On the homological mirror symmetry conjecture for pairs of pants

Nick Sheridan

Research output: Contribution to journalArticle

27 Scopus citations

Abstract

The n-dimensional pair of pants is defined to be the complement of n + 2 generic hyperplanes in CPn. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism A1 algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with n+2 generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the Homological Mirror Symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (Cn+2,W), where W = z 1...zn+2. We show that the endomorphism A1 algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.

Original languageEnglish (US)
Pages (from-to)271-367
Number of pages97
JournalJournal of Differential Geometry
Volume89
Issue number2
DOIs
StatePublished - Dec 2011

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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