1.5,0) X̅_2;
    	[above] at (0,1.5) X_2;
    	[below] at (-1.5,0) ^2;
    	, 
    	[thick,->] (3.1,1.5) – (4.4,0) node[midway,right] p_2;
    	[thick,->] (2.9,1.5) – (1.6,0) node[midway,left] g_2;
    	[dashed,->] (0.4,1.75) – (2.6,1.75) node[midway,above] ι;
    	[thick,->] (0.1,1.5) – (1.4,0) node[midway,right] g_1;
    	[thick,->] (-0.1,1.5) – (-1.4,0) node[midway,left] p_1;
    	[below] at (4.5,0) ^2;
    	[above] at (3,1.5) Y_2;
    	[below] at (1.5,0) Y̅_2;
    	[above] at (0,1.5) Y_2;
    	[below] at (-1.5,0) ^2;
    	.

Here the map π is an elliptic fibration on X_2 and f is a divisorial contraction which contracts a degree 5 del Pezzo surface to a point. 
The maps p_i   (i=1,2) are elliptic fibrations on Y_2 and g_i   (i = 1,2) are small contractions of Y_2. 
The rational map ι is a birational involution of Y_2 (see, for example, <cit.>). 

The following diagrams represent the sections of movable fans of X_3 and Y_3: 

    (A) at (0,sqrt(3)*1.5-1.5);
    	 (B) at (-1.5,-1.5);
    	 (C) at (1.5,-1.5);
    	 (0,sqrt(3)*1.5/3-1.5) node[circle] X_3;
    	[thick] (A)– (B)–(C)–(A);
    	 [above] at (A) ρ_3=_≥ 0H_3;
    	 [left] at (B) ρ_1=_≥ 0H_1;
    	 [right] at (C) ρ_2=_≥ 0H_2;
    	, 
    	 (A) at (0,0);
    	 (B) at (2,0);
    	 (C) at (3,sqrt(3));
    	 (D) at (2,sqrt(3)*2);
    	 (E) at (0,sqrt(3)*2);
    	 (F) at (-1,sqrt(3));
    	 (G) at (1,sqrt(3));
    	 (1,sqrt(3)/3) node[circle] Y_3;
    	 (1,5*sqrt(3)/3) node[circle] Y_3;
    	 (2,2*sqrt(3)/3) node[circle] Y'_3;
    	 (2,4*sqrt(3)/3) node[circle] Y”_3;
    	 (0,2*sqrt(3)/3) node[circle] Y”_3;
    	 (0,4*sqrt(3)/3) node[circle] Y'_3;
    	[thick] (A)– (B)–(C)–(D)–(E)–(F)–(A);
    	[thick] (A)– (G)–(B);
    	[thick] (C)– (G)–(D);
    	[thick] (E)– (G)–(F);
    	 [left] at (A) ρ'_1=_≥ 0 H_1';
    	 [right] at (B) ρ'_2=_≥ 0 H_2';
    	 [right] at (C) ρ'_3=_≥ 0 D_3';
    	 [right] at (D) ρ'_4=_≥ 0 D_4';
    	 [left] at (E) ρ'_5=_≥ 0 D_5';
    	 [left] at (F) ρ'_6=_≥ 0 D_6';
    	 [right] at (3.3,5*sqrt(3)/3) ρ'_0=_≥ 0L';
    	[dashed,->] (3.3,5*sqrt(3)/3) to [out=180,in=58] (G);

where Y'_3 and Y”_3 are birational models of Y_3. 
The divisors H_1, H_2, H_3 (resp. H_1', H_2', L') are the generator of the nef cone of X_3 (resp. Y_3). 
The divisors D'_3, D'_4, D'_5, D'_6 are L'-H'_1+H'_2, 2L'-H_1', 2L'-H_2', L'+H_1'-H_2', respectively. 
We describe the morphisms associated to each cone of the movable fans: 
Two dimensional cones ρ_i + ρ_j (1 ≤ i < j ≤ 3) correspond to elliptic fibrations on X_3. 
One dimensional cones ρ_i (i=1,2,3) correspond to K3 fibrations on X_3. 
Similarly, two dimensional cones ρ'_i + ρ'_i+1 (1 ≤ i ≤ 6) correspond to elliptic fibrations on some birational models Y_3, Y'_3, or Y”_3. 
Here we use the notation ρ'_7 = ρ'_1. 
One dimensional cones ρ'_i (1 ≤ i ≤ 6) correspond to K3 fibrations on some birational models Y_3, Y'_3, or Y”_3. 
Moreover, two dimensional cones ρ'_0 + ρ'_i (1 ≤ i ≤ 6) correspond to small contractions of birational models Y_3, Y'_3, or Y”_3. 
The one dimensional cone ρ'_0 corresponds to divisorial contractions of birational models Y_3, Y'_3, and Y”_3. 



