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Segre forms of singular metrics on vector bundles and Lelong numbers

Mats Andersson, Richard Lärkäng

Abstract

Let $E\to X$ be a holomorphic vector bundle. We consider a class of a singular Hermitian metrics on $E$ with analytic singularities that contains all Griffiths negative such metrics. One can define, given a smooth reference metric $h_0$, a current $s(E,h,h_0)$ called the associated Segre form, which defines the expected Bott-Chern class and coincides with the usual Segre form of $h$ where it is smooth. We prove that $s(E,h,h_0)$ is the limit of the Segre forms of a sequence of smooth metrics if the metric is smooth outside the degeneracy locus, and in general as a limit of Segre forms of metrics with empty degeneracy locus. One can also define an associated Chern form $c(E,h,h_0)$. We prove that the Lelong numbers of $s(E,h,h_0)$ and $c(E,h,h_0)$ are integers if the singularities are integral, and non-negative for $s(E,h,h_0)$.

Segre forms of singular metrics on vector bundles and Lelong numbers

Abstract

Let be a holomorphic vector bundle. We consider a class of a singular Hermitian metrics on with analytic singularities that contains all Griffiths negative such metrics. One can define, given a smooth reference metric , a current called the associated Segre form, which defines the expected Bott-Chern class and coincides with the usual Segre form of where it is smooth. We prove that is the limit of the Segre forms of a sequence of smooth metrics if the metric is smooth outside the degeneracy locus, and in general as a limit of Segre forms of metrics with empty degeneracy locus. One can also define an associated Chern form . We prove that the Lelong numbers of and are integers if the singularities are integral, and non-negative for .

Paper Structure

This paper contains 18 sections, 28 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Qpsh functions with analytic singularities
  4. qas bundles
  5. Generalized cycles and quasi-cycles
  6. Monge-Ampère type products as endomorphisms on ${\mathcal{QZ}}(Y)$
  7. Action of Segre and Chern forms associated with a smooth metric
  8. The submodule ${\mathcal{QZ}}_0(Y)$
  9. Multiplicities of quasi-cycles
  10. Regularization of qas metrics on a line bundle
  11. Segre forms and endomorphisms of qas bundles
  12. The case when $h$ is qas and non-degenerate
  13. The case with a general qas metric $h$
  14. Composition of Segre endomorphisms
  15. Chern forms of qas bundles
  16. ...and 3 more sections

Key Result

Theorem 1.3

Assume that $E\to X$ is a holomorphic vector bundle equipped with a qas metric $h$, and let $Z$ be the degeneracy locus of $h$. Let $h_0$ be a smooth reference metric, and let $s(E,h,h_0)$ denote the associated Segre form. (i) If $h_\epsilon$ is the metric defined by regular1, then regular2 holds. ( (iii) We have the decomposition where $M^{E,h,h_0}$ is closed and has support on $Z$ and $s'(E,h)=

Theorems & Definitions (83)

  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • ...and 73 more