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Homological mirror symmetry for higher-dimensional pairs of pants
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 7 / July 2020
- Published online by Cambridge University Press:
- 18 June 2020, pp. 1310-1347
- Print publication:
- July 2020
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- Article
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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.