§ FIBER PRODUCTS OF RATIONAL ELLIPTIC SURFACES

In this section, we will construct Calabi–Yau 3-folds which are mirror to X_i   (i=1,2,3) constructed in Section <ref>. 
In addition to these Calabi–Yau 3-folds, we also consider a Calabi–Yau 3-fold X_0 given as a linear section of the join J_0 = (, ). 
This Calabi–Yau 3-fold is known as intersections of two Grassmannians studied by <cit.>. 

The strategy of our construction is based on the following two observations. 





  *  There is a relation between projective joins and Hadamard products of corresponding I-functions <cit.>. 

  *  There is a geometric realization of a Hadamard product as a period integral of a fiber product of two families of elliptic curves. 




These observations naturally lead us to a family of Calabi–Yau 3-folds for each of our Calabi–Yau 3-folds X_i (or Y_i). 
To go into the details of mirror symmetry, we need to find suitable compactifications of the parameter spaces of the families. 
In this section, deferring this problem to the next section, we will restrict our attentions to constructing general members of such families and verifying the expected exchanges of the Hodge numbers. 



 §.§ Schoen's Calabi–Yau 3-folds

Let us recall the work of Schoen <cit.> to construct Calabi–Yau 3-folds as crepant resolutions of fiber products of two rational elliptic surfaces with sections. 

Let π_i : S_i ^1 (i=1,2) be relatively minimal rational elliptic surfaces with sections. 
We denote by D_i the image of singular fibers under the map π_i for i=1,2. 
For simplicity, we assume that all singular fibers of π_i (i=1,2) over D_1 ∩ D_2 are of type _b  (b ≥ 0) in the notation of Kodaira. 
We set  

    Y = S_1 ×_^1 S_2.

A point (x_1,x_2) ∈ Y is singular if and only if x_i is a singular point of the singular fiber π_i^-1(t) for both i=1, 2 where t := π_1(x_1) = π_2(x_2). 
By assumption, it is easy to check that these singular points are ordinary double points. 

Schoen proved that the dualizing sheaf of Y is trivial. 
A crepant resolution of Y is given by an iterated blow-up along smooth divisors on Y. 
Indeed, each node of Y is lying on a smooth divisor of Y which is a product of irreducible components of singular fibers π_1^-1(t) and π_2^-1(t) where t ∈ D_1 ∩ D_2. 
We denote by Ỹ a crepant resolution of Y. 
Then Ỹ is a Calabi–Yau 3-fold. 
Note that the condition h^1(_Ỹ) = 0 is proved by using the fact that Y is a divisor in S_1 × S_2. 



 §.§ Calabi–Yau 3-folds from rational elliptic modular surfaces

We consider Schoen's Calabi–Yau 3-folds in the case where S_i are rational elliptic modular surfaces. 

From the table of elliptic surfaces with four singular fibers due to Herfurtner <cit.>, we take the elliptic surfaces S(5), S(6), S(7) with the following types of singular fibers: 

These are relatively minimal rational elliptic surfaces with section. 
Note that S(5) and S(6) are Shioda's modular surfaces <cit.> associated to the congruence subgroups of SL(2, ) given by 

    Γ_1(n) = 
    {(
    [ a b; c d ]) ∈ SL(2, )  |   a, d ≡ 1, c ≡ 0   (mod  n)
    }

with n = 5,6, respectively. 

