Filtrations on quantum cohomology via Morse-Bott-Floer Spectral Sequences

Abstract: Using Morse-Bott-Floer spectral sequences, we describe 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. Our Morse-Bott-Floer spectral sequences give explicit descriptions of birth-death phenomena of the barcode of the persistence module associated to the $\mathbb{C}^*$-action, defined in our earlier paper. This paper contains the foundational work to rigorously construct a filtration on Floer complexes, announced in that earlier paper. We also include a substantial appendix on Morse-Bott-Floer theory, where a large part of the technical difficulties of the paper are dealt with. We compute a plethora of explicit examples, each highlighting various features, for Springer resolutions, ADE resolutions, and several Slodowy varieties of type A. We also consider certain Higgs moduli spaces, for which we compare our filtration with the well-known P=W filtration.