### 2021 Workshop on Algebraic and Analytic Geometry

##### MCM110&Online

The past decades have seen several major breakthroughs in our understanding of the geometry of algebraic varieties from the algebraic/arithmetic geometric side and from the complex analytic side. We wish to bring together some experts in these areas to expound the new ideas and to introduce the recent developments to graduate students, postdoctoral researchers and junior faculty members.
Due to COVID-19, we can not invite the interested participants outside Beijing to be present. The conference will be mixed -- online and offline. We will try to find the best way such that both the speakers online and offline and the participants can interact with each other.

Time:
2021.12.2-2021.12.4

Place:
MCM110&Online
Tencent Meeting Room ID: 804 3553 8041  Password: 120204

Organizing Committee:
Yifei Chen (AMSS CAS)
Baohua Fu (MCM AMSS)
Jie Liu (AMSS CAS)
Wenhao Ou (AMSS CAS)

Academy of Mathematics and Systems Science, CAS
Hua Luo-Keng Center for Mathematical Sciences, CAS
Morningside Center of Mathematics, CAS

Invited Speakers:
Jinxing Cai (Peking University)
Meng Chen (Fudan University)
Yi Gu (Soochow University)
Wenchuan Hu (Sichuan University)
Yong Hu (Shanghai Jiao Tong University)
Fangzhou Jin (Tongji University)
Sheng Rao (Wuhan University)
Mao Sheng (University of Science and Technology of China)
Lei Song (Sun Yat-Sen University)
Xiaotao Sun (Tianjin University)
Zhiwei Wang (Beijing Normal University)
Chuanhao Wei (Westlake University)
Jian Xiao (Tsinghua University)
Xiaokui Yang (Tsinghua University)
Zhiwei Zheng (Yanqi Lake Beijing Institute of Mathematical Sciences and Applications)

Schedule:
 Date Time Speaker Title Video 12.2 9:30 – 10:30 Xiaotao Sun A finite dimensional proof of the Verlinde formula (online) Video 10:50 – 11:50 Yong Hu Algebraic threefolds of general type with small volume (online) 11:50 – 13:40 Break 13:40 – 14:40 Jinxing Cai Automorphisms of an irregular surface of general type acting trivially in cohomology (online) Video 15:00 – 16:00 Meng Chen On projective varieties which are canonically fibred by points, curves and surfaces (online) Video 16:30 – 17:30 Wenchuan Hu The structure of Chow variety and recent results Video 12.3 9:30 – 10:30 Sheng Rao Deformation limit of Moishezon manifolds (online) Video 10:50 – 11:50 Fangzhou Jin Fundamental classes in motivic homotopy theory (online) Video 11:50 – 13:40 Break 13:40 – 14:40 Yi Gu Counterexamples to Fujita’s conjecture in positive characteristic (online) Video 15:00 – 16:00 Xiaokui Yang Geometric positivity and rational connectedness 16:30 – 17:30 Zhiwei Wang On some recent progress related to the extension of quasi-plurisubharmonic functions Video 12.4 9:30 – 10:30 Mao Sheng Tensor product theorem for semistable parabolic $\lambda$-connections (online) 10:50 – 11:50 Chuanhao Wei Kodaira-type vanishings via Nonabelian Hodge Theory (online) 11:50 – 13:40 Break 13:40 – 14:40 Lei Song On the image of Hitchin morphism for algebraic surfaces 15:00 – 16:00 Jian Xiao Geometric inequalities inspired by algebraic geometry 16:30 – 17:30 Zhiwei Zheng Ball quotients in algebraic geometry Video

Titles and Abstracts:
Speaker: Jinxing Cai (Peking University)
Title: Automorphisms of an irregular surface of general type acting trivially in cohomology
Abstracts: Let $S$ be a complex nonsingular minimal projective surface of general type with $q(S)>1$, and let $G$ be the subgroup of automorphisms of $S$ acting trivially on $H^2(S,\mathbb{Q})$. In this talk, we will give a survey on classifications for pairs $(S,G)$.