By using an automorphism of ^1, we define four elliptic surfaces T := T_0̅, T_0, T_1, T_2, T_3 with the following specified singular fibers at 0 and ∞: 


  * T_0̅≅ S(5), D = {0, x̅_1, x̅_2, ∞}, π^-1(0) is of type _5, and π^-1(∞) is of type _5, 

  * T_0 ≅ S(5), D = {0, x_1, x_2, ∞}, π^-1(0) is of type _5, and π^-1(∞) is of type _5, 

  * T_1 ≅ S(6), D = {0, y_1, y_2, ∞}, π^-1(0) is of type _6, and π^-1(∞) is of type _3, 

  * T_2 ≅ S(7), D = {0, z_1, z_2, ∞}, π^-1(0) is of type _7, and π^-1(∞) is of type _2, 

  * T_3 ≅ S(6), D = {0, w_1, w_2, ∞}, π^-1(0) is of type _6, and π^-1(∞) is of type _2, 

where π : T ^1 and D is the image of singular fibers under the map π. 
We assume that singular fibers other than 0 and ∞ are in general positions. 
Namely, all the points x̅_1, x̅_2, x_1, x_2, y_1, y_2, z_1, z_2, w_1, w_2 are mutually distinct. 



Based on the observations <ref>, <ref> in the beginning of this section, we claim the following. 

Let X_i^∗ = T_0̅×_^1 T_i (i = 0,1,2,3) be the fiber products of relatively minimal rational elliptic surfaces with sections. 
Let X̃_i^∗ (i=0,1,2,3) be the Schoen's Calabi–Yau 3-folds constructed in Section <ref>. 
Then X̃_0^∗, X̃_1^∗, X̃_2^∗, and X̃_3^∗, respectively, are mirror Calabi–Yau 3-folds of the linear section Calabi–Yau 3-folds of (G(2, V_5), G(2, V_5)), (G(2, V_5), ^2 ×^2), (G(2, V_5), _^3), and (G(2, V_5), ^1 ×^1 ×^1). 



We can find these elliptic surfaces T_i (1 ≤ i ≤ 3) by taking special one parameter families of Batyrev–Borisov toric mirror constructions of complete intersection elliptic curves in ^2 ×^2, _^3, and ^1 ×^1 ×^1, respectively. 


Below, we will provide convincing evidences for this conjecture. 



 §.§ Calculations of Hodge numbers

We calculate Hodge numbers of X̃_i^∗  (i=0,1,2,3) by the Schoen's results <cit.>. 


The Hodge numbers of X̃_i^∗  (i=0,1,2,3) are given as in the following table: 

In particular, these satisfy h^1,1(X_i) = h^2,1(X̃_i^∗) and h^2,1(X_i) = h^1,1(X̃_i^∗) for i=0,1,2,3. 
We refer to <cit.> for the Hodge numbers of X_0. 



We only calculate the Hodge numbers of X̃_1^∗. 
Since the other cases are similar, we leave them to the readers. 
First, we calculate the Euler number of X̃_1^∗. 
By the properties of the Euler number, we have 

    e(T_0̅×_^1 T_1)    = e(π^-1(0))e(π'^-1(0)) + e(π^-1(∞))e(π'^-1(∞)) = 45

where π : T_0̅^1 and π' : T_1 ^1 are projections. 
Since the resolution X̃_1^∗ T_0̅×_^1 T_1 is small, an ordinary double point of T_0̅×_^1 T_1 is replaced by ^1. 
Then we have 

    e(X̃_1^∗) = e(T_0̅×_^1 T_1) + 45 = 90.

Next, we calculate h^1,1(X̃_1^∗). 
We use the following result from <cit.>. 

Let S_1 and S_2 be relatively minimal rational elliptic surfaces with sections. 
Let π_i : S_i ^1 be the projection. 
Let D_i be the image of singular fibers under the projection π_i. 
We define D' =D_1 ∩ D_2. 
Assume that singular fibers of S_i over D' are of type _b for i=1,2. 
For s ∈ D_i, we denote by b_i(s) the number of irreducible components of the singular fiber π_i^-1(s). 
Let η be the generic point of ^1. 
We define d=1 if (S_1)_η and (S_2)_η are isogenous and d=0 if otherwise. 
Then h^1,1 of Schoen's Calabi–Yau 3-fold is given by 

    d + 19 + ∑_s ∈ D' b_1(s)b_2(s) - ∑_s ∈ D' b_1(s) - ∑_s ∈ D' b_2(s) + # D'.


