Log Bott localization with non-isolated lci zero varieties

TL;DR

This paper develops a logarithmic Bott localization formula for holomorphic sections of a logarithmic tangent bundle on complex manifolds with divisors, accommodating non-isolated zero schemes and linking to Coleff-Herrera currents.

Contribution

It introduces a new localization formula for logarithmic tangent bundles with non-isolated zero schemes, extending Bott's classical results.

Findings

Established a logarithmic Bott localization formula.

Connected local residue terms with Coleff-Herrera currents.

Extended localization techniques to non-isolated zero schemes.

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.