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Non-pluripolar products on vector bundles and Chern--Weil formulae

Mingchen Xia

Abstract

In this paper, we develop several pluripotential-theoretic techniques for singular metrics on vector bundles. We first introduce the theory of non-pluripolar products on holomorphic vector bundles on complex manifolds. Then we define and study a special class of singularities of Hermitian metrics on vector bundles, called $\mathcal{I}$-good singularities, partially extending Mumford's notion of good singularities. Next, we derive a Chern--Weil type formula expressing the Chern numbers of Hermitian vector bundles with $\mathcal{I}$-good singularities in terms of the associated b-divisors. We also define an intersection theory on the Riemann--Zariski space and apply it to reformulate our Chern--Weil formula.

Non-pluripolar products on vector bundles and Chern--Weil formulae

Abstract

In this paper, we develop several pluripotential-theoretic techniques for singular metrics on vector bundles. We first introduce the theory of non-pluripolar products on holomorphic vector bundles on complex manifolds. Then we define and study a special class of singularities of Hermitian metrics on vector bundles, called -good singularities, partially extending Mumford's notion of good singularities. Next, we derive a Chern--Weil type formula expressing the Chern numbers of Hermitian vector bundles with -good singularities in terms of the associated b-divisors. We also define an intersection theory on the Riemann--Zariski space and apply it to reformulate our Chern--Weil formula.
Paper Structure (46 sections, 106 theorems, 242 equations)

This paper contains 46 sections, 106 theorems, 242 equations.

Table of Contents

  1. Background
  2. Main results in \ref{['part:1']}
  3. Main results in \ref{['part:2']}
  4. Auxiliary results in pluripotential theory
  5. A potential extension of Kudla's program
  6. Conventions
  7. Non-pluripolar products on vector bundles
  8. Preliminaries
  9. Singular metrics on vector bundles
  10. Singular Hermitian forms
  11. Singular metrics on vector bundles
  12. The projective bundle
  13. Finsler metrics
  14. Special singularities on vector bundles
  15. Pull-backs of currents
  16. ...and 31 more sections

Key Result

Theorem 2

Let $T\in \widehat{Z}_a(X)$, $\hat{E}$ be a Griffiths positive Hermitian vector bundles on $X$ having small unbounded locus. Assume that $T$ is transversal to $\hat{E}$. Then for any $i>\rank E$, $c_i(\hat{E})\cap T=0$.

Theorems & Definitions (266)

  • Theorem 2: \ref{['cor:vanishingChern']}
  • Theorem 3: \ref{['thm:ChernrepChern']}
  • Theorem 4: \ref{['thm:Igoodvect']}
  • Theorem 5: =\ref{['prop:Igoodtensor']}+\ref{['thm:Igoodcancel']}
  • Theorem 6: \ref{['thm:nefbvolume']}
  • Theorem 7: =\ref{['thm:mixedDDD']}
  • Theorem 8: =\ref{['cor:ref2']}
  • Theorem 9: \ref{['thm:convdsmeasures']}
  • Theorem 10: \ref{['thm:NAdatacontdS']}
  • Definition 1.1
  • ...and 256 more