Green currents for holomorphic automorphisms of compact Kähler manifolds

Dinh, Tien-Cuong; Sibony, Nessim

doi:10.1090/S0894-0347-04-00474-6

Green currents for holomorphic automorphisms of compact Kähler manifolds

Authors: Tien-Cuong Dinh and Nessim Sibony
Journal: J. Amer. Math. Soc. 18 (2005), 291-312
MSC (2000): Primary 37F10, 32H50, 32Q15, 32U40
DOI: https://doi.org/10.1090/S0894-0347-04-00474-6
Published electronically: December 7, 2004
MathSciNet review: 2137979
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Abstract: Let be a holomorphic automorphism of a compact Kähler manifold $(X,\omega)$ of dimension $k\geq 2$ . We study the convex cones of positive closed -currents , which satisfy a functional relation

$\begin{displaymath}f^* T_p=\lambda T_p, \ \lambda>1,\end{displaymath}$

and some regularity condition (PB, PC). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy

is positive, then for some $m\geq 1$ , there is a positive closed

-current

which satisfies the relation

$\begin{displaymath}f^* T_m=\exp({h}(f)) T_m.\end{displaymath}$

Moreover, every quasi-p.s.h. function is integrable with respect to the trace measure of

. When the dynamical degrees of

are all distinct, we construct an invariant measure $\mu$ as an intersection of closed currents. We show that this measure is mixing and gives no mass to pluripolar sets and to sets of small Hausdorff dimension.

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