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

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 spectral sequences give explicit descriptions of birth-death phenomena of the barcode of the persistence module associated to the $\mathbb{C}^*$-action. This paper contains the foundational work to rigorously construct a filtration on Floer complexes from the $\mathbb{C}^*$-action, announced in our earlier paper. A substantial appendix on Morse-Bott-Floer theory deals with several of the technical difficulties of the paper. 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 famous P=W filtration.