Homological mirror symmetry for higher-dimensional pairs of pants
Lekili, Yanki, Polishchuk, Alexander, Lekili, Yanki
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 $k\geqslant n$ , 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 $x_{1}x_{2}\ldots x_{n+1}=0$ . By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$ -dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$ . We also prove similar equivalences for finite abelian covers of the $n$ -dimensional pair of pants.