Applying the above lemma to our case, we obtain h^1,1(X̃_1^∗) = 47. 
By the relation e(X̃_1^∗) = 2(h^1,1 - h^2,1), we obtain h^2,1(X̃_1^∗) = 2 as desired. 




§ FAMILIES OF SCHOEN'S CALABI–YAU 3-FOLDS

In this section, we will construct families of Calabi–Yau 3-folds of X̃_0^∗ and X̃_1^∗ which are expected to be mirror families of X_0 and X_1, respectively. 
When constructing these families, we will use the relation between Hadamard products and fiber products in <cit.>. 



 §.§ Mirror families of linear sections of (, )

Let S(5) be the Shioda's modular surface associated with the congruence subgroup Γ_1(5). 
Let π:S(5) ^1 be the projection to the modular curve. 
We fix a homogeneous coordinate of ^1 as done in <cit.>. 

Then the map π has _5-fibers over the points 0 and ∞ and _1-fibers over the points [1,z] where z^2 + 11z - 1 = 0. 



We set S_i = S(5) and S_i ^1   (i=1,2) as above. 
Let μ : ^1 ×^1 ^1 be the multiplication map defined by 

    ([s_0, s_1], [t_0, t_1]) ↦ [s_0 t_0, s_1 t_1].

The map μ is defined outside of B = {(0, ∞), (∞, 0)}. 
Let ^1 ×^1 be the blow-up of ^1 ×^1 along B. 
Then the rational map μ extends to the morphism ^1 ×^1^1. 
We define S_1 × S_2 by the fiber product of ^1 ×^1^1 ×^1 and S_1 × S_2 ^1 ×^1: 

    S_1 × S_2dr    S_1 × S_2 [d] 
    ^1 ×^1r   ^1 ×^1

We define a morphism p : S_1 × S_2^1 by the composition of the projection S_1 × S_2^1 ×^1 and the morphism ^1 ×^1^1. 


Let D_0 = { 0, ∞} and D_1 = {[1, z] | (z+1)(z^2 - 123 z + 1) = 0}. 
For [z_0, z_1] ∈^1 ∖ (D_0 ∪ D_1), the fiber of the map p : S_1 × S_2^1 over [z_0, z_1] is isomorphic to X_0^∗ constructed in Section <ref>. 




For each [1,z] ∈^1 ∖ D_0, the fiber p^-1([1,z]) is the fiber product of rational elliptic surfaces which are isomorphic to S(5). 
Indeed, the fiber  p^-1([1,z]) is isomorphic to the fiber product S(5) ×_^1 S(5)' where S(5)' is isomorphic to S(5) but a morphism S(5)' ^1 is given by the composition of π:S(5) ^1 and the map ^1 ^1 where  

    [s_0, s_1] ↦ [s_1,  z s_0].

The condition that singular fibers of S(5) ^1 and S(5)' ^1, other than over 0 and ∞, are located over the different positions is equivalent to [1, z] ∉ D_1. 


For [z_0, z_1] ∈^1 ∖ (D_0 ∪ D_1), the fiber p^-1([z_0, z_1]) has a small resolution, which is a Calabi–Yau 3-fold X̃^∗_0. 
Therefore we obtain a family of Schoen's Calabi–Yau 3-folds over ^1. 
For [z_0, z_1] ∈ D_1, the fiber p^-1([z_0, z_1]) has an extra nodal point, so D_1 is a discriminant locus of the family. 



This family of Calabi–Yau 3-folds have two large complex structure limit points at 0 and ∞. 
This result reflects the fact that there exists Calabi–Yau 3-folds X and Y which are derived equivalent <cit.> but not isomorphic to each other <cit.>. 

Note that both Calabi–Yau 3-folds are linear sections of (, ). 


A mirror family of linear section Calabi–Yau 3-folds in (, ) was already studied by <cit.>. 
Their constructions are based on the toric degeneration of (, ) and conifold transitions of linear section Calabi–Yau 3-folds. 




 §.§ Mirror families of linear sections of (, ^2 ×^2)

First we construct a family of elliptic curves whose period integrals coincide with the I-function of linear sections of ^2 ×^2 (see Appendix <ref>). 
We denote by ^1_s, ^2_t, ⋯, etc, the projective spaces with homogeneous coordinates s=[s_0, s_1], t=[t_0, t_1, t_2], and so on. 


