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Actions of a group of prime order without equivariantly simple germs

Ivan Proskurnin

Abstract

We prove that equivariantly simple invariant singularities can only exist for very few representations of a group of prime order: for real representations and some ``almost, but not quite real'' representations.

Actions of a group of prime order without equivariantly simple germs

Abstract

We prove that equivariantly simple invariant singularities can only exist for very few representations of a group of prime order: for real representations and some ``almost, but not quite real'' representations.
Paper Structure (7 sections, 7 theorems, 2 equations)

This paper contains 7 sections, 7 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Definitions, notation and auxiliary statements
  3. Equivariantly simple and equivariantly stable germs
  4. Real group actions and Roberts' (in)equality
  5. Real representation corresponding to an arbitrary representation
  6. Equivariant Milnor number of a $\mathbb{Z}_p$-invariant singularity
  7. Proof of main theorem

Key Result

Theorem 1.1

Let $\tau$ be a linear action of $\mathbb{Z}_p$ on $(\mathbb{C}^n,0)$, $rk(\tau)$ the maximal rank of a $\tau$-invariant quadratic form, $p$ -- a prime number. Germs equivariantly simple with respect to $\tau$ may only exist in one of the two cases: 1)$det(\tau) \neq 1, n - rk(\tau) \leq log_2 (p+1)

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • ...and 1 more