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Green currents of holomorphic correspondences on compact Kähler manifolds

Muhan Luo, Marco Vergamini

Abstract

Consider a holomorphic correspondence $f$ on a compact Kähler manifold $X$ of dimension $k$. Let $1\le q\le k$ be any integer such that the dynamical degrees of $f$ satisfy $d_{q-1}<d_q$. We construct the Green currents $T_c$ of $f$ associated with the classes $c$ belonging to the dominant eigenspace for the action of $f^*$ on $H^{q,q}(X,\R)$. We also show that the super-potential of $T_c$ is $\log$-Hölder-continuous. When $f$ has simple action on cohomology and its graph satisfies an assumption on the local multiplicity, we prove the exponential equidistribution of all positive closed currents towards the main Green current, i.e., the only one associated to the unique maximal degree $d_q$.

Green currents of holomorphic correspondences on compact Kähler manifolds

Abstract

Consider a holomorphic correspondence on a compact Kähler manifold of dimension . Let be any integer such that the dynamical degrees of satisfy . We construct the Green currents of associated with the classes belonging to the dominant eigenspace for the action of on . We also show that the super-potential of is -Hölder-continuous. When has simple action on cohomology and its graph satisfies an assumption on the local multiplicity, we prove the exponential equidistribution of all positive closed currents towards the main Green current, i.e., the only one associated to the unique maximal degree .
Paper Structure (13 sections, 118 equations)

This paper contains 13 sections, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Super-potentials of currents
  4. Holomorphic correspondences
  5. Non-invertible linear maps
  6. Construction and regularity of Green currents
  7. Construction of Green currents
  8. Regularity of Green currents
  9. Inverse multiplicity and equidistribution results
  10. Examples
  11. Families of holomorphic correspondences
  12. Symmetric product and polynomial correspondences
  13. Properties of the pullback operator

Theorems & Definitions (22)

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  • proof : Proof of Theorem \ref{['thm: green-exist']}
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  • ...and 12 more