There exists a three dimensional projective variety T and a morphism π : T^2_t such that 


  * A general fiber of π is an elliptic curve. 

  * For [z_1, z_2] ∈^1, let ℓ_[z_1,z_2] be a line in ^2_t defined by {[t_0, t_1, t_2] ∈^2_t| z_2 t_1 - z_1 t_2 = 0}. Then the inverse image π^-1(ℓ_[z_1, z_2]) is either the Shioda's modular surface S(6) associated with the congruence subgroup Γ_1(6) if [z_1, z_2] = [1,1] or a rational elliptic surface with section if [z_1, z_2] ≠ [1,0], [1,-1], [1,1],[0,1]. 
The latter case has five singular fibers. 

  * Let us fix [z_1, z_2] ≠ [1,0], [1,-1], [0,1]. 
For the map π^-1(ℓ_[z_1, z_2]) ℓ_[z_1,z_2], the fiber over the point [1,0,0] is of type _6 and the fiber over the point [0, z_1, z_2] is of type _3. 




We will sketch our construction of T. 
Let us consider the following complete intersection T_0 in ^2_x×^2_y×^2_c: 

    c_0 x_0 y_0 + c_1 x_1 y_2 + c_2 x_2 y_1 = 0,  c_0 x_1 y_1 + c_1 x_2 y_0 + c_2 x_0 y_2 = 0, 
       c_0 x_2 y_2 + c_1 x_0 y_1 + c_2 x_1 y_0 = 0.

There is a natural projection π_0 : T_0 ^2_c coming from the projection to the third factor. 
The variety T_0 is smooth outside of 81 ordinary double points, which are on the fibers of π_0 over {[1, -ζ^i, 0], [1, 0, -ζ^j], [0,1,-ζ^k] | 0 ≤ i,j,k ≤ 2 } where ζ = e^2 π i / 3. 

Let G ≅_3^ 3 be the subgroup of Aut(^2_x×^2_y×^2_c) generated by 

    ([x_0,x_1,x_2],[y_0,y_1,y_2],[c_0,c_1,c_2])    ↦ ([x_0,ζ x_1,x_2],[y_0,y_1, ζ y_2],[c_0,ζ c_1,c_2]), 
    
    ([x_0,x_1,x_2],[y_0,y_1,y_2],[c_0,c_1,c_2])    ↦ ([x_0,x_1,ζ x_2],[y_0,ζ y_1,y_2],[c_0,c_1,ζ c_2]), 
    
    ([x_0,x_1,x_2],[y_0,y_1,y_2],[c_0,c_1,c_2])    ↦ ([x_0,ζ x_1,ζ^2 x_2],[y_0,ζ y_1, ζ^2 y_2],[c_0,c_1,c_2]).

The group G acts on T_0. 
We define T by the quotient T_0/G of T_0 by G. 
Then the projection π_0 :  T_0 ^2_c naturally induces the morphism π :  T^2_c/G =: ^2_t where [t_0, t_1, t_2] = [c_0^3, c_1^3, c_2^3]. 

It is straightforward to show that π^-1(ℓ_[1,1]) is isomorphic to the modular surface S(6) by comparing the functional invariants and also homological invariants of π^-1(ℓ_[1,1]) with those of S(6). 



The above special family of equations (<ref>) has been chosen so that the period integrals of the family of elliptic curves coincide with the period integrals of toric mirror construction of the complete intersection (1,1) ∩ (1,1) ∩ (1,1) in ^2 ×^2 by Batyrev–Borisov <cit.> (also <cit.>). 


We will construct a family of Schoen's Calabi–Yau 3-folds X̃^∗_1. 
Consider the above family T^2_t and also S := S(5) ^1_s. 
By the analogy of the construction in Section <ref>, we first consider a rational map μ : ^1_s×^2_t^2_z given by 

    ([s_0, s_1] , [t_0, t_1, t_2]) ↦ [s_0 t_0, s_1 t_1, s_1 t_2].

This map μ is defined outside of B_1 ⊔ B_2 where 

    B_1    = {([1,0], [0, t_1, t_2]) | (t_1, t_2) ∈^2 ∖{(0,0)}}, 
    
    B_2    = {([0,1], [1, 0, 0]) }.

