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Dimension of fixed loci of diagonalizable groups via algebraic cobordism

Olivier Haution

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.

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.
Paper Structure (35 sections, 74 theorems, 160 equations)

This paper contains 35 sections, 74 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Existence of fixed points
  3. Fixed locus dimension
  4. Remarks
  5. Strategy of the proof
  6. Organization of the paper
  7. The Lazard ring
  8. The algebraic Hurewicz morphism
  9. Cobordism classes of varieties
  10. Landweber ideals
  11. Actions on cobordism generators
  12. Group actions on varieties
  13. Generators of the Landweber ideals
  14. Milnor hypersurfaces
  15. The q-filtration
  16. ...and 20 more sections

Key Result

Theorem 1

Let $G$ be a finite abelian $p$-group of rank $r$. Then $J(G)$ is the subgroup $I_p({r})$ of $\mathop{\mathrm{\mathbb{L}}}\nolimits$ generated by the classes of smooth projective varieties of dimension $< p^{r-1}$ having all Chern numbers divisible by $p$.

Theorems & Definitions (175)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Lemma 2.1.1
  • proof
  • Definition 2.2.1
  • Lemma 2.2.1
  • proof
  • ...and 165 more