Automorphism groups and anti-pluricanonical curves (Q932930)

For an algebraic variety \(X\) we define \(\text{Aut}(X)\) as a group of automorphisms and \(\text{Aut}^*(X)\) as the image of a natural map \(\text{Aut}\to \text{Aut}(\text{Oic}(X))\). The question, the author adresses, is the following (Question 1.1): If X is a smooth projective surface with infinite group \(\text{Aut}^*\), does there exists a birational morphism \(\phi:X\to Y\), such that \(Y\) is a surface has an anticanonical curve (i.e. member of \(|-K_Y|\)) and an infinite subgroup \(G\subset \text{Aut}^*Y\), such that \(G\) can be lifted via \(\phi\) to \(G\)? The author answers positively to this question if \(\text{Aut}^*X\) contains an infinite subgroup of null-entropy. Here the entropy of an automorphism g is the logarithm of the largest eigenvalue of the linear map \(g_*:H^*(X,\mathbb{C})\to H^*(X,\mathbb{C})\). A subgroup \(G\) of automorphism has null-entropy if each member has entropy zero. The author proves this results by, roughly speaking, studying the action of the group \(G\) on a Neron-Severi cone of \(X\). Later in the paper the author proves some geometric criteria for the surface \(X\) so that \(\text{Aut}^*X\) has an infinite group of null-entropy (Theorem 1.3) and studies the case of surfaces \(X\) such that there exists an automorphism with positive entropy (Theorems 1.6 and 1.7).