Seminar on Geometric Complex Analysis

Seminar information archive ~06/22Next seminarFuture seminars 06/23~

Date, time & place Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Kengo Hirachi, Shigeharu Takayama

2014/01/27

11:00-12:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Junjiro Noguchi (The University of Tokyo)
Logarithmic 1-forms and distributions of entire curves and integral points (JAPANESE)
[ Abstract ]
The Log-Bloch-Ochiai Theorem says, in the most general form so far, that every entire curve in a Zariski open $X$ of a compact Kahler manifold $\bar{X}$ must be degenerate, if $\bar{q}(X)> \dim X$ ([NW02] Noguchi-Winkelmann, Math.\ Z. 239, 2002). If $X$ is defined a quasi-projective algebraic variety defined over a number field, then there is no Zariski dense $(S, D)$-integral subset in $X$ ($D=\partial X=\bar{X}\subset X$). We discuss this kind of properties more.
In the talk we will fix an error in an application in [NW02], and we will show
Theorem 1. (i) Let $M$ be a complex projective algebraic manifold, and let $D=\sum_{j=1}^l D_j$ be a sum of divisors on $M$ which are independent in supports. If $l> \dim M+r(\{D_j\})-q(M)$, then every entire curve $f:\mathbf{C} \to M\setminus D$ must be degenerate.
(ii) Let $M$ and $D_j$ be defined over a number field. If $l> \dim M+r(\{D_j\})-q(M)$, then there is no Zariski-dense $(S,D)$-integral subset of $M\setminus D$.
For the finiteness we obtain
Theorem 2. Let the notation be as above.
(i) If $l \geq 2 \dim M+r(\{D_j\})$, then $M\setminus D$ is completehyperbolic and hyperbolically embedded into $M$.
(ii) Let $M$ and $D_j$ be defined over a number field. If $l> 2\dim M+r(\{D_j\})$, then every $(S,D)$-integral subset of $M\setminus D$ is finite.

Precise definitions will be given in the talk. We will also discuss an application of Theorem 1 (ii) to generalize Siegel's Theorem on integral points on affine curves,
recent due to A. Levin.