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On the Chern filtration for the moduli of bundles on curves

Woonam Lim, Miguel Moreira, Weite Pi

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.

On the Chern filtration for the moduli of bundles on curves

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 -actions that categorify this symmetry. Our study of the Chern filtration is motivated by the phenomena in several related geometries.

Paper Structure

This paper contains 25 sections, 37 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Tautological classes and Chern filtrations
  3. Main results and conjectures
  4. Relation to the $P=C$ phenomenon
  5. Acknowledgements
  6. Preliminaries
  7. Moduli stacks and moduli spaces
  8. Tautological classes and the descendent algebra
  9. Chern filtration and refined Poincaré series
  10. Proof of Theorem \ref{['thm: topcherndegree']}
  11. The associated graded and its (non-degenerate) pairing
  12. The refined Poincaré polynomial of N21
  13. Chern filtration of intermediate stacks
  14. Strategy of the proof
  15. A basis of Mumford relations
  16. ...and 10 more sections

Key Result

Theorem 2

The top Chern degree of $N_{r,d}$ equals $(r+2)(r-1)(g-1)$.

Theorems & Definitions (76)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6: = Corollary \ref{['cor: f determines I gr']}
  • Conjecture 7
  • Conjecture 9
  • Remark 1.1
  • Definition 1.2: $\textup{SL}_r$ descendent algebra
  • ...and 66 more