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Hochschild cohomology of the Fukaya category via Floer cohomology with coefficients (2308.03438v3)

Published 7 Aug 2023 in math.SG

Abstract: Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map $CO_\lambda : QH*(X) \to HH*(\mathcal{F}(X)_\lambda)$ to a variant of the length-zero closed-open map on $L$ incorporating $\mathbf{k}[H_1(L; \mathbb{Z})]$ coefficients, denoted $CO0_\mathbf{L}$. The former is categorically important but very difficult to compute, whilst the latter is geometrically natural and amenable to calculation. We further show that, after a suitable completion, injectivity of $CO0_\mathbf{L}$ implies injectivity of $CO_\lambda$. Via Sheridan's version of Abouzaid's generation criterion, this gives a powerful tool for proving split-generation of the Fukaya category. We illustrate this by showing that the real part of a monotone toric manifold (of minimal Chern number at least 2) split-generates the Fukaya category in characteristic 2. We also give a short new proof (modulo foundational assumptions in the non-monotone case) that the Fukaya category of an arbitrary compact toric manifold is split-generated by toric fibres.

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