代数曲面上的叶层化由一阶常微分方程定义，它是代数曲面纤维化的推广。上世纪80年代，Miyaoka将Mori的极小模型理论引入到叶层化双有理分类的研究，近30年来非一般型叶层化的 分类有了很大的进展。为了一般型叶层化的分类，我们引进了叶层化的陈数和斜率，利用肖刚的斜率不等式，对斜率小于4的叶层化给出了亏格不等式，并用于判别非一般型叶层化的代 数可积性（庞加莱问题）。对于一般型叶层化，陈数的下界问题，斜率的下界问题，庞加莱问题等都是未解决问题。我们将围绕这些问题，介绍最新的研究进展。

In this talk, I will introduce a recent research progress on the following conjecture: any \(3\)-fold of general type has the volume lower bound \(1/420\). I will also report some new results on the classification of \(3\)-folds attaining minimal volumes. The context covers joint works with Jungkai Chen and Zhi Jiang.

We construct infinitely many non-isotrivial families of abelian varieties of \(\textrm{GL}_2\)-type over four punctured projective lines with bad reduction of type-\((1/2)\infty\) via p-adic Hodge theory and Langlands correspondence. They lead to algebraic solutions of the Painleve VI equation. Recently Lin-Sheng-Wang proved the conjecture on the torsioness of zeros of Kodaira-Spencer maps of those type families. Based on their theorem we show the set of those type families of abelian varieties is \emph{exactly} parameterized by torsion sections of the universal family of elliptic curves modulo the involution.

In this talk, we discuss multiplicativity of Chow motives of complete intersections. We show that any Fano or Calabi-Yau complete intersection admits a self-dual multiplicative Chow-K\"{u}nneth decomposition, by studying isogenous autocorrespondences. Moreover, the third relative power of the corresponding universal families satisfies the Franchetta property.

Given a spherical Mukai vector \(v\) on a smooth projective K3 surface \(S\), when \(S\) is of Picard number \(1\), there is a unique bundle \(E\) with \(v(E)=v\), and \(E\) is always stable with the unique polarization on \(S\). When $S$ is of Picard number greater than \(2\), there are usually more spherical bundles with the fixed spherical Mukai vector.
In his book `Lectures on K3 surfaces', at the end of the chapter on the bounded derived categories of K3, Huybretchs asks if every spherical bundle is semistable with respect to some polarization and if there is a `counting theory' for spherical bundles. Unfortunately, both problems fail in a naive way. More precisely,
(1) there exists an example of a spherical vector bundle that is never semistable;
(2) there exists an example of K3 surface and infinitely many spherical vector bundles with the same spherical Mukai vector \(v\). Moreover, each of the vector bundles is stable with respect to some polarization.
However, we may put some assumptions on S so that the `counting theory' can make sense. In particular, when \(Nef(S)\) is rational polyhedral, there are finitely many spherical vector bundles with \(v(E)=v\) that can be stable with respect to some polarization. Denote the counting by \(H(v)\), we make a detailed study in the case that \(S\) is a generic elliptic K3 surface admitting a section. We show that \(H(rk,\sigma,*)=rk\), where \(\sigma\) stands for the divisor of the \(\mathbb{P}^1\) section. We conjecture that on average \(H(rk - , -) ~ (\ln rk)^2\) and reduce the conjecture to a problem in analytic number theory on the estimation of a certain sum of divisor functions.

An algebraic variety defined over the real numbers is called maximal if it satisfies the Smith-Thom inequality. In this talk, I will present several new constructions of maximal real varieties by using moduli spaces of vector bundles or coherent sheaves on maximal varieties.

