On the Chern filtration for the moduli of bundles on curves

TL;DR

This paper introduces the Chern filtration on the cohomology of moduli spaces of bundles on curves, providing new insights into their structure and symmetries, especially in rank two cases, with connections to the $P=C$ phenomena.

Contribution

It defines and computes the Chern filtration for moduli of bundles, revealing symmetries and constructing categorifications, advancing understanding of cohomological invariants in algebraic geometry.

Findings

Full computation of the Chern filtration for rank two stable bundles

Discovery of a symmetry in the Chern filtration for rank two cases

Construction of $rak{sl}_2$-actions categorifying the observed symmetry

Abstract

We introduce and study the Chern filtration on the cohomology of the moduli of bundles on curves. This can be viewed as a natural cohomological invariant defined via tautological classes that interpolates between additive Betti numbers and the multiplicative ring structure. In the rank two case, we fully compute the Chern filtration for moduli of stable bundles and all intermediate stacks in the Harder--Narasimhan stratification. We observe a curious symmetry of the Chern filtration on the moduli of rank two stable bundles, and construct $\mathfrak{sl}_2$-actions that categorify this symmetry. Our study of the Chern filtration is motivated by the $P=C$ phenomena in several related geometries.