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.

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