tailieunhanh - Đề tài " Extension properties of meromorphic mappings with values in non-K¨ahler complex manifolds "

Statement of the main result. Denote by Δ(r) the disk of radius r in C, Δ := Δ(1), and for 0 | Annals of Mathematics Extension properties of meromorphic mappings with values in non-K ahler complex manifolds By S. Ivashkovich Annals of Mathematics 160 2004 795 837 Extension properties of meromorphic mappings with values in non-Kahler complex manifolds By S. Ivashkovich 0. Introduction . Statement of the main result. Denote by A r the disk of radius r in C A A 1 and for 0 r 1 denote by A r 1 A A r an annulus in C. Let An r denote the polydisk of radius r in Cn and An An 1 . Let X be a compact complex manifold and consider a meromorphic mapping f from the ring domain An X A r 1 into X. In this paper we shall study the following Question. Suppose we know that for some nonempty open subset U c An our map f extends onto U X A. What is the maximal u D U such that f extends meromorphically onto u X A This is the so-called Hartogs-type extension problem. If u An for any f with values in our X and any initial nonempty u then one says that the Hartogs-type extension theorem holds for meromorphic mappings into this X. For X C . for holomorphic functions the Hartogs-type extension theorem was proved by F. Hartogs in Ha . If X CP1 . for meromorphic functions the result is due to E. Levi see Lv . Since then the Hartogs-type extension theorem has been proved in at least two essentially more general cases than just holomorphic or meromorphic functions. Namely for mappings into Kahler manifolds and into manifolds carrying complete Hermitian metrics of nonpositive holomorphic sectional curvature see Gr Iv-3 Si-2 Sh-1 . The goal of this paper is to initiate the systematic study of extension properties of meromorphic mappings with values in non-Kahler complex manifolds. Let h be some Hermitian metric on a complex manifold X and let Wh be the associated 1 1 -form. We call Wh and h itself pluriclosed or ddc-closed if ddcwh 0. In the sequel we shall not distinguish between Hermitian metrics and their associated forms. The latter we shall call simply metric forms. This .