Abstract: A classical theorem of C. Jordan asserts that finite subgroups in a general linear group over a field of characteristic zero contains normal abelian subgroups of bounded index. In general, a group G has Jordan property, if any finite subgroup of G contains a normal abelian subgroup of index at most J, where J is a constant only depending on G. J.P. Serre proves Cremona group of rank 2 has Jordan property, and he conjectures Cremona group of any rank has Jordan property. The conjecture is proved by Prokhorov-Shramov and Birkar. In this talk, we give explicit bounds for Cremona group of rank 2 in odd characteristic. This is a joint work with C. Shramov.

Abstract: The Mordell-Lang conjecture, now a theorem, describes the structure of rational points on subvarieties of semiabelian varieties. It states that for a semiabelian variety G over an algebraically closed field K of characteristic zero and a finitely generated subgroup Γ of G(K), the intersection X(K)∩Γ of any subvariety X⊆G with Γ is a finite union of cosets of algebraic subgroups. A fundamental uniformity question asks whether the number and the structure of these cosets can be bounded exclusively in terms of the dimension of G, the degree d of X, and the rank r of Γ.
    In this work we establish such a uniform bound. More precisely, we prove that there exists a constant c=c(dim G,d)^r such that X∩Γ is contained in at most c translates of algebraic subgroups of G.
    The proof combines several novel ingredients. We first define the Betti map associated with a family of semiabelian varieties using the mixed Hodge structure, which eventually gives a non-degeneracy condition for family of subvarieties in semiabelian varieties, via functional transcendence. We establish an equidistribution result for the Galois orbits of small points in a family of semiabelian varieties. Finally, we apply techniques from Diophantine approximation to control the number of algebraic points of large heights.
    Our theorem unifies and extends earlier work on abelian varieties and tori. Due to the twist given by the extension type, our case deal with the negativity of canonical line bundle and canonical currents which are absent in previous works. This is a joint work in progress with Zhaobo Han and Jiawei Yu.

Abstract: Given a variety of general type, the slope is the ratio between its canonical volume and its Euler characteristic. One classical topic in Algebraic Geometry is the search for lower and upper bounds for the slope. In the case of irregular surfaces, one breakthrough moment has been Pardini's proof of the Severi inequality. I will start by recalling this argument, then will describe other results in higher dimension. In particular, I will introduce the methods developed in collaboration with Barja and Pardini and discuss the case when the eventual map is generically birational, giving some new inequalities.