Automatically generating Fukaya categories and computing quantum cohomology

Abstract

Suppose one has found a non-empty sub-category $\mathcal{A}$ of the Fukaya category of a compact Calabi-Yau manifold $X$ which is homologically smooth in the sense of non-commutative geometry, a condition intrinsic to $\mathcal{A}$. Then, we show $\mathcal{A}$ split-generates the Fukaya category and moreoever, that our hypothesis implies (and is therefore equivalent to the assertion that) $\mathcal{A}$ satisfies Abouzaid's geometric generation criterion [Abo]. An immediate consequence of earlier work [G1, GPS1, GPS2] is that the open-closed and closed-open maps, relating quantum cohomology to the Hochschild invariants of the Fukaya category, are also isomorphisms. Our result continues to hold when $c_1(X) \neq 0$ (for instance, when $X$ is monotone Fano), under a further hypothesis: the 0th Hochschild cohomology of $\mathcal{A}$ $\mathrm{HH}^0(\mathcal{A})$ should have sufficiently large rank: $\mathrm{rk}\ \mathrm{HH}^0(\mathcal{A}) \geq \mathrm{rk}\ \mathrm{QH}^0(X)$. Our proof depends only on formal properties of Fukaya categories and open-closed maps, the most recent and crucial of which, compatibility of the open-closed map with pairings, was observed independently in ongoing joint work of the author with Perutz and Sheridan [GPS2] and by Abouzaid-Fukaya-Oh-Ohta-Ono [AFO+]; a proof in the simplest settings appears here in an Appendix. Because categories Morita equivalent to categories of coherent sheaves or matrix factorizations are homologically smooth, our result applies to resolve the split-generation question in homological mirror symmetry for compact symplectic manifolds (generalizing a result of Perutz-Sheridan [PS2] proven in the case $c_1(X) = 0$): any embedding of coherent sheaves or matrix factorizations into the split-closed derived Fukaya category is automatically a Morita equivalence when it has large enough $\mathrm{HH}^0$ (which it always does if $c_1(X)=0$).