Speaker: Meng Chen (Fudan University)
Title: On projective varieties which are canonically fibred by points, curves and surfaces
Abstracts: Let $n\geq 2$ be any integer. We prove the optimal lower bound $\nu_{n,n-i}$ of the canonical volume and the optimal upper bound $r_{n,n-i}$ of the canonical stability index for minimal projective $n$-folds, of general type, which are canonically fibered by $i$-folds $(i=0,1,2)$. Apart from the fact $\nu_{n,n}=2$, experts know that $r_{n,n}=n+2$. In this talk, we show that $\nu_{n,n-1}=6/(2n+mod(n,3))$ and $r_{n,n-1}=(5n+mod(n,3)+3)/3$. The values of $r_{n,n-2}$ and $\nu_{n,n-2}$ are obtained up to $n\leq 11$, and are effectively estimated for $n\geq 12$.

Speaker: Yi Gu (Soochow University)
Title: Counterexamples to Fujita’s conjecture in positive characteristic
Abstracts: Let X be a smooth complex projective variety of dimension n, and A be an ample divisor on X. Fujita conjectured that K_X+mA is free and very ample if m≥n+1 and m≥n+2 respectively. In this talk, we will show the counterpart of Fujita’s conjecture in positive characteristic fails. For any integer m and characteristic p>0, we will present an example of a projective surface S that K_X+mA is not free for some ample divisor A. This is a adjoint work with Lei Zhang and Yongmin Zhang.

Speaker: Wenchuan Hu (Sichuan University)
Title: The structure of Chow variety and recent results
Abstracts: We will talk about the structure of Chow variety $C_{p,d(P^n)}$ parameterizing the effective algebraic $p$-cycles of degree $d$ in the projective space $P^n$ and focus on their recent progress, especially a problem of Shafarevich about the rationality of the irreducible component of the Chow variety.

Speaker: Yong Hu (Shanghai Jiao Tong University)
Title: Algebraic threefolds of general type with small volume
Abstracts: It is known that the optimal Noether inequality $\vol(X) \ge \frac{4}{3}p_g(X)- \frac{10}{3}$ holds for every $3$-fold $X$ of general type with $p_g(X) \ge 11$. In this talk, we give a complete classification of $3$-folds $X$ of general type with $p_g(X) \ge 11$ satisfying the above equality by giving the explicit structure of a relative canonical model of $X$. This model coincides with the canonical model of $X$ when $p_g(X) \ge 23$. I would also introduce the second and third optimal Noether inequalities for $3$-folds $X$ of general type with $p_g(X) \ge 11$.This is a joint work with Tong Zhang.

Speaker: Fangzhou Jin (Tongji University)
Title: Fundamental classes in motivic homotopy theory
Abstracts: We generalize Fulton’s intersection theory to the setting of motivic homotopy theory by developing the theory of fundamental classes, and establish (refined) Gysin maps and an excess intersection formula. The construction applies to some non-orientable cohomology theories such as hermitian K-theory and higher Chow-Witt groups. We also develop a theory of Euler classes of vector bundles and prove a motivic Gauss-Bonnet formula, which computes refined Euler characteristics in quadratic forms. This is a joint work with F. Déglise and A. Khan.

Speaker: Sheng Rao (Wuhan University)
Title: Deformation limit of Moishezon manifolds
Abstracts: Let $\pi: \mathcal{X}\rightarrow \Delta$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber $\pi^{-1}(t)$ for each nonzero $t$ in an uncountable subset of $\Delta$ is Moishezon and the reference fiber $X_0$ satisfies the local deformation invariance for Hodge number of type $(0,1)$, then $X_0$ is still Moishezon. This talk is based on the joint works with Yi Li and I-Hsun Tsai.

Speaker: Lei Song (Sun Yat-Sen University)
Title: On the image of Hitchin morphism for algebraic surfaces
Abstracts: It is conjectured by T. H. Chen and B. C. Ngô that the image of Hitchin morphism for the moduli stack of Higgs bundles on a  smooth projective variety is the space of spectral data, generalizing the picture for curves. I will talk about the generalization, and show this is the case for surfaces, by decomposition of spectral data. Some properties of the space for certain surfaces will be discussed. This is a joint work with Hao Sun at SCUT.

