# Homological Algebra of Mirror Symmetry

@article{Kontsevich1995HomologicalAO, title={Homological Algebra of Mirror Symmetry}, author={Maxim Kontsevich}, journal={arXiv: Algebraic Geometry}, year={1995}, pages={120-139} }

Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has
$$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$
.

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#### 1,195 Citations

Introduction to Homological Mirror Symmetry

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Mirror symmetry states that to every Calabi-Yau manifold \(X\) with complex structure and symplectic symplectic structure there is another dual manifold \(X^\vee \), so that the properties of \(X\)… Expand

Homological mirror symmetry with higher products

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We construct an $A_{\infty}$-structure on the Ext-groups of hermitian holomorphic vector bundles on a compact complex manifold. We propose a generalization of the homological mirror conjecture due to… Expand

Mirror Symmetry for hyperkaehler manifolds

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We prove the Mirror Conjecture for Calabi-Yau manifolds equipped with a holomorphic symplectic form. Such manifolda are also known as complex manifolds of hyperkaehler type. We obtain that a complex… Expand

Local Calabi–Yau manifolds of typeA˜via SYZ mirror symmetry

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- Journal of Geometry and Physics
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We carry out the SYZ program for the local Calabi--Yau manifolds of type $\widetilde{A}$ by developing an equivariant SYZ theory for the toric Calabi--Yau manifolds of infinite-type. Mirror geometry… Expand

Local mirror symmetry and type IIA monodromy of Calabi-Yau manifolds

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- 2000

We propose a monodromy invariant pairing Khol(X) H3(X _ ;Z) ! Q for a mirror pair of Calabi-Yau manifolds, (X; X _ ). This pairing is utilized implicitly in the previous calculations of the… Expand

Homological mirror symmetry of $\mathbb{F}_1$ via Morse homotopy

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- 2020

This is a sequel to our paper arXiv:2008.13462, where we proposed a definition of the Morse homotopy of the moment polytope of toric manifolds. Using this as the substitute of the Fukaya category of… Expand

Homological mirror symmetry at large volume

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A typical large complex-structure limit for mirror symmetry consists of toric varieties glued to each other along their toric boundaries. Here we construct the mirror large volume limit space as a… Expand

Mirror symmetry and deformation quantization

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- 2002

In homological mirror symmetry (see [Ko1]) and in the theory of D-modulesone meets similar objects. They are pairs (L,ρ) where L is a Lagrangianmanifold, and ρ is a ﬂat bundle on L (local system). In… Expand

Mirror symmetry for log Calabi-Yau surfaces I

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- 2011

We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family is… Expand

On the homological mirror symmetry conjecture for pairs of pants

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The n-dimensional pair of pants is defined to be the complement of n+2 generic hyperplanes in CP^n. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism… Expand

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