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Actions of diagonalizable $p$-groups and Chern numbers modulo $p$

Olivier Haution

Abstract

We obtain lower bounds for the dimension of fixed loci of diagonalizable $p$-groups acting on smooth projective varieties. Those bounds depend on the modulo $p$ Chern numbers of the ambient variety, and are expressed in a natural way by introducing an appropriate filtration on the "modulo $p$ cobordism ring" (for $p=2$ this is Thom's unoriented cobordism ring $MO^*$). They are obtained using equivariant localization methods, via the concentration theorem for the Chow ring, and by a technique of "partition dividing". As applications we derive statements in the spirit of Boardman's Five-Halves Theorem for involutions on manifolds.

Actions of diagonalizable $p$-groups and Chern numbers modulo $p$

Abstract

We obtain lower bounds for the dimension of fixed loci of diagonalizable -groups acting on smooth projective varieties. Those bounds depend on the modulo Chern numbers of the ambient variety, and are expressed in a natural way by introducing an appropriate filtration on the "modulo cobordism ring" (for this is Thom's unoriented cobordism ring ). They are obtained using equivariant localization methods, via the concentration theorem for the Chow ring, and by a technique of "partition dividing". As applications we derive statements in the spirit of Boardman's Five-Halves Theorem for involutions on manifolds.

Paper Structure

This paper contains 6 sections, 16 theorems, 99 equations.

Table of Contents

  1. Generalized Conner--Floyd Chern classes
  2. µp-actions
  3. Dividing partitions
  4. Fixed locus dimension
  5. The cobordism ring
  6. Applications

Key Result

Theorem 1

If $G$ acts on a smooth projective $k$-variety $X$, we have

Theorems & Definitions (52)

  • Theorem
  • Definition 1.1
  • Example 1.1
  • Lemma 1.1
  • proof
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • ...and 42 more