Speaker: Mao Sheng (University of Science and Technology of China)
Title: Tensor product theorem for semistable parabolic $\lambda$-connections
Abstracts: Let $k$ be an algebraically closed field and $X$ be a smooth projective variety over $k$. Let $D\subset X$ be a reduced effecitve normal crossing divisor. Fix an ample line bundle $L$ over $X$. We prove that the tensor product of $\mu_L$-semistable parabolic $\lambda$-connections $V_i, i=1,2$ over $(X,D)$ is again $\mu_L$-semistable if $char(k)=0$ or $char(k)=p>0,rk(V_1)+rk(v_2)\leq p+1$. This is a joint work with Jianping Wang.

Speaker: Xiaotao Sun (Tianjin University)
Title: A finite dimensional proof of the Verlinde formula
Abstracts: A formula of dimensions for the spaces of generalized theta functions on moduli spaces of parabolic bundles on a curve of genus g , the so called Verlinde formula,  was predicted by Rational Conformal Field Theories. The proof of Verlinde formula by identifying the spaces of generalized theta functions with the spaces of conformal blocks from physics was given in last century mainly by Beauville and Faltings (so called infinite dimensional proof). Under various conditions, many mathematicians tried to give proofs of Verlinde formula without using of conformal blocks, which are called finite dimensional proofs by Beauville. In this talk, we give unconditionally a purely algebro-geometric proof of Verlinde formula.
Our proof is based on two recurrence relations, one of which establishs an inductive procedure for the genus of curves, another one provides an inductive procedure for the number of parabolic points. This is a joint work with Mingshuo Zhou.

Speaker: Chuanhao Wei (Westlake University)
Title: Kodaira-type vanishings via Nonabelian Hodge Theory
Abstracts: In the past decade, Mochizuki has completed the spectacular theory of mixed Twistor D-modules. In this talk, I will first briefly introduce this result. Then, I will show that Kodaira-type vanishing still holds under the setting of mixed Twistor D-modules, which generalizes Saito’s vanishing under the setting of mixed Hodge Modules. Furthermore, I will also talk about a version of Kawamata-Viehweg vanishing, under this general setting.

Speaker: Zhiwei Wang (Beijing Normal University)
Title: On some recent progress related to the extension of quasi-plurisubharmonic functions
Abstracts: In this talk, I would like to introduce the problem on the extension of quasi-plurisubharmonic functions on compact Kähler manifolds posed by Coman-Guedj-Zeriahi, and some recent progress related to this question. These are based on joint works and helpful discussions with Prof. Fusheng Deng, Jiafu Ning, Xiankui Meng and Xiangyu Zhou.

Speaker: Jian Xiao (Tsinghua University)
Title: Geometric inequalities inspired by algebraic geometry
Abstracts: Geometric inequalities reveal relation between different geometric invariants, such as volume, surface area, width, diameter, etc. By the correspondences between convexity and positivity, such as mixed volumes of convex bodies and intersection numbers of divisors, we present a series of new geometric inequalities inspired by positivity results in algebraic and analytic geometry.

Speaker: Xiaokui Yang (Tsinghua University)
Title: Geometric positivity and rational connectedness
Abstracts: In this presentation, we shall talk about several characterizations of rationally connected varieties by using positivity notions in differential geometry and algebraic geometry. Parts of this talk are based on joint works with D. Li, J. Liu, W. Ou, J. Wang and G. Zhong.

Speaker: Zhiwei Zheng (Yanqi Lake Beijing Institute of Mathematical Sciences and Applications)
Title: Ball Quotients in Algebraic Geometry
Abstracts: Quotients of complex hyperbolic balls appear as moduli spaces in algebraic geometry naturally. For example, moduli spaces of cubic surfaces, cubic threefolds, non-hyperelliptic curves of genus 3 and 4 are arithmetic ball quotients of dimension 4, 10, 6, 9 respecitively. The famous theory by Deligne and Mostow realizes moduli spaces of points on the projective line with specified weight as ball quotients (with dimension at most 9) with finite volume. In this talk I will discuss a ball quotient (with dimension 6) from both perspectives of Deligne-Mostow and periods of K3 surfaces, and the unification of those two constructions. This is a joint work with Yiming Zhong. I will also introduce a new construction of ball quotients from Calabi-Yau threefolds, which is a joint work with Chenglong Yu.