This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the degenerate fibre is a union of toric varieties meeting along toric strata. In addition, the total space must look locally toric outside of some singular set. This is a generalisation of both Mumford-type degenerations of abelian varieties and degenerations of Calabi-Yau hypersurfacesin toric varieties. From such a degeneration, we construct a dual intersection graph along with a singular affine structure. We also explain how to reverse this procedure, using logarithmic structures. We then use a discrete Legendre transform on these affine manifolds to recover mirror symmetry. Finally, we discuss the relationship between this construction and more classical forms of SYZ.
The details of the ideas presented here will appear elsewhere.
We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.
The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti [arXiv:0709.1453, arXiv:0807.0597] relates all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifolds/3-orbifolds to the Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. In this paper, we present a proof of the Remodeling Conjecture for open-closed orbifold Gromov-Witten invariants of an arbitrary affine toric Calabi-Yau 3-orbifold relative to a framed Aganagic-Vafa Lagrangian brane. This can be viewed as an all genus open-closed mirror symmetry for affin…
We study integral dlt models of a proper C((t))-variety X along a toric stratum of the special fiber. We prove that the associated Berkovich retraction - from the non-archimedean analytification of X onto the dual complex of the model - is an affinoid torus fibration around the simplex corresponding to the toric stratum, which extends results from Nicaise-Xu-Yu. This allows us to construct new types of non-archimedean retractions for maximally degenerate families of quartic K3 surfaces and quintic 3-folds, by gluing several non-archimedean SYZ fibrations, each one toric along a codimension one…
This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault c…
Mirror symmetry of Calabi-Yau manifolds can be understood via a Legendre duality between a pair of certain affine manifolds with singularities called tropical manifolds. In this article, we study conifold transitions from the point of view of Gross and Siebert. We introduce the notions of tropical nodal singularity, tropical conifolds, tropical resolutions and smoothings. We interpret known global obstructions to the complex smoothing and symplectic small resolution of compact nodal Calabi-Yaus in terms of certain tropical $2$-cycles containing the nodes in their associated tropical conifolds.…
We show that (quantum) D-modules of Givental type on affine flag varieties are transformed to D-modules of Beilinson-Drinfeld type on $n$-reduced Sato Grassmannians via affine Miura transformation.
This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certain sorts of degenerations of Calabi-Yau manifolds a log Calabi-Yau space, which is a log structure on the degenerate fibre. Then many statements about mirror symmetry which one hopes to be true for the general fibre should first be proved for this log CY space. In this paper we be…
Given any smooth cubic curve $E\subseteq \mathbb{P}^2$, we show that the complex affine structure of the special Lagrangian fibration of $\mathbb{P}^2\setminus E$ constructed by Collins--Jacob--Lin arXiv:1904.08363 coincides with the affine structure used in Carl--Pomperla--Siebert for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl--Pomperla--Siebert.
Following the work of Castano-Bernard and Matessi on conifold transition in the Gross-Siebert program, we construct orbi-conifold transitions of the Shoen's Calabi-Yau threefold and their mirrors. The construction glues together the local models for orbi-conifold transitions in the previous work with Kanazawa.
We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dxdy/xy over 2-chains whose boundaries lie on B_ϕwhere {B_ϕ} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for projective plane in a Calabi-Yau 3-fold by Chiang-Klemm-Yau-Zaslow.
We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. W…
A classical result of Miyanishi-Sugie and Keel-McKernan asserts that for smooth affine surfaces, affine-uniruledness is equivalent to affine-ruledness, both properties being in fact equivalent to the negativity of the logarithmic Kodaira dimension. Here we show in contrast that starting from dimension three, there exists smooth affine varieties which are affine-uniruled but not affine-ruled.
In this paper we classify all the compactifications of affine homology $3$-cells into the blow-ups of the projective $3$-space along smooth curves such that the log canonical divisors are linearly trivial. As a result, we prove that each embedded affine $3$-fold is isomorphic to the affine $3$-space except one example.
The 2-dimensional Lyness map is a 5-periodic birational map of the plane which may famously be resolved to give an automorphism of a log Calabi-Yau surface, given by the complement of an anticanonical pentagon of $(-1)$-curves in a del Pezzo surface of degree 5. This surface has many remarkable properties and, in particular, it is mirror to itself. We construct the 3-dimensional big brother of this surface by considering the 3-dimensional Lyness map, which is an 8-periodic birational map. The variety we obtain is a special (non-$\mathbb Q$-factorial) affine Fano 3-fold of type $V_{12}$, and we…
We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which…
The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over a polynomial ring. Here we give the mirror to this description, and in particular, a clean new proof of mirror symmetry for smooth toric stacks.
We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of $A_\infty$-structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an $n$-dimensional affine hypersurface of degree $n + 2$ and the double cover of the $n$-dimensional affine space br…
T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this article, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.
