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

TL;DR

This paper computes the $C_2$-equivariant ordinary cohomology of smooth antisymmetric complex quadrics, providing new insights into their topological structure and applications to classical algebraic geometry problems.

Contribution

It introduces the first calculation of $C_2$-equivariant cohomology for antisymmetric quadrics, including a key application to cubic surface lines.

Findings

Calculated the $C_2$-equivariant cohomology of antisymmetric quadrics.

Proved an equivariant refinement of the 27 lines on a cubic surface.

Identified a $C_2$-equivariant Grassmannian within the quadrics.

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 $\mathbb{P}^3$.