Let ^1_s×^2_t be the blow-up of ^1_s×^2_t along B_1 ⊔ B_2. 
Then the rational map μ extends to the morphism ^1_s×^2_t^2_z. 
We define S × T by the fiber product expressed in the following diagram: 

    S × Tdr    S × T[d] 
    ^1_s×^2_tr   ^1_s×^2_t

We define a morphism p : S × T^2_z by the composition of the projection S × T^1_s×^2_t and the morphism μ : ^1_s×^2_t^2_z. 
Note that μ^-1([z_0, z_1, z_2]) ≅^1 for general [z_0, z_1, z_2] ∈^2_z. 


Let D_2 = {dis(z_0,z_1,z_2) = 0} be a divisor in ^2_z where 

    dis(z_0,z_1,z_2) = ∏_ζ^2+11ζ-1=0 ((ζ z_0 + z_1 + z_2)^3 - 27 ζ z_0 z_1 z_2).

We also define D_0 = {z_0 z_1 z_2 = 0} and D_1 = {z_1 + z_2 = 0 }. 
Then for any [z_0, z_1, z_2] ∈^2_z∖ (D_0 ∪ D_1 ∪ D_2), a fiber of the morphism p : S × T^2_z over [z_0, z_1, z_2] is the fiber product of rational elliptic surfaces with sections. 
A crepant resolution of the fiber p^-1([z_0, z_1, z_2]) is deformation equivalent to Schoen's Calabi–Yau 3-folds X̃^∗_1 constructed in Section <ref>. 



Let us take a point [z_0, z_1, z_2] ∈^2_z∖ (D_0 ∪ D_1). 
A fiber p^-1([z_0, z_1, z_2]) is isomorphic to S(5) ×_^1_s T_z_1, z_2 where T_z_1,z_2 is the inverse image of {z_2 t_1 - z_1 t_2 = 0}⊂^2_t under the map π :  T^2_t. 
Note that we identify ^1_s≅{ z_2 t_1 - z_1 t_2 = 0 }⊂^2_t by the embedding: 

    [s_0, s_1] ↦ [z_0 s_1, z_1 s_0, z_2 s_0].

By Proposition <ref>, if z_1 = z_2 then T_z_1, z_2 is a modular surface S(6), and if z_1 ≠± z_2 then T_z_1,z_2 is an elliptic surface with five singular fibers. 
We define by F_1 ⊂^1_s and F_2 ⊂^1_s the images of the singular fibers of the elliptic fibrations S(5) ^1_s and T_z_1, z_2^1_s, respectively. 
Then the condition F_1 ∩ F_2 = {[1,0], [0,1]} is equivalent to the condition [z_0, z_1, z_2] ∉ D_2. 
Hence for [z_0, z_1, z_2] ∈^2_z∖ (D_0 ∪ D_1 ∪ D_2) satisfying z_1 = z_2, the fiber p^-1([z_0, z_1, z_2]) describes X_1^∗ in Conjecture <ref>. 
Therefore the claim follows. 


Let [1, -z_1, -z_2] be the affine coordinate of ^2_z around the point [1,0, 0]. 
By the construction of the above family of Calabi–Yau 3-folds, there exists a period ω_0(z_1, z_2) given by the Hadamard product of period integrals of two families of elliptic curves. 
The explicit form of ω_0 is given in (<ref>) where we set x_i=z_i. 



It is straightforward to compute the Picard–Fuchs operators which annihilate ω_0(z_1,z_2). 
They are given by 


    P_1    = p_20θ_1^2 + p_11θ_1 θ_2 + p_02θ_2^2 + p_10θ_1 + p_01θ_2 + p_00, 
    
    P_2    = z_1 θ_2^3 - z_2 θ_1^3

where we set θ_i = z_i ∂/∂ z_i. 
The coefficients of θ_1^i θ_2^j   (0 ≤ i + j ≤ 2) in P_1 are 

    p_20   = z_1^2 + 11z_1 z_2 + 10 z_2^2 + 11z_1 + 11z_2 -1, 
    
    p_11   = 5z_1^2 + 10 z_1 z_2 + 5z_2^2 + 22z_1 + 22z_2 + 1, 
    
    p_02   = 10z_1^2 + 11z_1 z_2 + z_2^2 + 11z_1 + 11z_2 -1, 
    
    p_10   = (z_1 + z_2) (2z_1 + 5z_2 + 11), 
    
    p_01   = (z_1 + z_2) (5z_1 + 2z_2 + 11), 
    
    p_00   = (z_1 + z_2) (z_1 + z_2 + 3).

