論文の公開元へHigher dimensional case of sharper estimates of Ohsawa–Takegoshi L^2-extension theorem
概要
The Ohsawa–Takegoshi L 2 -extension theorem [21] states the following: Let Ω ⊂ C n be a bounded pseudoconvex domain, V be a closed complex submanifold of Ω and φ be a plurisubharmonic function on Ω. Suppose that a holomorphic function f ∈ O(V ) is given. Then there exists an F ∈ O(Ω) satisfying the L 2 -estimate
∫ Ω |F| 2 e −φ ≤ C ∫ V |f| 2 e −φ ,
where C is the positive constant that is independent of the weight φ and a given f ∈ O(V ). The Ohsawa–Takegoshi L 2 -extension theorem and its generalizations are applied widely in the studies of complex analysis as well as complex geometry and algebraic geometry. For example, Demailly’s approximation of plurisubharmonic functions [9], Siu’s invariance of plurigenera [23] and so forth. The Green function on a domain in C is a solution of the Laplace equation. More precisely, this function is the upper envelope of negative subharmonic functions with a logarithmic pole at the given point. Since the Green function have many information of the domain on which this is defined, many studies were conducted in complex analysis. The Suita conjecture was a long-standing conjecture about a relationship between Bergman kernels and logarithmic capacities. The study of the interplay of the Ohsawa–Takegoshi L 2 -extension theorem and the Green function was essential in its resolution.
In [5], [6] and [11], the optimal L 2 -extension theorem was proved. This means that we can determine the positive constant C in the best possible way. Many problems including Suita conjecture were solved by using the optimal L 2 -estimate. Here we state the Berndtsson– Lempert type optimal L 2 -extension theorem [5].
