Equidistribution of positive closed currents for certain birational maps on P^k

초록

In this talk, we will consider birational maps $f_{+}:{\mathbb{P}}^k\to {\mathbb{P}}^k$ with a splitting condition ${\bigcup }_{n\ge 0}{f}_{-}^n(I^{+})\cap {\bigcup }_{n\ge 0}{f}_{+}^n(I^{-})=\varnothing$, where $f_{-}$ is the inverse of $f_{+}$ and $I^{\pm }$ are the sets of indeterminacy for $f_{\pm }$, respectively. For more information about this class, we refer the reader to \cite{DV}. This class of birational maps can be considered as a natural generalization of birational maps induced from a regular polynomial automorphism of ${\mathbb{C}}^k$. We will discuss a generalization of an equidistribution theorem for regular polynomial automorphisms of ${\mathbb{C}}^k$ to birational maps of this class. More precisely, we will find a sufficient condition of a positive closed $(p,p)$-current $S$ of unit mass on ${\mathbb{P}}^k$ in terms of the set of indeterminacy of $f_{-}$ so that the sequence of properly normalized pull-backs $d^{-pn}(f^n{)}^{\ast }SS$ of $S$ converges to the Green $(p,p)$-current.