Let us study these differential operators over ^2_z. 
Around the points [0,1,0] and [0,0,1], we have the following facts. 


Let [w_0, 1, -w_2] be the affine coordinate of ^2_z around the point [0,1,0]. 
Let Q_1 and Q_2 be the following gauge transforms of the operators P_1, P_2 around the point [0,1,0]: 

    Q_1 = w_0 P_1(w_0, w_2) w_0^-1, Q_2 = P_2(w_0, w_2).

Then Q_1 and Q_2 determine the period integrals of the above family around the point [0,1,0]. 
Similar results hold for the point [0,0,1]. 


This is a standard fact. We refer <cit.>, for example, for details. 



There exists a period integral which is holomorphic around the point [0, 1, 0]. 
More precisely, it is given by 

    ∑_d_0', d_2' ≥ 0( ∑_k=0^d_0'd_0'k^2d_0' + kk) (d_0' + d_2')!^3/(d_0'!)^3 (d_2'!)^3 w_0^d_0' w_2^d_2'

where [w_0, 1, -w_2] is the affine coordinate of ^2_z. 

Similar results hold for the point [0,0,1]. 


This follows by the argument in <cit.>. 




By calculating the monodromies around the coordinate hyperplanes of ^2_z, we obtain the following result. 

The points [1,0,0], [0,1,0], and [0,0,1] are large complex structure limit points in the sense of Morrison <cit.>. 


We can calculate the g=0 Gromov–Witten invariants for X_1 (resp. Y_1) by applying J. Brown's result <cit.> or the abelian/nonabelian correspondence <cit.> to resolved joins _1 and (L)^ 8 (resp. '_1 and (L')^ 11). 
The tables of BPS numbers of X_1 and Y_1 are shown in Appendix <ref>. 


We can extract enumerative invariants from Picard–Fuchs operators for each large complex structure limit points [1,0,0], [0,1,0], and [0,0,1] (see <cit.> and references therein for the details). 
By construction, the invariants obtained from the point [1,0,0] coincide with the g=0 Gromov–Witten invariants of X_1. 
By computation, we can check that the invariants obtained from the points [0,1,0] and [0,0,1] coincide with the g=0 Gromov–Witten invariants of Y_1 for sufficiently large degree. 





§ LISTS OF ENUMERATIVE INVARIANTS

In this Appendix, we give the lists of the BPS numbers obtained from the g=0 Gromov–Witten invariants of X_1 and Y_1. 





 §.§ 
 
Let X_1 be the Calabi–Yau 3-fold constructed in Section <ref>. 
We fix a basis of H^2(X_1, ) as D_1, D_2 where D_i is the restriction of the pull-back of the hyperplane class of the i-th factor of ^2 ×^2. 
We represent a homology class C ∈ H_2(X_1, ) by the intersection numbers (d_1, d_2):=(D_1.C, D_2.C). 





 §.§ 

Let Y_1 be the Calabi–Yau 3-fold constructed in Section <ref>. 
We fix a basis of H^2(Y_1, ) as L', D' where L' is the restriction of the relative hyperplane class of the resolved join and D' is the pull-back of the hyperplane class of ^2 by the map of the elliptic fibration on Y_1. 
We represent a homology class C' ∈ H_2(Y_1, ) by the intersection numbers (d'_1, d'_2):=(L'.C', D'.C'). 






§ I-FUNCTIONS OF COMPLETE INTERSECTIONS OF RESOLVED JOINS

We calculate I-functions of Calabi–Yau 3-folds X_0 and X_1 discussed in Section <ref> and <ref>. 
We refer to <cit.> for definitions and properties of I-functions. 
Note that the I-functions of Calabi–Yau 3-folds are related to period integrals of mirror families. 



 §.§ The I-function of X_0

To calculate the I-function of X_0, we use J. Brown's formula about I-functions of toric bundles. 
Let us recall that X_0 is a complete intersection of ten relative hyperplanes in the resolved join _×((-H_1) (-H_2)). 

