A Note on Schoen's Calabi--Yau Threefolds
Hülya Argüz Thomas Prince


Schoen's Calabi–Yau Threefolds]Cohomology of the fixed point locus of an anti-symplectic involution on Schoen's Calabi–Yau


Labaratoure de Mathématiques de Versailles, Université de Versailles Saint-Quentin-en-Yvelines
Versailles, 78000
France
nuromur-hulya.arguz@uvsq.fr


Mathematical Institute
University of Oxford
Woodstock Road
Oxford
OX2 6GG
UK
thomas.prince@magd.ox.ac.uk




We study the topology of a real Lagrangian in Schoen's Calabi–Yau threefold X and compute its mod 2 cohomology using two methods; first via a concrete Mayer–Vietoris calculation, then by an exact sequence relating the mod 2 cohomology of the real Lagrangian to the cohomology of X. We conclude that these two methods agree. This in particular corrects a previous computation made by Castaño-Bernard–Matessi.

[
    Thomas Prince
    December 30, 2023
=====================





§ INTRODUCTION

Schoen's Calabi–Yau threefold X <cit.> is the fibred product of two rational elliptic surfaces over ^1, and is of particular interest from the point of view of the Strominger–Yau–Zaslow conjecture in mirror symmetry <cit.>. Kovalev has described a Lagrangian 3-torus fibration on X <cit.>, which has further been studied by Gross in <cit.> from the perspective of toric degenerations of Calabi–Yau toric complete intersections. In particular, Gross describes a singular torus fibration on X using constructions given in <cit.>. 

Later on, Castaño-Bernard–Matessi–Solomon studied anti-symplectic involutions on the total space of such torus fibrations in <cit.>. The fixed point set of such involutions are real Lagrangians. As these Lagrangians provide an algebro-geometric path to open Gromov-Witten invariants and the Fukaya category <cit.>, understanding their topology attracts significant interest from both symplectic and algebraic geometers. The mod 2 cohomology of real Lagrangians is particularly widely studied in the literature due to its deep connections to real enumerative algebraic geometry  <cit.>. 

For the case of the real Lagrangian in Schoen's Calabi–Yau X, Castaño-Bernard–Matessi outline the computation of the cohomology of a real Lagrangian in <cit.> via an argument based on the Mayer–Vietoris theorem. Here we first detail this Mayer-Vietoris computation by further investigating the topology of the torus fibration on Schoen's Calabi–Yau threefold, noting that our result differs slightly from the one obtained in <cit.> (in which H^1(Σ, _2) ≅_2^34). We explain this discrepancy below, and give a detailed account of the Mayer–Vietoris calculation outlined in <cit.>. Our main result is the following:


The mod 2 cohomology groups of the real Lagrangian Σ⊂ X are given by

    H^0(Σ,_2) = H^3(Σ,_2) ≅_2,    and    H^1(Σ,_2) = H^1(Σ,_2) ≅_2^36.


We provide an alternative proof to this result using techniques developed in <cit.>, where we give a general framework to compute the mod 2 cohomology groups of real Lagrangians in a Calabi–Yau X, and its mirror X̆, which admit (singular) torus fibrations

    f X → B,    and   f̆X̆→ B

over a _2 homology sphere B. We let ι denote the anti-symplectic involution on X sending x ↦ -x in each fibre. Our main result in <cit.> involves the comparison of the following maps:

	
  * The map D ↦ D^2, where D ∈ H^1(B, R^1f̆_⋆_2).
	
  * The map β H^1(B, R^2f_⋆_2) → H^2(B, R^1f_⋆_2); the connecting homomorphism introduced by Castaño-Bernard–Matessi in <cit.> associated to the short exact sequence 

    0 → R^1f_⋆_2⊕^2_2 →π_⋆_2 → R^2f_⋆_2 → 0,

where π denotes the restriction of f to the fixed locus L_ of ι.

We prove in <cit.> that the linear maps β and  coincide. Applying this result to the case of Schoen's Calabi–Yau, we verify Theorem <ref>. This result also provides the first instance of a real Lagrangian in Calabi–Yau with Picard number greater than one which provides positive evidence for Conjecture 1.3 in <cit.>, stating that the cohomology of real Lagrangians in Calabi–Yau toric hypersurfaces are 2-primary.






 §.§ Acknowledgements

TP was supported by a Fellowship by Examination at Magdalen College, Oxford and HA was supported by the Fondation Mathématiques Jacques Hadamard.



§ A REAL LAGRANGIAN IN SCHOEN'S CALABI–YAU




Take an elliptically fibred K3 surface over ^1, with 24 singular fibres
of Kodaira type I_1. Let γ be a simple closed curve bounding a disk D^2 containing the image of 12 singular fibers and not intersecting any other critical value on the base. Let M^4 be the preimage of D^2, and set X := M^4 × T^2, where T^2 denotes the standard 2-torus. It is shown in <cit.> that Schoen's Calabi–Yau X is diffeomorphic to a fibered coproduct of two copies of X along their boundary.



