Equidistribution toward the Green current in big cohomology classes (Q2863010)

The Green current of a holomorphic endomorphism \(f:\mathbb{CP}^k\rightarrow\mathbb{CP}^k\) is defined as the weak limit of the sequence \(\lambda^{-n}(f^n)^*\omega_{FS}\), where \(\lambda\) stands for the degree of \(f\) and \(\omega_{FS}\) for the Fubiny-Study form. In a more general setting, this sort of current might not exist and, in this paper, the author proves its existence for certain cases.NEWLINENEWLINEThe main result in this article asserts that if \(f:X\rightarrow X\) is a \(1\)-regular dominant meromorphic map of a compact Kähler manifold and the spectral radius of the induced linear map in \((1,1)\) cohomology \(f^*:H^{1,1}(X,\mathbb{R})\rightarrow H^{1,1}(X,\mathbb{R})\) is a simple eigenvalue \(\lambda>1\) such that \(f^*\alpha=\lambda\alpha\) for \(\alpha\) a cohomology class which can be represented by a positive closed current \(T_\alpha\) with identically zero Lelong numbers on \(X\), then there exists a current \(T_\alpha\in \alpha\) such that every smooth form \(\theta\in \alpha\) equidistributes towards it. Namely, NEWLINE\[NEWLINE\frac{1}{\lambda^n}(f^n)\theta\rightarrow T_\alpha.NEWLINE\]NEWLINE In addition, the same equidistribution property holds starting with any current \(S_\alpha\in\alpha\) with identically zero Lelong numbers on \(X\).NEWLINENEWLINEA cohomology class is called big if it can be represented by a strictly positive current \(T\). Actually, for every big and numerically effective class \(\alpha\), there is always a representative current as stated in the hypotheses of the main theorem but, if the class is merely big, such a current might not exist. However, if this is the case, the author obtains results on the equidistribution by adding extra assumptions.NEWLINENEWLINEThese proofs rely on certain volume estimates that allow to prove a criterion of existence of Green currents given in a previous article by the author [J. Geom. Anal. 23, No. 2, 970--998 (2013; Zbl 1285.37011)]. Actually, similar arguments are used, in the present paper as well, to prove the equidistribution property of different categories of maps.NEWLINENEWLINEFinally, the author shows that if \(f:X\rightarrow X\) is a dominant \(1\)-regular meromorphic map of a compact Kähler surface \(X\) with small topological degree and \(\alpha\) is a big cohomology class such that \(f^*\alpha=\lambda\alpha\), then \(X\) must be a rational surface.