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Elliptic Curves Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 024, 13 pages Doran{Harder{Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves Atsushi KANAZAWA Department of Mathematics, Kyoto University, Kitashirakawa-Oiwake, Sakyo, Kyoto, 606-8502, Japan E-mail: [email protected] Received December 20, 2016, in final form April 06, 2017; Published online April 11, 2017 https://doi.org/10.3842/SIGMA.2017.024 Abstract. We prove the Doran{Harder{Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi{ Yau manifold X degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi{Yau manifold of X can be constructed by gluing the two mirror Landau{ Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau{Ginzburg superpotentials. Key words: Calabi{Yau manifolds; Fano manifolds; SYZ mirror symmetry; Landau{Ginz- burg models; Tyurin degeneration; affine geometry 2010 Mathematics Subject Classification: 53D37; 14J33; 14J32; 14J45; 14D06 1 Introduction The aim of this short article is to prove the Doran{Harder{Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. Given a Tyurin degeneration of a Calabi{Yau manifold X to a union of two quasi-Fano manifolds X1 [Z X2 intersecting along a common smooth anti-canonical divisor Z 2 |−KXi j for i = 1; 2, it is natural to investigate a potential relationship between geometry of the Calabi{ Yau manifold X and that of the quasi-Fano manifolds X1 and X2. Motivated by the works of Dolgachev [5], Tyurin [21] and Auroux [3], recently Doran{Harder{Thompson proposed a re- markable conjecture (Conjecture 2.3), which builds a bridge between mirror symmetry for the Calabi{Yau manifold X and that for the quasi-Fano manifolds X1 and X2 [6]. It claims that we should be able to glue together the mirror Landau{Ginzburg models Wi : Yi ! C of the pair (Xi;Z) for i = 1; 2 to construct a mirror Calabi{Yau manifold Y of X equipped with a fibration 1 W : Y ! P . They provided supporting evidence for the conjecture in various different set- tings. For instance it was shown that under suitable assumptions we can glue together the 1 Landau{Ginzburg models Wi : Yi ! C for i = 1; 2 to obtain a C -manifold Y with the expected topological Euler number χ(Y ) = (−1)dim X χ(X). Thus the topological version of the conjecture is essentially proven. However, the real difficulty of this conjecture lies in constructing Y as a complex manifold, which should be mirror to the symplectic manifold X (or vice versa). In this article, we will prove the conjecture in the case of elliptic curves, beginning with a sym- plectic elliptic curve X and constructing the mirror complex elliptic curve Y . In order to obtain the correct complex manifold Y by gluing the mirror Landau{Ginzburg models Wi : Yi ! C for i = 1; 2, we need to keep track of the subtle complex structures of the (not necessarily algebraic) K¨ahlermanifold Yi for i = 1; 2. To this end, we find ideas of SYZ mirror symmetry very useful. 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.html 2 A. Kanazawa The key idea in our proof is twofold. The first is to obtain the correct complex structure by gluing the underlying affine base manifolds of X1 and X2 in SYZ mirror symmetry. This is based on the philosophy that a Tyurin degeneration of a Calabi{Yau manifold X can be thought to be fibred over a Heegaard splitting of the base B of a special Lagrangian torus fibration φ: X ! B. The second is to construct theta functions out of the Landau{Ginzburg superpotentials. As a corollary, we observe that the product formulae of the theta functions are the manifestation of quantum corrections appearing in SYZ mirror symmetry. It is worth mentioning that a variant of the conjecture was discussed in the work of Auroux [3]. He studied a Z=2Z-invariant version of mirror symmetry (Calabi{Yau double covers) and inves- tigated elliptic curves from a different perspective from ours [3, Example 3.2]. Advantages of the present work are firstly to work with not necessarily identical affine manifolds (doubling) and secondly to construct theta functions out of the Landau{Ginzburg superpotentials in an interesting way based on the geometry of the conjecture. Structure of article We will provide a self-contained description of the subjects for completeness. Section2 builds basic setup and formulates the main conjecture. Section3 reviews basics of SYZ mirror symme- try both in the Calabi{Yau and quasi-Fano settings. Section4 is the main part of this article and proves the conjecture in the case of elliptic curves. Section5 comments on further research directions. 