Filtrations on equivariant quantum cohomology and Hilbert-Poincar\'e series

TL;DR

This paper demonstrates how Floer theory induces a filtration on equivariant quantum cohomology, leading to new Hilbert-Poincaré polynomials and explicit presentations in specific geometric contexts.

Contribution

It establishes a filtration by ideals on equivariant quantum cohomology via Floer theory and develops structural properties of these filtrations across different Floer cohomology theories.

Findings

Floer theory induces a filtration on equivariant quantum cohomology.

Hilbert-Poincaré polynomials depend on Floer theory.

Explicit presentations obtained for equivariant symplectic cohomology in Calabi-Yau and Fano cases.

Abstract

We prove that Floer theory induces a filtration by ideals on equivariant quantum cohomology of symplectic manifolds equipped with a $\mathbb{C}^*$-action. In particular, this gives rise to Hilbert-Poincar\'e polynomials on ordinary cohomology that depend on Floer theory. En route, the paper develops structural properties of filtrations on three versions of equivariant Floer cohomology. We obtain an explicit presentation for equivariant symplectic cohomology in the Calabi-Yau and Fano settings.