Generating the Fukaya categories of compact toric varieties

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.

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

References in corpus (3)

• Quantum Structures for Lagrangian Submanifolds
• Generating the Fukaya categories of Hamiltonian G-manifolds
• Higher rank local systems in Lagrangian Floer theory