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| 開催情報 | 月曜日 10:30～12:00 数理科学研究科棟(駒場) 128号室 |
|---|---|
| 担当者 | 平地 健吾, 高山 茂晴 |
2014年01月27日(月)
11:00-12:00 数理科学研究科棟(駒場) 126号室
野口 潤次郎 氏 (東大数理)
Logarithmic 1-forms and distributions of entire curves and integral points (JAPANESE)
野口 潤次郎 氏 (東大数理)
Logarithmic 1-forms and distributions of entire curves and integral points (JAPANESE)
[ 講演概要 ]
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 ˉX must be degenerate, if ˉq(X)>dimX ([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=∂X=ˉX⊂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=∑lj=1Dj be a sum of divisors on M which are independent in supports. If l>dimM+r({Dj})−q(M), then every entire curve f:C→M∖D must be degenerate.
(ii) Let M and Dj be defined over a number field. If l>dimM+r({Dj})−q(M), then there is no Zariski-dense (S,D)-integral subset of M∖D.
For the finiteness we obtain
Theorem 2. Let the notation be as above.
(i) If l≥2dimM+r({Dj}), then M∖D is completehyperbolic and hyperbolically embedded into M.
(ii) Let M and Dj be defined over a number field. If l>2dimM+r({Dj}), then every (S,D)-integral subset of M∖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.
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 ˉX must be degenerate, if ˉq(X)>dimX ([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=∂X=ˉX⊂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=∑lj=1Dj be a sum of divisors on M which are independent in supports. If l>dimM+r({Dj})−q(M), then every entire curve f:C→M∖D must be degenerate.
(ii) Let M and Dj be defined over a number field. If l>dimM+r({Dj})−q(M), then there is no Zariski-dense (S,D)-integral subset of M∖D.
For the finiteness we obtain
Theorem 2. Let the notation be as above.
(i) If l≥2dimM+r({Dj}), then M∖D is completehyperbolic and hyperbolically embedded into M.
(ii) Let M and Dj be defined over a number field. If l>2dimM+r({Dj}), then every (S,D)-integral subset of M∖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.
