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The $C_2$-equivariant ordinary cohomology of complex quadrics II: The symmetric case

Steven R. Costenoble, Thomas Hudson

TL;DR

This work advances the understanding of $C_2$-equivariant ordinary cohomology by computing the $RO(\Pi BU(1))$-graded cohomology of smooth symmetric complex quadrics with Burnside Mackey-functor coefficients. The authors develop a Mackey-functor framework, extend the grading, and establish divisibility phenomena that enable splitting of long exact sequences, culminating in explicit additive and multiplicative structures for all quadric types. For the four symmetric-types $(B,B)$, $(B,D)$, $(D,B)$, and $(D,D)$, they introduce canonical generators $x_{p,q}$ (type ${C_2}/{C_2}$) and $y$ (type ${C_2}/e$), provide detailed divisibility relations via elements $\varphi$ and $\varphi^\chi$, and derive precise multiplicative relations that encode the interaction between nonequivariant and equivariant data. A notable outcome is the first natural instance where cohomology contains ${\underline {H}}_{C_2}^{RO(C_2)}({C_2}/e)$-summands, illustrating a richer equivariant refinement beyond sums of point cohomologies and informing future work (e.g., QuadricsIII) on deeper geometric phenomena like refined $27$-lines counts.

Abstract

In this, the second of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth symmetric quadrics graded on the representation ring of $ΠBU(1)$ and with coefficients in the Burnside Mackey functor. These calculations exhibit various interesting properties, including the first naturally occurring example we are aware of where the cohomology is not just the sum of shifted copies of the cohomology of a point, but also has summands that are shifted copies of the cohomology of the free orbit $C_2/e$.

The $C_2$-equivariant ordinary cohomology of complex quadrics II: The symmetric case

TL;DR

This work advances the understanding of -equivariant ordinary cohomology by computing the -graded cohomology of smooth symmetric complex quadrics with Burnside Mackey-functor coefficients. The authors develop a Mackey-functor framework, extend the grading, and establish divisibility phenomena that enable splitting of long exact sequences, culminating in explicit additive and multiplicative structures for all quadric types. For the four symmetric-types , , , and , they introduce canonical generators (type ) and (type ), provide detailed divisibility relations via elements and , and derive precise multiplicative relations that encode the interaction between nonequivariant and equivariant data. A notable outcome is the first natural instance where cohomology contains -summands, illustrating a richer equivariant refinement beyond sums of point cohomologies and informing future work (e.g., QuadricsIII) on deeper geometric phenomena like refined -lines counts.

Abstract

In this, the second of three papers about -equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth symmetric quadrics graded on the representation ring of and with coefficients in the Burnside Mackey functor. These calculations exhibit various interesting properties, including the first naturally occurring example we are aware of where the cohomology is not just the sum of shifted copies of the cohomology of a point, but also has summands that are shifted copies of the cohomology of the free orbit .

Paper Structure

This paper contains 8 sections, 25 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Mackey functor-valued ordinary cohomology
  3. Divisibility
  4. The excess intersection formula
  5. Quadrics of type (B,B)
  6. The $RO({C_2})$-graded part
  7. Quadrics of type (B,D) and (D,B)
  8. Quadrics of type (D,D)

Key Result

Proposition 1.2

The cohomology of a quadric of type B is given by where $\mathop{\mathrm{grad}}\nolimits c = 2$ and $\mathop{\mathrm{grad}}\nolimits y = 2p$. A basis is given by with one basis element in each even grading from $0$ to $2(2p-1)$. (If $p = 0$, $Q_{1} = \emptyset$ and these relations force the cohomology to be $0$.) If $p > 1$, the cohomology of a quadric of type D is given by where Here, $\matho

Theorems & Definitions (58)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 3.1: Lemma 16.2 of Co:InfinitePublished
  • Proposition 3.2
  • proof
  • Proposition 4.1: Excess Intersection Formula
  • proof
  • Definition 5.2
  • ...and 48 more