 There is a fibration on X over the solid torus D^2 × S^1, obtained by taking the product of the fibration on M^4 with the standard S^1 fibration on T^2. This gives a torus fibration on X over the three sphere S^3, expressed as the union of the solid tori 
 
    B_i=D^2 × S^1,

 for 1≤ i ≤ 2, the bases of the fibrations on the two copies of X forming X. 

Consider an involution on M^4 obtained by restricting a fiber-preserving anti-symplectic, and anti-holomorphic involution on the K3 surface and take the involution on T^2 which preserves the fibres of the fibration on S^1. The product of these two involutions induces an anti-symplectic involution on X, and the fixed point locus Σ, which is naturally a real Lagrangian, is an 8-to-1 cover

    σΣ→ B ≅ S^3,
 
branched over 24 circles, as described in <cit.>. 



 §.§ Proof of Theorem <ref> via Mayer-Vietoris

 We set A_i := σ^-1(B_i) for each i ∈{1,2}; we will apply the Mayer–Vietoris theorem to the decomposition Σ = A_1 ∪ A_2.

First note that the fixed point locus on the fiber-preserving anti-symplectic involution on M^4 is obtained, as in <cit.>, as the dishoint uunion of a 2-disc E and a genus 4 surface with three disks removed, denoted by S. The boundary components of S are contained in the preimage of the boundary of D^2. Recalling that B_i ≅ D^2 × S^1 for each i ∈{1,2} we obtain that each A_i is the disjoint union of two copies of S× S^1, and two copies of E × S^1.





We let E^1_i × S^1, S^1_i × S^1, E^2_i × S^1, S^2_i × S^1 denote the four components of A_i for each i ∈{1,2}. Following <cit.> we may identify fibres of σ with the half integral points in ^2/^2 and ^3/^3. Note that, similarly as in <cit.>, since the monodromy always leaves one point on each fiber invariant, the fixed point locus Σ has two connected components: one is homeomorphic to the base S^3 and the other is a 7-to-1 branched cover over S^3.





For each i ∈{1,2}, the boundary of A_i consists of four two-dimensional tori (given by ∂ (E^j_i × S^1) and ∂ (S^j_i× S^1) for j ∈{1,2} respectively). Following the constructions described in <cit.> and <cit.>, the components of A_1∩ A_2 can be indexed by T_i, for 0≤ i ≤ 7, where each T_i is the 2-torus defined as follows:

	
  * T_0 bounds a copy of E^1_i × S^1 in A_i for each i ∈{1,2}.
	
  * T_4 bounds an the solid torus E^2_1 × S^1 in A_1.
	
  * T_1, T_2, and T_3 are boundary components of S^1_1× S^1 ⊂ A_1.
	
  * T_5, T_6, and T_7 are boundary components of S^2_1 × S^1 ⊂ A_1.
	
  * T_3 bounds the solid torus E^2_2 × S^1 in A_2.
	
  * T_1, T_4, and T_5 are boundary components of S^1_2× S^1 ⊂ A_2.
	
  * T_2, T_6, and T_7 are boundary components of S^2_2× S^1 ⊂ A_2.
































	

	



Now consider the following part of the Mayer–Vietoris sequence for the decomposition X = A_1 ∪ A_2:

    H_1(Σ,_2) → H_0(A_1∩ A_2,_2) → H_0(A_1,_2)⊕ H_0(A_2,_2) → H_0(Σ,_2) → 0.

Using the descriptions of A_1, A_2, and X given above, this sequence has the form

    H_1(Σ,_2) →_2^8 →^4_2 ⊕^4_2 →_2^2 → 0.

The preceding part of the Mayer–Vietoris sequence has the form

    H_1(A_1∩ A_2,_2) → H_1(A_1,_2)⊕ H_1(A_2,_2) → H_1(Σ,_2).

Computing these spaces, this sequence becomes

    ^16_2 φ^24_2 ⊕^24_2  H_1(Σ,_2).

We let φ = φ_1⊕φ_2 denote the components of the map φ in (<ref>). Note that it follows directly from (<ref>) and (<ref>) that H_1(Σ,_2) = 34 + k, where k =(φ). We now describe a system of linear equations defining the kernel of φ.

Since A_1 ∩ A_2 is the disjoint union of eight two-dimensional tori, the vector space H_1(A_1∩ A_2,_2) is generated by eight pairs x_i,y_i ∈ H_1(T_i,_2) for i ∈{0,…,7}.  Hence any element of H_1(A_1∩ A_2,_2) can be expressed as a sum of the form ∑_i=0^7(a_ix_i+b_iy_i), where a_i and b_i are elements of _2.

We choose the bases {x_i,y_i} of each component of H_1(A_1∩ A_2,_2) such that φ_1(y_i) generates the first homology group of the S^1 factor of the corresponding connected component of A_1 (recalling that this component is homeomorphic to either E^j_1 × S^1 or S^j_1 × S^1 for some j ∈{1,2}) and φ_2(x_i) generates the first homology group of the S^1 factor of the corresponding connected component of A_2. Moreover, the homology classes φ_1(x_i) and φ_2(y_i) are either meridian curves on a solid torus, or the classes of boundary circles of one of the three discs removed from the punctured genus four surface S.


