中科院数学与系统科学研究院
数学研究所
学术报告
多复变与复几何研讨班
报告人： Viet-Anh Nguyen教授（法国里尔大学）
题 目： The generalized Lelong numbers and intersection theory
时 间：2024.02.19（星期一）下午1：30-3:00
地 点：Zoom Meeting : 857 8027 2443 密码：315719
https://us06web.zoom.us/j/85780272443?pwd=rkark0a1DFAbT4pNfEaEtcqq4hHEad.1
摘 要：
The notion of Lelong number $\nu(T,x)$ of a positive closed current $T$ at a single point $x$ in an ambient complex manifold $X$ plays a fundamental role in Complex Analysis and Complex Geometry. In 1982 Henri Skoda formulated this notion for the more general class of positive plurisubharmonic currents. In this lecture we introduce a new concept of the generalized Lelong numbers $\nu_j(T,V),$ where $V$ is a submanifold in $X$ and $T$ is a positive plurisubharmonic current in $X$. In general, we have $\dim V +1$ generalized Lelong numbers associated to $T$ along $V.$ The classical case where $V=\{x\}$ corresponds to $\dim V=0.$
Our present research is inspired by two works. The first one is the theory of tangent currents for positive closed currents which were developed by Tien-Cuong Dinh and Nessim Sibony (2018). The second work is he theory of the Lelong number for positive plurisubharmonic currents along a complex linear subspace in $\C^n$ which were developed by Lucia Alessandrini and Giovanni Bassanelli (1996).
Next, we study these new numerical values and establish their basic properties. In particular, we obtain geometric characterizations as well as an upper-semicontinuity of the generalized Lelong numbers in the sense of Yum-Tong Siu (1974). When the current $T$ is positive closed, we also establish some links between the generalized Lelong numbers and Dinh-Sibony cohomology classes.
Finally, as an application we give an effective condition (in terms of the generalized Lelong numbers) ensuring that $m$ positive closed currents $T_1,\ldots,T_m$ of possibly different bidegrees $(p_j,p_j)$ for $1\leq j\leq m$ on a compact Kähler manifold $X$ are wedgeable in the sense of Dinh-Sibony.
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