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The $C_2$-equivariant ordinary cohomology of complex quadrics I: The antisymmetric case

Steven R. Costenoble, Thomas Hudson

TL;DR

The paper computes the additive and multiplicative structure of the $C_2$-equivariant ordinary cohomology of smooth antisymmetric complex quadrics, using an extended grading and a presentation over $\mathcal{R}^{\diamond}$. The main result identifies $H_{C_2}^{\Diamond}(\chi Q_{2p})$ as generated by the equivariant fundamental classes $m_s$ with explicit relations and divisibility properties, and it specializes to a Grassmannian case $\mathrm{Gr}_{2}(\mathbb{C}^{3+\sigma})$ via a $C_2$-diffeomorphism to $\chi Q(\mathbb{C}^{3+3\sigma})$, yielding concrete generators and relations. The paper then applies this to refine the classical 27 lines on a smooth cubic by computing the Euler class $e(\mathrm{Sym}^3(\pi^{\vee}))$ and decomposing the 27 lines into three invariant lines and twelve pairs under the $C_2$-action, with a careful comparison to Brazelton’s $S_4$-set results. Overall, the work demonstrates how extended grading and equivariant Thom-style constructions yield tractable, explicit descriptions of equivariant cohomology rings and their geometric consequences.

Abstract

In this, the first of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth antisymmetric quadrics. One of these quadrics coincides with a $C_2$-equivariant Grassmannian, and we use this calculation to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in $\mathbb{P}^3$.

The $C_2$-equivariant ordinary cohomology of complex quadrics I: The antisymmetric case

TL;DR

The paper computes the additive and multiplicative structure of the -equivariant ordinary cohomology of smooth antisymmetric complex quadrics, using an extended grading and a presentation over . The main result identifies as generated by the equivariant fundamental classes with explicit relations and divisibility properties, and it specializes to a Grassmannian case via a -diffeomorphism to , yielding concrete generators and relations. The paper then applies this to refine the classical 27 lines on a smooth cubic by computing the Euler class and decomposing the 27 lines into three invariant lines and twelve pairs under the -action, with a careful comparison to Brazelton’s -set results. Overall, the work demonstrates how extended grading and equivariant Thom-style constructions yield tractable, explicit descriptions of equivariant cohomology rings and their geometric consequences.

Abstract

In this, the first of three papers about -equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth antisymmetric quadrics. One of these quadrics coincides with a -equivariant Grassmannian, and we use this calculation to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in .

Paper Structure

This paper contains 10 sections, 12 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The cohomology of antisymmetric quadrics
  3. Restriction maps
  4. A Grassmannian
  5. 27 lines
  6. Comparison with Brazelton’s result
  7. Equivariant ordinary cohomology
  8. Ordinary cohomology
  9. The cohomology of projective spaces
  10. Pushforwards and representation by singular manifolds

Key Result

Theorem 1.5

As an algebra over $\mathcal{R}^\diamond$, the ordinary cohomology $H_{C_2}^\Diamond(\chi Q_{2p})$ is generated by the fundamental classes with $s\in\{0,\dots,p\}$. These classes are such that $\widehat{c}_\omega^{} m_s$ is infinitely divisible by $\zeta_0$, while $\widehat{c}_{\chi\omega}^{} m_s$ is infinitely divisible by $\zeta_1$. Furthermore, the following relations hold: These relations ge

Figures (2)

  • Figure 1: ${C_2}$-affine planes and their associated projective lines
  • Figure 2: $\mathbb{H}^{RO({C_2})}$

Theorems & Definitions (30)

  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 20 more