Let \(X\) be a smooth complex projective variety of dimension \(d\) and \(f\) an automorphism of \(X\). Suppose that the pullback \(f^*|_{N^{1}(X)_{\mathbb{R}}}\) of \(f\) on the real Néron--Severi space \(N^{1}(X)_{\mathbb{R}}\) is unipotent and denote the index of the eigenvalue \(1\) by \(k+1\). We prove an upper bound for the polynomial volume growth \(\textrm{plov}(f)\) of \(f\) as follows:
\(\textrm{plov}(f) \leq (k/2 + 1)d\). Combining with the inequality $k \leq 2(d-1)$ due to Dinh--Lin--Oguiso--Zhang, we obtain an optimal inequality that \(\textrm{plov}(f) \leq d^2\), which affirmatively answers questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang.

代数曲面纤维化的斜率不等式是复代数曲面中的一个重要工具，我们将讨论如何在正特征建立相关的斜率不等式,它与复数域上的联系与区别。作为应用， 我们将介绍由斜率不等式来得到正特征一般型曲面上的Miyaoka-Yau型不等式等。这一报告基于与孙笑涛、周明铄的合作工作。

The coniveau filtration was first defined by Grothendieck to understand the generalized Hodge conjecture. He also raised the question of comparing two versions of such filtrations. For rational coefficients, Deligne's theory of weights proves the equality of the coniveau filtration and a stronger version of the filtration. In this talk I will report on some recent work on the integral version of Grothendieck's question, and indicate some of its arithmetic applications. The approach is based on a study of the topology of the space of cycles.

The moduli of Higgs bundles on a curve can either be viewed as a variant of the moduli of vector bundle on a curve --- a very classical moduli space that has been studied for decades, or the non-abelian Dolbeault cohomology of the curve in view of the non-abelian Hodge theory. In this talk, I will discuss some interesting symmetries of the cohomology of the moduli of Higgs bundles that do not show up for the moduli of vector bundles. I will then explain how different viewpoints lead to completely different proofs of this statement. If time permits, I may discuss some open questions.

I will try to explain Faltings' p-adic Simpson correspondence via the exponential twisting approach. This is a joint work with Zhaofeng Yu in progress.

我们计划讨论代数拓扑的方法在代数cycle理论，特别是周群和Lawson同调的结构方面的一些应用。

近十年来特征 \(p\) 上的三维的极小模型理论正趋近完善，这为代数簇的具体分类奠定了理论基础。接下来我们试图考虑特征 \(p\) 上三维代数簇的具体结构，一类典型的代数簇是 典范除子 \(K\) 数值平凡的代数簇。我们将介绍特征 \(p\) 上一些新的现象和目前得到的一些结果。

For low degrees, the moduli space of quasi-polarized K3 surfaces has several natural compactifications such as the Satake-Baily-Borel compactification, GIT compactification(s), or compactifications based on K-stability. It is natural to reconcile these different compactifications from a geometric and birational point of view. This was done in the eighties by Shah and Looijenga for the degree \(2\) case, and more recently by Laza-O’Grady and Ascher-DeVleming-Liu for degree \(4\). In this talk, I will discuss the degree \(6\) K3 surfaces and make some predictions on the general degree case. Specifically, we give an interpolation between the Baily-Borel and GIT models for degree \(6\) K3 surfaces following the general outlines of the Hassett-Keel-Looijenga (HKL) program. We then discuss the connection to the K-stability point of view.

By a recent result of Wenhao Ou, we have a Miyaoka type inequality \(c_2(X)\cdot c_1(X)^{n-2}\geq 0\) for terminal varieties with nef anti-canonical divisors. In this talk, I will discuss some cases when strict inequality holds and give a rough classification in dimension \(3\). This is joint work with Masataka Iwai and Chen Jiang.

The classical Noether inequality states that \(K_S^2\ge 2p_g(S)-4\) for all minimal surfaces of general type. For irregular minimal surfaces of general type, the stronger Noether inequality \(K_S^2\ge 2p_g(S)\) was proved by O. Debarre. Recently, the optimal Noether inequality for threefolds of general type was proved by Jungkai Chen, Meng Chen and Chen Jiang. In this talk, I will introduce an optimal Noether inequality for almost all irregular threefolds of general type. This is a joint work in progress with Tong Zhang.