Beijing Institute of Mathematical Sciences and Applications

Kawaguchi--Silverman Conjecture (KSC) predicts an interesting relation between arithmetic and algebraic dynamics that for a birational self-map $f$ of a smooth projective variety $X$ defined over an algebraic closure $L$ of the field of rational numbers, that the arithmetic degree $\alpha_f(x)$ exists and coincides with the first dynamical degree $\delta_f$ for any $L$-point $x$ of $X(L)$ with a Zariski dense orbit. In this talk, after a brief introduction of arithmetic degree and KSC , we show that KSC holds when $X$ has Kodaira dimension zero and irregularity $q(X) \ge \dim X -1$ or $X$ is an irregular threefold (modulo one possible exception). We also study the existence of Zariski dense orbits, with explicit examples. This is a joint work with Professors Jungkai Chen and Hsueh-Yung Lin.

The minimal log discrepancy (MLD) is an invariant of singularity defined in the context of the minimal model program. In this talk, we will discuss the minimal log discrepancies of quotient singularities. I will explain that the PIA (precise inversion of adjunction) conjecture and Shokurov's index conjecture hold for quotient singularities. For the PIA conjecture, the theory of the arc space of a quotient singularity established by Denef and Loeser is an essential tool. For the index conjecture, Jordan's theorem on finite linear groups is essentially used. This is joint work with Kohsuke Shibata.

The problem of classifying birational involutions of the complex plane goes back to Bertini, whose classification was however incomplete, which was completed by Bayle-Beauville more than twenty years ago. They prove that if an involution has a fixed irrational curve, then the isomorphism class of the curve determines the conjugacy class of the involution. It turns out that this is not true over the real numbers. In this talk I want to motivate the classification of birational involutions of the real projective plane.

The Seshadri constant of a polarized variety (X,L) at a point x measures how positive is the polarization L at x. If x is very general, the Seshadri constant does not depend on x, and captures global information on X. Inspired by ideas from the Geometry of Numbers, we introduce in this talk successive Seshadri minima, such that the last one is the Seshadri constant at a point, and the first one is the width of the polarization at the point. Assuming the point is very general, we obtain two results: a) the product of the successive Seshadri minima is proportional to the volume of the polarization; b) if X is toric, the i-th successive Seshadri constant is proportional to the i-th successive minima of a suitable 0-symmetric convex body. Based on joint work with Atsushi Ito.