Super-potentials, densities of currents and number of periodic points for holomorphic maps

TL;DR

This paper develops methods using super-potentials and densities to analyze positive closed currents on compact Kähler manifolds, providing new bounds on the number of periodic points of holomorphic maps.

Contribution

It introduces two compatible methods for defining wedge-products of currents and applies these to derive sharp bounds on periodic points for holomorphic maps.

Findings

Boundedness of currents is inherited by currents with similar super-potentials.

Two methods for wedge-products of currents are shown to be equivalent.

Sharp upper bounds for periodic points of holomorphic maps are established.

Abstract

We prove that if a positive closed current is bounded by another one with bounded, continuous or Hoelder continuous super-potentials, then it inherits the same property. There are two different methods to define wedge-products of positive closed currents of arbitrary bi-degree on compact Kaehler manifolds using super-potentials and densities. When the first method applies, we show that the second method also applies and gives the same result. As an application, we obtain a sharp upper bound for the number of isolated periodic points of holomorphic maps on compact Kaehler manifolds whose actions on cohomology are simple. A similar result still holds for a large class of holomorphic correspondences.