Dimension of fixed loci of diagonalizable groups via algebraic cobordism

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.