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Cohomology of configuration spaces on punctured varieties

Yifeng Huang

Abstract

In the theory of configuration spaces, "splitting" usually refers to the phenomenon that the configuration spaces on a manifold and those on its punctured version are closely related cohomologically. We prove a splitting theorem that is equivariant and mixed-Hodge-theoretic; both are new features in such results. As an application, we determine the generating function for the mixed Hodge numbers of the unordered configuration spaces of a multi-punctured elliptic curve.

Cohomology of configuration spaces on punctured varieties

Abstract

In the theory of configuration spaces, "splitting" usually refers to the phenomenon that the configuration spaces on a manifold and those on its punctured version are closely related cohomologically. We prove a splitting theorem that is equivariant and mixed-Hodge-theoretic; both are new features in such results. As an application, we determine the generating function for the mixed Hodge numbers of the unordered configuration spaces of a multi-punctured elliptic curve.

Paper Structure

This paper contains 22 sections, 14 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Punctured Riemann surfaces
  4. Remarks about our results
  5. Methods and organizations
  6. Preliminaries
  7. Arrangements and Orlik--Solomon algebra
  8. Hyperplane-like arrangements
  9. A spectral sequence
  10. Degeneration
  11. Braid-puncture arrangements
  12. An explicit isomorphism
  13. Technical notes
  14. Main derivation
  15. Compatibility with differential
  16. ...and 7 more sections

Key Result

Theorem \oldthetheorem

Let $Y$ be a possibly punctured non-compact pure variety of complex dimension $d$. Then for $p,q,i\geq 0$ and $P\in Y$, where the summand is zero if any of $p-kd,q-kd,i-k(2d-1),n-k$ is negative.

Theorems & Definitions (35)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem: Splitting, non-equivariant
  • Theorem \oldthetheorem: Splitting, equivariant
  • Corollary \oldthetheorem
  • Conjecture \oldthetheorem
  • Corollary \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem: Totaro totaro1996configuration
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 25 more