We correct the definitions and descriptions of the integral structures in [U14]. The previous flat basis in [ibid] is characterized by the Frobenius solutions and integral in the first approximation by mean of the graded quotients of monodromy filtration, but not integral in the strict sense. In this article, we use the integral structure of Iritani in [I11] for A-model. Using this precise version, we study open mirror symmetry for quintic threefolds through log mixed Hodge theory, especially the recent result on Neron models for admissible normal functions with non-torsion extensions in the j…
A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half…
Mirror symmetry for del Pezzo surfaces was studied by Auroux, Katzarkov and Orlov who suggested that the mirror should take the form of a Landau-Ginzburg model with a particular type of elliptic fibration. This problem was then considered again but from an algebro-geometric perspective by Gross, Hacking and Keel. Their construction allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. In the case of $S$ a Fano surface they proved that this family may be l…
We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.
We describe mirror symmetry for weak toric Fano manifolds as an equivalence of D-modules equipped with certain filtrations. We discuss in particular the logarithmic degeneration behavior at the large radius limit point, and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauss-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D-module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined …
Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This proves a variant of the Frobenius structure conjecture by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is supposed to give the hypothetical mirror family. Although the statement of our theorem involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constan…
We introduce tropical corals, balanced trees in a half-space, and show that they correspond to holomorphic polygons capturing the product rule in Lagrangian Floer theory for the elliptic curve. We then prove a correspondence theorem equating counts of tropical corals to punctured log Gromov--Witten invariants of the Tate curve. This implies that the homogeneous coordinate ring of the mirror to the Tate curve is isomorphic to the degree-zero part of symplectic homology, confirming a prediction of homological mirror symmetry.
In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological Mirror conjecture and geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from "flat pieces" an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group $SO(1,18)$ by piecewise-linear transformations on the 2-dimensional sphere $S^2$ equipped with naturally defined singular affine structure.
Given a punctured Riemann surface with a pair-of-pants decomposition, we compute its wrapped Fukaya category in a suitable model by reconstructing it from those of various pairs of pants. The pieces are glued together in the sense that the restrictions of the wrapped Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree. The $A_\infty$-structures are given by those in the pairs of pants. The category of singularities of the mirror Landau-Ginzburg model can also be constructed in the same way from local affine pieces that are mirrors of the pairs of pants.
We give a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. The main result of this article is the proof that the topological Calabi-Yau compactifications of M. Gross' "Topological Mirror Symmetry", can be made into symplectic compactifications. As an example, we obtain a pair of compact symplectic 6-manifolds together with Lagrangian fibrations whose underlying affine structures are dual. The symplectic manifolds obtained are homeomorphic to a smooth quintic Calabi-Yau 3-fold and its mirro…
We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a special Lagrangian torus fibration in the Strominger-Yau-Zaslow conjecture. We study the topological torus fibration in the large complex structure limit and show that it coincides with our combinatorial model.
We use a formula of Bultot to compute the motivic zeta function for the toric degeneration of the quartic K3 and its Gross-Siebert mirror dual degeneration. We check for this explicit example that the identification of the logarithm of the monodromy and the mirror dual Lefschetz operator works at an integral level.
The mirror of a toric orbifold is an affine curve called the mirror curve. In this paper, firstly, we recall the basic tools in tropical geometry and give a definition of the mirror curve. Then we calculate the tropical spine of the mirror curve for a smooth toric Calabi-Yau 3-fold near the large radius limit. Finally, we recall a special kind of real algebraic curves called the cyclic-M curve and show that under some special choices of parameters near the large radius limit, the mirror curve is a cyclic-M curve. Applying Mikhalkin's result of cyclic-M curves, we show that the mirror curve is …
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.
This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal boundary D or to maximally unipotent degenerations of Calabi-Yau manifolds. The new ingredient is a notion of "punctured Gromov-Witten invariant", currently in progress with Abramovich and Chen. The mirror to a pair (X,D) is constructed as the spectrum of a ring defined using the punctured invariants of (X,D). An analogous construction leads to mirrors of Calabi-Yau manifolds. This can be viewed a…
We prove that if Y is a hypersurface of degree d in P^n with isolated singularities, then the log canonical threshold of (P^n,Y) is at least min{n/d,1}. Moreover, if d is at least n+1, then we have equality if and only if Y is the projective cone over a (smooth) hypersurface in P^{n-1}. In the case when Y is a hyperplane section of a smooth hypersurface in P^{n+1}, Cheltsov and Park have proved that Y has isolated singularities and they have obtained the above lower bound for the log canonical threshold. Moreover they made the conjecture about the equality case (for d=n+1) and they proved that…
Gross, Hacking, and Keel have constructed mirrors of log Calabi-Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher genus log Gromov-Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding noncommutative algebras of functions.
