## 研究生多复变与复几何短期课程

课程名称： Some aspects related to hyperbolicity problem

课程安排：分两部分，每部分8课时，共计16课时

第一部分： Density of positive closed currents and some applications

主讲人： Dinh Tuan HUYNH (AMSS Beijing)

时间：2021年5月12/19/26， 6月2日下午14:00-16:00.

地点：后主楼1220.

摘要：

Recently, Dinh-Sibony introduced the notion of density currents associated to a family $\{T_i\}_{i=1}^q$ of finite positive closed currents on a compact K\"{a}hler manifold. In the case where such density current is uniquely determined, it could be used to define a suitable wedge product of $T_i$, called {\sl Dinh--Sibony's} product. The notion of density currents extends the classical theory of intersection for positive closed currents, and has several deep applications on complex dynamical systems.

In the first part, we will recall some fundamental works on theory of intersection of positive closed currents. We then describe the main result in the theory of density currents. If time permits, we will explain some key applications of this theory.

In the second part, we will report our recent work on studying the Monge--Amp\`{e}re operator and comparing it with Dinh-Sibony's product defined via density currents. We show that if $u$ is a plurisubharmonic (p.s.h) function belonging to the B\l ocki-Cegrell class, which is the largest subset of p.s.h functions on some domain in $\mathbb{C}^n$ where one can define a Monge--Amp\`{e}re operator that coincides with the usual one for smooth p.s.h functions and which is continuous under decreasing sequences, then the Dinh-Sibony $n$-fold self-product of $\ddc u$ exists and coincides with $(\ddc u)^n$. This means that the domain of definition of the Monge--Amp\`{e}re operator defined via Dinh-Sibony's product contains the B\l ocki-Cegrell class.

In the third part, we will give a brief introduction to value distribution theory. We will present some classical results such as First Main Theorem, Cartan's Second Main Theorem. If time permits, we will also talk about the application of jet differential forms in hyperbolicity problem.

In the fourth part, we will discuss about an application of density theory of currents in value distribution theory. Let $f :\mathbb{C}\rightarrow X$ be a transcendental holomorphic curve into a projective manifold $X$. Based on the theory of density currents, we show that given a very ample line bundle $L$, there is an exceptional set of divisors which is a countable union of proper algebraic subsets of the space of effective divisors generated by global sections of $L$ such that for every divisor $D$ outside this set, the geometric defect of $D$ (i.e, the defect of truncation $1$) with respect to $f$ is zero. This result could be regarded as a generalization of the classical Casorati-Weierstrass Theorem, as well as a weak version of the fundamental conjecture for entire holomorphic curves into projective varieties in the case where the canonical line bundle is $\leq 0$.

主讲人简介：Dinh Tuan Huynh got Ph.D. in Orsay 2016 under the supervision of Julien Duval. After that he spend two years in Osaka University and 1 year in Bonn. Now he is a postdoc in AMSS, Beijing. He is one of the leading experts in Nevanlinna theory.