Intrinsic mirror symmetry and categorical crepant resolutions

Abstract

The main result of the present paper concerns finiteness properties of Floer theoretic invariants on affine log Calabi-Yau varieties $X$. Namely, we show that: (a) the degree zero symplectic cohomology $S{H}^0(X)$ is finitely generated and is a filtered deformation of a certain algebra defined combinatorially in terms of a compactifying divisor $\mathbf{D}.$ (b) For any Lagrangian branes $L_0,L_1$, the wrapped Floer groups $W{F}^{\ast }(L_0,L_1)$ are finitely generated modules over $S{H}^0(X).$ We then describe applications of this result to mirror symmetry, the first of which is an ``automatic generation" criterion for the wrapped Fukaya category $\mathcal{W}(X)$. We also show that, in the case where $X$ is maximally degenerate and admits a ``homological section", $\mathcal{W}(X)$ gives a categorical crepant resolution of the potentially singular variety $\mathrm{Spec}(S{H}^0(X))$. This provides a link between the intrinsic mirror symmetry program of Gross and Siebert and the categorical birational geometry program initiated by Bondal-Orlov and Kuznetsov.

arXiv comment90 pages

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Cited by in corpus (2)

• The canonical wall structure and intrinsic mirror symmetry
• Homological Mirror Symmetry for local SYZ singularities