Homological mirror symmetry for log Calabi-Yau surfaces

Abstract

Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w:M\to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b\text{Coh}(Y)$, and the wrapped Fukaya category $D^b\mathcal{W}(M)$ is isomorphic to $D^b\text{Coh}(Y\backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D$. We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y,D)$, notably in work of Auroux-Katzarkov-Orlov and Abouzaid.

arXiv commentThis is the final version before publication, incorporating the appendix by Lutz, previously missing from the arxiv. Main article by Hacking and Keating. Comments welcome!

References in corpus (24)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Uniqueness of enhancement for triangulated categories
• Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
• Homogeneous coordinate rings and mirror symmetry for toric varieties
• Moduli of surfaces with an anti-canonical cycle
• On partially wrapped Fukaya categories
• Infinitely many exotic monotone Lagrangian tori in CP^2
• Legendrian Fronts for Affine Varieties
• Lagrangian isotopy of tori in $S^2\times S^2$ and $\mathbb{C}{P}^2$
• On the combinatorics of exact Lagrangian surfaces
• Lecture notes on stabilization of contact open books
• SYZ mirror symmetry for toric Calabi-Yau manifolds
• Existence of Lefschetz fibrations on Stein and Weinstein domains
• Arithmetic mirror symmetry for the 2-torus
• On the geometry of anticanonical pairs
• Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds
• Smoothings and Rational Double Point Adjacencies for Cusp Singularities
• Floer cohomology in the mirror of the projective plane and a binodal cubic curve
• Combinatorial aspects of exceptional sequences on (rational) surfaces
• Mirror Symmetry And Quantum Geometry
• SYZ mirror symmetry for toric varieties
• An introduction to Weinstein handlebodies for complements of smoothed toric divisors
• Exact Lefschetz fibrations associated with dimer models

Cited by in corpus (11)

• The canonical wall structure and intrinsic mirror symmetry
• Stable maps to Looijenga pairs
• An introduction to Weinstein handlebodies for complements of smoothed toric divisors
• Intrinsic mirror symmetry and categorical crepant resolutions
• Mirror Symmetry for Truncated Cluster Varieties
• Weinstein handlebodies for complements of smoothed toric divisors
• Symplectomorphisms and spherical objects in the conifold smoothing
• Mirrors to Del Pezzo Surfaces and the Classification of $T$-Polygons
• Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings
• A cone conjecture for log Calabi-Yau surfaces
• Symplectomorphisms of some Weinstein 4-manifolds