Dynamical degrees of birational transformations of projective surfaces (Q2792315)

The paper under review studies the properties of the set of dynamical degrees of a fixed projective surface. If \(X\) is a projective surface and \(f\) is a birational transformation of \(X\), its dynamical degree is defined as NEWLINE\[NEWLINE \lambda(f)=\lim_{n\to\infty}||(f^n)_*||^{1/n} NEWLINE\]NEWLINE where \(||\cdot||\) is any norm on \(\mathrm{End}(NS_{\mathbb R}(X))\). The set of all dynamical degrees \(\Lambda(X):= \{ \lambda(f)|f \text{ birational transformations of } X \}\) is called the dynamical spectrum of \(X\). By a theorem of Diller and Favre a dynamical degree different from 1 is either a Pisot or a Salem number. Those are algebraic integers whose Galois conjugates lie in the open, resp. closed, unit disk.NEWLINENEWLINEThe first part of the paper is devoted to a survey of properties of the dynamical degrees, with proofs and several examples, and to the proof ofNEWLINENEWLINETheorem A: Let \(k\) be an algebraically closed field. Let \(f\) be a birational transformation of \(X\) defined over \(k\). If \(\lambda(f)\) is a Salem number, there exists a surface \(Y\) and a birational mapping \(\varphi: Y\dashrightarrow X\) such that \(\varphi^{-1}\circ f\circ \varphi \) is an automorphism of \(Y\).NEWLINENEWLINEAs a corollary, they obtain a spectral gap property: there is no dynamical degree between 1 and the Lehmer number \(\lambda_L\), which is conjecturally the infimum of the set of Salem numbers.NEWLINENEWLINESecondly, the authors study the set \(\Lambda(X)\) for a non-rational surface \(X\) and they prove thatNEWLINENEWLINETheorem B: (1) \(\Lambda(X)\) is made of quadratic integers and Salem numbers of degree at most 6 (resp. 22, resp. 10) if \(X\) is birationally equivalent to an abelian surface (resp. a \(K3\) surface, resp. an Enriques surface); (2) \(\Lambda(X)= \{1 \}\) otherwise.NEWLINENEWLINEThe final part of the paper is about rational surfaces. In this case, the situation is more complicated and the results rely on the study of the Picard-Manin space and the hyperbolic space.NEWLINENEWLINEThe minimal degree \(\mathrm{mcdeg}(f)\) of a birational transformation \(f\) of the projective plane is the minimum of the set of degrees of all its conjugates.NEWLINENEWLINEThe main results of this part are a bound on \(\mathrm{mcdeg}(f)\) in terms of \(\lambda(f)\) and the fact that \(\Lambda(\mathbb{P}^2)\) is well ordered and it is also closed if the base field is algebraically closed.