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Variations on cohomology rings and zero schemes

Kamil Rychlewicz

Abstract

We extend the theorem of Hausel and the author from arXiv:2212.11836 that relates equivariant cohomology rings and algebras of functions on zero schemes. This paper combines three separate results. We prove that for a reductive group G acting on a smooth projective variety one can see the equivariant cohomology ring as the ring of functions on the zero scheme over the Kostant section, provided that some transversality condition is satisfied. In particular, we show that the conclusion holds for spherical varieties. We then show a version for singular varieties, e.g. discriminant varieties, where in general we only recover a part of equivariant cohomology ring, generated by Chern classes. We also show that an analogous result, connecting equivariant K-theory to the ring of functions on the fixed-point scheme, holds for GKM spaces. This is a concise version of some results from the PhD thesis arXiv:2407.14659, which contains a broader introduction to the topic.

Variations on cohomology rings and zero schemes

Abstract

We extend the theorem of Hausel and the author from arXiv:2212.11836 that relates equivariant cohomology rings and algebras of functions on zero schemes. This paper combines three separate results. We prove that for a reductive group G acting on a smooth projective variety one can see the equivariant cohomology ring as the ring of functions on the zero scheme over the Kostant section, provided that some transversality condition is satisfied. In particular, we show that the conclusion holds for spherical varieties. We then show a version for singular varieties, e.g. discriminant varieties, where in general we only recover a part of equivariant cohomology ring, generated by Chern classes. We also show that an analogous result, connecting equivariant K-theory to the ring of functions on the fixed-point scheme, holds for GKM spaces. This is a concise version of some results from the PhD thesis arXiv:2407.14659, which contains a broader introduction to the topic.
Paper Structure (8 sections, 13 theorems, 69 equations, 2 figures)

This paper contains 8 sections, 13 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgements
  4. Regular actions and zero schemes
  5. A short recap on Kostant sections and equivariant cohomology
  6. Zero schemes and equivariant cohomology for non-regular actions
  7. Singular varieties
  8. K-theory conjecture for GKM spaces

Key Result

Theorem 1.1

Let $X$ be a smooth projective complex algebraic variety of dimension $n$. Consider an algebraic vector field $V\in\mathrm{Vect}(X)$ and assume that its zero set is isolated, but nonempty. Let $Z$ denote the zero scheme of $V$. Then the coordinate ring $\mathbb C[Z]$ admits an increasing filtration such that there is an isomorphism of graded algebras. The degree on the left is twice the degree o

Figures (2)

  • Figure 1: Zero scheme $\mathcal{Z}$ from Example \ref{['exthick']}. An affine patch $\mathcal{S}\times \{[x:y:1-x]|x,y\in\mathbb C\}$ shown.
  • Figure 2: The discriminantal variety $Y\subset \mathbb P^3$. Affine real part shown, for $a=1$. It is the locus of polynomials $x^3+bx^2+cx+d$ with a double root. Singular locus in red.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • ...and 28 more