Filtrations on quantum cohomology from the Floer theory of $\mathbb{C}^*$-actions

Abstract

We construct a filtration by ideals on quantum cohomology for symplectic manifolds with a Hamiltonian $S^1$-action that extends to a pseudoholomorphic $\mathbb{C}^*$-action. These spaces include all Conical Symplectic Resolutions, in particular all Quiver Varieties. In particular, we obtain a family of filtrations on singular cohomology for any Conical Symplectic Resolution, that is sensitive to the choice of $\mathbb{C}^*$-action. The symplectic form is rarely exact at infinity for these spaces, so substantial foundational work is carried out to rigorously define Floer theory, in particular symplectic cohomology. Using Floer theory, we construct a periodic persistence module, giving rise to a graded periodic barcode associated to the $\mathbb{C}^*$-action. This encodes birth-death phenomena of Floer invariants. Our filtrations can be viewed as a Floer-theoretic analogue of Atiyah-Bott filtrations, arising from stratifying a manifold by gradient flowlines of a Morse-Bott function, but they are distinct from those and they can detect non-topological properties of the quantum product.