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Log Bott localization with non-isolated lci zero varieties

Maurício Corrêa, Elaheh Shahsavaripour

Abstract

We establish a logarithmic Bott localization formula for global holomorphic sections of $T_X(-\log D)$ on a compact complex manifold $X$ with simple normal crossings divisor $D$. The zero scheme is allowed to have non-isolated compact components, assumed to be local complete intersections and to satisfy the natural Bott nondegeneracy condition. We further give a current-theoretic formulation and, in the local complete intersection case, identify the local residue term with a Coleff-Herrera current.

Log Bott localization with non-isolated lci zero varieties

Abstract

We establish a logarithmic Bott localization formula for global holomorphic sections of on a compact complex manifold with simple normal crossings divisor . The zero scheme is allowed to have non-isolated compact components, assumed to be local complete intersections and to satisfy the natural Bott nondegeneracy condition. We further give a current-theoretic formulation and, in the local complete intersection case, identify the local residue term with a Coleff-Herrera current.
Paper Structure (5 sections, 12 theorems, 158 equations)

This paper contains 5 sections, 12 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Logarithmic preliminaries and the conormal Bott action
  3. Fundamental currents and Coleff--Herrera currents
  4. Tubes around compact lci components
  5. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.4

In the standing setup above, assume that for every irreducible component $Z_j$ of the zero scheme of $v$ the following hold: Then, for every $\mathrm{GL}_n(\mathbb C)$-invariant polynomial $\Phi$ of degree $n$, where

Theorems & Definitions (30)

  • Definition 1.1: Bott-nondegeneracy condition
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4: Normal bundle in the lci case
  • ...and 20 more