Siu's inequality and its applications to birational automorphisms

Let $X$ be a smooth projective variety of dimension $d$, $L$ be an ample line bundle on $X$, and $f$ be a birational automorphism of $X$. Then we can define the $i$-th degree of f with respect to $L$ as the number $\operatorname{deg}_i(f) = f^*(L^i).L^{d-i}$. We define the $i$-th dynamical degree of $f$ as the limit of the sequence $((\operatorname{deg}_i(f^n))^{1/n})$. Dynamical degrees are invariants of birational automorphisms, independent of the choice of the line bundle $L$. Many interesting properties of automorphisms can be described in terms of dynamical degrees. However, calculating or even estimating the dynamical degree of a fixed automorphism often turns out to be a complicated problem. Nevertheless, Junyi Xie showed that there exists an algorithm which, for any $\epsilon$, allows one to construct a number differing from the $i$-th dynamical degree by no more than $\epsilon$. Surprisingly, the only prerequisite for proving this very non-trivial result is Siu's inequality (a classical result at the beginning of Lazarsfeld's book). We will discuss Xie's result and its corollaries.