2 Doran{Harder{Thompson conjecture In this section, we will provide background materials and review the Doran{Harder{Thompson conjecture, following the original article [6]. In general mirror symmetry is a conjecture about a Calabi{Yau manifold near a large complex structure limit, which is thought to be a maximal degeneration, in the complex moduli space. However in this article we will be interested in another class of loci in the complex moduli space, where a Calabi{Yau manifold degenerates to a union of two quasi-Fano manifolds. 2.1 Tyurin degeneration A Calabi{Yau manifold X is a compact K¨ahlermanifold such that the canonical bundle is i trivial KX = 0 and H (X; OX ) = 0 for 0 < i < dim X. A quasi-Fano manifold X is a smooth i variety X such that |−KX j contains a smooth Calabi{Yau member and H (X; OX ) = 0 for 0 < i. A Tyurin degeneration is a degeneration X! ∆ of Calabi{Yau manifolds over the unit disc ∆ = fjzj < 1g ⊂ C, such that the total space X is smooth and the central fibre X0 = X1 [Z X2 is a union of two quasi-Fano manifolds X1 and X2 intersecting normally along a common anti-canonical divisor Z 2 |−KXi j for i = 1; 2. Conversely, we have the following result of Kawamata{Namikawa [15], which is slightly modified for our setting. Theorem 2.1 ([15, Theorem 4.2]). Let X1 and X2 be quasi-Fano manifolds and Z 2 |−KXi j a common smooth anti-canonical divisor for i = 1; 2. Assume that there exist ample divisors Di 2 Pic(Xi) for i = 1; 2 which restrict to an ample divisor D1jZ = D2jZ on Z. Then the union X1 [Z X2 of X1 and X2 intersecting normally along Z is smoothable to a Calabi{Yau manifold X if and only if N =∼ N −1 (d-semistability). Moreover the resulting Calabi{Yau Z=X1 Z=X2 manifold X is unique up to deformation. A Tyurin degeneration is thought to be a complex analogue of a Heegaard splitting of a com- pact oriented real 3-fold without boundary. Based on this analogy, in his posthumous article [21] Doran{Harder{Thompson Conjecture via SYZ Mirror Symmetry: Elliptic Curves 3 Tyurin proposed to study geometry of a Calabi{Yau 3-fold by using that of quasi-Fano 3-folds when they are related by a Tyurin degeneration. 2.2 Mirror symmetry for quasi-Fano manifolds We consider a pair (X; Z) consisting of a quasi-Fano manifold X and an anti-canonical divisor Z 2 |−KX j. The complement X n Z can be thought of as a log Calabi{Yau manifold as there exists a nowhere vanishing volume form Ω on X n Z with poles along Z. Example 2.2. Let X be a toric Fano n-fold and Z the toric boundary, which is the complement × n × n of the dense torusp (C ) ⊂ X. Then X n Z = (C ) carries a standard holomorphic volume n × n form Ω = ^i=1 −1d log zi, where (zi) are the coordinates of (C ) . Definition 2.3. A Landau{Ginzburg model is a pair (Y; W ) of a K¨ahlermanifold Y and a holo- morphic function W : Y ! C, which is called a superpotential. It is classically known that there is a version of mirror symmetry for Fano manifolds together with an anti-canonical divisor. We expect that such mirror symmetry should hold also for quasi-Fano manifolds (or even for varieties with effective anti-canonical divisors [2]). Here we formulate a mirror conjecture for quasi-Fano manifolds (see for example Katzarkov{Kontsevich{ Pantev [14], Harder [9]). Conjecture 2.4. For a pair (X; Z) of a quasi-Fano n-fold X and a smooth anti-canonical divisor Z 2 |−KX j, there exists a Landau{Ginzburg model (Y; W ) such that P n−i+j;j i −1 1) j h (X) = h (Y; W (s)) for a regular value s 2 C of W , 2) the generic fibres of W and the generic anti-canonical hypersurfaces in X are mirror families of compact Calabi{Yau (n − 1)-folds, where hi(Y; W −1(s)) is the rank of the relative cohomology group Hi(Y; W −1(s)). The pair (Y; W ) is called a mirror Landau{Ginzburg model of (X; Z). The anti-canonical divisor Z can be thought of as an obstruction for the quasi-Fano mani- fold X to be a Calabi{Yau manifold and W is an obstruction (or potential function) for the Floer homology of a Lagrangian torus in X to be defined in the sense of Fukaya{Oh{Ohta{Ono [4,7] as we will see in the next section.
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