Generating the Fukaya categories of compact toric varieties

Jack Smith

Generating the Fukaya categories of compact toric varieties

Symplectic Geometry 2023-08-10 v2

Authors: Jack Smith

Abstract

Let $X$ be a compact toric variety. The quantum cohomology of $X$ decomposes as a direct sum, and associated to each summand $Q$ is a toric fibre $L_Q$ with rank $1$ local system. By building an explicit twisted-complex-like object, we show that on $Q$ the Kodaira-Spencer isomorphism of Fukaya-Oh-Ohta-Ono factors through the closed-open string map to the Hochschild cohomology of $L_Q$ . We deduce that the latter is injective and hence, assuming an appropriate version of Abouzaid's criterion, that $L_Q$ split generates the corresponding summand of the Fukaya category.

Keywords

fukaya category toric variety quantum cohomology kodaira-spencer isomorphism closed-open map hochschild cohomology split generation

Cite

@article{arxiv.1804.06386,
  title  = {Generating the Fukaya categories of compact toric varieties},
  author = {Jack Smith},
  journal= {arXiv preprint arXiv:1804.06386},
  year   = {2023}
}

Comments

There are gaps in the discussion of the perturbation scheme in Section 3. The main results have subsequently been proved by different (better!) methods in arXiv:1910.01096 and arXiv:2308.03438

Generating the Fukaya categories of compact toric varieties

Abstract

Keywords

Cite

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