Dimension of fixed loci of diagonalizable groups via algebraic cobordism

TL;DR

This paper characterizes the possible dimensions of fixed loci of diagonalizable group actions on smooth projective varieties using algebraic cobordism, providing bounds, explicit examples, and maximality results.

Contribution

It introduces a method to determine fixed locus dimensions via algebraic cobordism and constructs explicit actions to realize and prove bounds are maximal.

Findings

Derived restrictions on fixed locus dimensions from Chern numbers.

Constructed explicit group actions realizing lower bounds.

Proved maximality of the constructed family in the cobordism ring.

Abstract

We determine all restrictions on the dimension of the fixed locus of a diagonalizable group acting on a smooth projective variety that arise from the Chern numbers of the ambient variety. We reduce the problem to finding lower bounds for actions of p-groups, which we achieve by analyzing the equivariant cobordism ring with the help of the concentration theorem. To do so, we construct enough explicit examples of actions that realize the expected lower bound. We then prove that this family is maximal in the equivariant cobordism ring, in an appropriate sense.