Homological mirror symmetry for higher dimensional pairs of pants

Abstract

Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in ${\mathbb{CP}}^n$, for $k\ge n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $(n+2)$-generic hyperplanes in $\mathbb{C}{P}^n$ ($n$-dimensional pair-of-pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_1{x}_2..x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of $n$-dimensional pants is equivalent to the derived category of $x_1{x}_2...x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair-of-pants.

arXiv comment41 pages, 10 figures. Typographical edits. To appear in Compositio Mathematica

References in corpus (7)

• Flat surfaces and stability structures
• An open string analogue of Viterbo functoriality
• A Geometric Approach to Orlov's Theorem
• Categorical resolutions of irrational singularities
• Fukaya categories of the torus and Dehn surgery
• Auslander orders over nodal stacky curves and partially wrapped Fukaya categories
• Mirror symmetry for honeycombs

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• Compatibility in Ozsvath-Szabo's bordered HFK via higher representations