Fukaya categories of higher-genus surfaces and pants decompositions

Abstract

In this paper we prove a local-to-global principle for the Fukaya category of a closed Riemann surface $\Sigma$ of genus $g\ge 2$. We show that $\mathrm{Fuk}(\Sigma )$ can be glued from the Fukaya categories of the pairs-of-pants making up a pants decomposition of $\Sigma$. This extends our earlier results for the case of punctured Riemann surfaces. Our result has several interesting consequences: we obtain simple proofs of old and new HMS statements for Riemann surfaces, and establish a geometrization theorem for the objects of $\mathrm{Fuk}(\Sigma )$.

arXiv commentA correction has been made to the statement of Theorem 5.6. This affects Theorem 7.1 and its corollaries. 46 pages

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• Line fields on punctured surfaces and twisted derived categories