Algebraic sheaves of Floer homology groups via algebraic torus actions on the Fukaya category

TL;DR

This paper establishes the algebraicity of certain symplectic and Floer homology actions on Fukaya categories, with applications to Lagrangian flux, sheaf tameness, and mirror symmetry, under specific geometric assumptions.

Contribution

It proves the algebraicity of the $H^1(M,\mathbb{G}_m)$ action on Fukaya categories and connects it with geometric actions, advancing understanding in symplectic geometry and mirror symmetry.

Findings

Algebraicity of the $H^1(M,\mathbb{G}_m)$ action on Fukaya categories.

Tameness of Lagrangian Floer sheaves under symplectic isotopies.

Construction of a Zariski chart in mirror symmetry context.

Abstract

Let $(M,\omega_M)$ be a monotone or negatively monotone symplectic manifold, or a Weinstein manifold. One can construct an "action" of $H^1(M,\mathbb{G}_m)$ on the Fukaya category (wrapped Fukaya category in the exact case) that reflects the action of $Symp^0(M,\omega_M)$ on the set of Lagrangian branes. A priori this action is only analytic. The purpose of this work is to show the algebraicity of this action under some assumptions. We use this to prove a tameness result for the sheaf of Lagrangian Floer homology groups obtained by moving one of the Lagrangians via global symplectic isotopies. We also show the algebraicity of the locus of $z\in H^1(M,\mathbb{G}_m)$ that fix a Lagrangian brane in the Fukaya category. The latter has applications to Lagrangian flux. Finally, we prove a statement in mirror symmetry: in the Weinstein case, assume that $M$ is mirror to an affine or projective variety $X$, that there exists an exact Lagrangian torus $L\subset M$ such that $H^1(M)\to H^1(L)$ is surjective, and that $L$ is sent to a smooth point of $x\in X$ under the mirror equivalence. Then we construct a Zariski chart of $X$ containing $x$, that is isomorphic to $H^1(L,\mathbb{G}_m)$, and such that other points of this chart correspond to non-exact deformations of $L$ (possibly equipped with unitary local systems). In particular, this implies rationality of the irreducible component containing $x$; however, it is stronger. Under our assumptions, one can construct an algebraic action of $H^1(M,\mathbb{G}_m)$, namely the action by non-unitary local systems. By combining techniques from family Floer homology and non-commutative geometry, we prove that this action coincides with the geometric action mentioned in the first paragraph. We use this to deduce the theorems above.