The following proposition is an immediate consequence of J. Brown's formula. 
This result is already mentioned by S. Galkin in his talk <cit.>. 


Let J^(τ,z) = e^τ H/z∑_d=0^∞ J^_d(τ,z) Q^d be the J-function of  where H is the Schubert divisor of . 
We denote by G(2, V_5)_i the i-th factor of G(2, V_5) × G(2, V_5). 
Then the I-function of X_0 is 

    I_X_0(τ, z) = e^τ L / z∑_d = 0^∞∏_1 ≤ i ≤ 2∏_m=1^d(L + m z)^5 J^_i_d(τ/2, z)|_X_0 Q^d

where J^_i_d(τ/2, z)|_X_0 is the restriction of cohomology classes of _i to X_0. 



Applying J. Brown's formula <cit.> and quantum Lefschetz theorem <cit.> to a complete intersection X_0 of the resolved join := _×((-H_1) (-H_2)), we have the twisted I-function 

    I(τ, z) = e^τ /z∑_d_1, d_2, d_3 ≥ 0∏_m=1^d_3 (L + mz)^10/∏_1 ≤ i ≤ 2∏_m=1^d_3-d_i(L - H_i + mz)J^G(2, V_5)_1_d_1(τ_1, z) J^G(2, V_5)_2_d_2(τ_2, z) e^d_3 τ_3 Q_1^d_1 Q_2^d_2 Q_3^d_3

where τ = τ_1 H_1 + τ_2 H_2 + τ_3 L. 
Since (L - H_i)|_X_0∼ 0, we have an injective homomorphism of NE(X_0) to NE() and the injective homomorphism of Novikov rings given by Q ↦ Q_1 Q_2 Q_3 where Q is a generator of the Novikov ring of X_0. 
Passing from the twisted I-function to the I-function of X_0, we obtain the desired form. 



The coefficient of z^0 in the I-function of X_0 is 

    ∑_d=0^∞( ∑_k=0^ddk^2 d+kk)^2 x^d

where x = e^τ Q. 
This is a Hadamard product of the period integrals of Shioda's modular surface associated with Γ_1(5) (see, for example, <cit.>). 




 §.§ The I-function of X_1

Similar to X_0, we can calculate the I-function of X_1. 
Let us recall that X_1 is a complete intersection of eight relative hyperplanes in the resolved join _×^2 ×^2((-H_1) (-H_2)) where H_1 is the pull-back of the Schubert divisor class of  and H_2 is the pull-back of the (1,1) class of ^2 ×^2. 


Let J^(τ, z) be the J-function of  as above. 
Let J^^2 ×^2(τ_1, τ_2, z) = e^τ_1 D_1+ τ_2 D_2/z∑_d_1, d_2 ≥ 0 J^^2 ×^2_d_1, d_2(τ_1, τ_2, z) Q_1^d_1 Q_2^d_2 be the J-function of ^2 ×^2 where D_i   (i=1,2) is the hyperplane class of the i-th factor of ^2 ×^2. 
Then the I-function of X_1 denoted by I_X_1(τ_1, τ_2, z) is 

    e^τ_1 D_1 + τ_2 D_2/z∑_d_1, d_2 ≥ 0∏_m=1^d_1+d_2 (L + m z)^5 J^_d_1 + d_2(0, z)|_X_1∏_m=1^d_1 + d_2 (L + m z)^3 J^^2 ×^2_d_1, d_2(τ_1, τ_2, z)|_X_1 Q_1^d_1 Q_2^d_2

where D_i is the divisor class on X_1 corresponding to the elliptic fibrations on X_1. 



The proof is similar to Proposition <ref>. We omit the details. 



The coefficient of z^0 in the I-function of X_1 is 

    ∑_d_1, d_2 ≥ 0( ∑_k=0^d_1 + d_2d_1 + d_2k^2 d_1 + d_2 + kk) (d_1 + d_2)!^3/(d_1!)^3 (d_2!)^3 x_1^d_1 x_2^d_2

where x_1 = e^τ_1 Q_1 and x_2 = e^τ_2 Q_2. 
This is a generalization of the usual Hadamard product to the case between the power series of one variable and the power series of two variables. 


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