D-bar Spark Theory and Deligne Cohomology

TL;DR

This paper explores the relationship between spark characters, Deligne cohomology, and holomorphic vector bundles on complex manifolds, providing explicit formulas, refined invariants, and applications to algebraic cycles and foliations.

Contribution

It introduces an explicit analytic product formula for Deligne cohomology using spark characters and defines refined Chern classes, advancing the understanding of cohomological invariants in complex geometry.

Findings

Explicit product formula for Deligne cohomology

Refined Chern classes for holomorphic bundles

Vanishing theorem for holomorphic foliations

Abstract

We study the Harvey-Lawson spark characters of level p on complex manifolds. Presenting Deligne cohomology classes by sparks of level $p$, we give an explicit analytic product formula for Deligne cohomology. We also define refined Chern classes in Deligne cohomology for holomorphic vector bundles over complex manifolds. Applications to algebraic cycles are given. A Bott-type vanishing theorem in Deligne cohomology for holomorphic foliations is established. A general construction of Nadel-type invariants is given together with a new proof of Nadel's conjecture.