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Title: Jordan property of automorphism groups of surfaces of positive characteristic Speaker: Yifei Chen Abstract: A classical theorem of C. Jordan asserts the general linear group $G$ over a field of characteristic zero is Jordan. That is, any finite subgroup of $G$ contains a normal abelian subgroup of index at most $J$, where $J$ is an integer only depends on the group $G$. J.-P. Serre proved that the same property holds for the Cremona group of rank $2$. In this talk, we will discuss Jordan property for automorphism groups of surfaces of positive characteristic. This is a joint work with C. Shramov. Title: Algebraic surfaces on the Severi line in positive characteristic Speaker: Yi Gu Abstract: Let $X$ be a minimal surface of maximal Albanese dimension. The Severi inequality asserts that $K_X^2 \ge 4 \chi(\omega_X)$. People call $X$ on the Severi line if $X$ is of general type and the equality in Severi inequality is attained. It is conjectured that $X$ is on the Severi line if and only if its canonical model admits a flat double cover over an Abelian surface. In characteristic zero, Barja-Pardini-Stoppino and Lu-Zuo have independently proved this conjecture. In this talk, we will show that the same conjecture holds in arbitrary characteristic. This is a joint work with Prof. Xiaotao Sun and Mingshuo Zhou. Title: Syzygies of certain homogeneous varieties Speaker: Zhi Jiang Abstract: We will discuss properties $(N_p)$ of ample line bundles on homogeneous varieties, which include abelian varieties and rational homogeneous varieites. Title: On the relative Kawamata-Morrison cone conjecture Speaker: Zhan Li Abstract: Kawamata-Morrison cone conjecture predicts that the number of minimal models of Calabi-Yau varieties are finite. We show that for a Calabi-Yau fiber space, the number of relative minimal models is finite assuming the Kawamata-Morrison cone conjecture for geometric fiber and standard conjectures in the minimal model program. This is an ongoing joint work with Hang Zhao. Title: Twisted functoriality in nonabelian Hodge theory over char $p$ Speaker: Mao Sheng Abstract: In the classical nonabelian Hodge theory, it is a remarkable fact that the Simpson correspondence is independent of the choice of a background Kähler metric and is compatible with pullback. However, it goes wrong in the analogous theory over char $p>0$. Inspired by the work of Faltings in $p$-adic nonabelian Hodge theory, and based on a recent joint work with Yupeng Wang, I am going to argue that the correct functoriality with respect to pullback is the so-called twisted functorality. This generalizes a recent result of A. Langer regarding this issue in a vast way. Title: On the space of spectral data Speaker: Lei Song Abstract: The Hitchin map gives a fibration of the moduli of Higgs bundles over a complex smooth variety. Its image is conjectured to be the space of spectral data by Chen and Ngô generalizing the picture for curves. I will talk about the generalization, and show the space is a birational invariance. This is a work in progress with H. Sun at SCUT. Title: Chow groups and unramified cohomology of quadrics Speaker: Peng Sun Abstract: In the 1990's, Karpenko computed Chow groups of quadrics based on the work of Swan from the K-theoretic point of view. In a series of papers, Kahn, Rost and Sujatha studied unramified cohomology of quadrics based on the work of Karpenko and tools from motivic cohomology. However, their results are valid with the exception of characteristic two. In this talk, I will explain the main ideas and show how these results can be adopted in characteristic two. Title: Finding rational curves by forgetful maps Speaker: Runhong Zong Abstract: In this talk, we are mainly concerned with the geometry of rational curves on algebraic varieties. Our main object of study is rationally connected varieties, a kind of varieties that contain "plenty" of rational curves. It was shown by Kollár-Miyaoka-Mori that a kind of "smoothing of comb" technique is available on this kind of varieties. We firstly show a most general form of this kind of "smoothing of comb" technique exists, which extends all the "smoothing of comb" techniques in the literature before. Then we apply this generalized technique to study the geometry of forgetful maps between moduli spaces of stable maps and twisted stable maps to rationally connected varieties. As a result, we solve several open questions on topology, geometry and arithmetic of rationally connected varieties. |