	
	Consider a class ∑_i=0^7(a_ix_i+b_iy_i), for a_i and b_i in _2. This class is contained in the kernel of φ if and only if the following equations hold:

	
  * a_0=b_0=0.
	
  * b_4=0.
	
  * b_1+b_2+b_3=0 and a_1=a_2=a_3.
	
  * b_5+b_6+b_7=0 and a_5=a_6=a_7.
	
  * a_3=0.
	
  * a_1+a_4+a_5=0 and b_1=b_4=b_5.
	
  * a_2+a_6+a_7=0 and b_2=b_6=b_7.



	These equations each follow directly from the above description of the 3-manifolds bounded by each T_i for i ∈{0,…, 7}. For example, considering the equations in item (1), both x_0 and y_0 are the classes of longitudes of a solid torus and hence both a_0 and b_0 must vanish.
	
	Considering the equations in item (3), we recall that T_1, T_2, and T_3 are the boundary components of a copy of S × S^1 ⊂ A_1. The classes y_1, y_2, and y_3 are all mapped (by φ_1) to the multiples of the S^1 factor of S^1_1 × S^1, and hence the component of φ_1(∑_i=0^7(a_ix_i+b_iy_i)) corresponding to this factor is equal to the class of φ_1(b_1y_1+b_2y_2+b_3y_3) = b_1+b_2+b_3, which must vanish. The classes x_1, x_2, and x_3 correspond to boundary components of the surface S^1_1. The only linear combination ∑^3_i=1a_ix_i of these classes which are boundaries of 2-chains in S^1_1 are multiples of x_1+x_2+x_3 (which is the class of the boundary of S^1_1 itself). That is, ∑^3_i=1a_ix_i = λ (x_1+x_2+x_3) for some λ; hence a_1=a_2=a_3. Similar computations verify the equations in items (4), (6), and (7).


Taken as equations over the integers, the equations described in Lemma <ref> have no non-zero solutions, and hence H^1(Σ,) ≅^34 (in agreement with the calculation of <cit.>). However, over _2, we find a 2-dimensional space of solutions generated by:

	
  * a_i = 0 for all i ∈{0,…,7}, b_0=b_1=b_4=b_5=0, and b_2=b_3=b_6=b_7=1,
	
  * b_i = 0 for all i ∈{0,…,7}, a_0=a_1=a_2=a_3=0, and a_4=a_5=a_6=a_7=1.


That is, we find that k=2, H_1(Σ,_2) ≅_2^36, and hence H^1(Σ,_2) ≅_2^36.



 §.§ Proof of Theorem <ref> via the squaring map


Consider the following exact sequence of sheaf cohomology groups introduced in <cit.>:

    0 → H^1(B,R^1f_⋆_2) → H^1(B,σ_⋆_2) → H^1(B,R^2f_⋆_2) β→ H^2(B,R^1f_⋆_2).

Noting that Schoen's Calabi–Yau X can be described as a complete intersection in a toric variety <cit.>, it is simply-connected by the Lefschetz hyperplane theorem; in this case, H^1(B,R^1f̆_⋆_2) ≅ H_2
(X, _2). Moreover, the mirror to Schoen's Calabi–Yau X is itself homeomorphic to X (see <cit.>). Hence the ranks of H^1(B,R^1f̆_⋆_2) and H^1(B,R^2f_⋆_2) agree and <cit.> implies that the rank of the connecting homomorphism

    β H^1(B,R^2f_⋆_2) → H^2(B,R^1f_⋆_2)

is equal to the rank of the squaring map D ↦ D^2 from H^2(X,_2) to H^4(X,_2) which takes an element D ∈ H^2(X,_2) to D⌣ D ∈ H^4(X,_2). Note that this map is linear over _2.

As discussed in the introduction, Schoen's Calabi–Yau is the fibre product of a pair of rational elliptic surfaces E_1 and E_2
	
    X ^π_2[r] ^π_1[d]     E_2 [d] 
    
    	E_1 [r]    ^1.

	In particular, H^2(X,_2) ≅_2^19 is spanned by vectors in the 10 dimensional subspaces π_1^⋆(H^2(E_1,_2)) ≅_2^10 and π_2^⋆(H^2(E_2,_2)) ≅_2^10. These spaces intersect in the class of a fibre of either composition X →^1 in (<ref>).
	
	Since (π_i^⋆(α))^2 = π_i^⋆(α^2), for any class α∈ H^2(E_i,_2) and i ∈{1,2}, the image of  is spanned by π_1^⋆(H^4(E_1,_2)) ≅_2 and π_2^⋆(H^4(E_2,_2)) ≅_2. That is, the rank of  is equal to 2 and, by <cit.>, the rank of β is also equal to 2.

In particular, we can conclude from (<ref>) that H^1(B,σ_⋆_2) has dimension 19+(19-2)=36. Hence, from the Leray spectral sequence for σ, we have that H^1(B,σ_⋆_2) ≅ H^1(Σ, _2) ≅_2^36, verifying Theorem <ref>.


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