Development of Geometric Complex Analysis

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In spite of a lot of work in complex analysis and conplex analytic geometry in the last century, there seem to remain unnoticed important questions in the basic theory of several complex variables. Relation between the extension and the sivision problems is very likely one of them. The purpose of this research project was to get a new viewpoint of relating extension and division problems on complex manifolds after the author's previous works on the L^2extention theorems for holomorphic functions. As a result we succeeded in improving a well known L^2division theory of H.Skoda. On the other hand, there was a progress concering Levi flat hypersurfaces : Let X be a complex manifold of dimension n and let M be a real hypersurface of X. M is called Levi flat if it locally separates X into two Stein domains i.e.if M is locally psendoconvex from both sides. In recent works of Lins-Nets and Ohsawa, it was proved that complex projective space of dimension n contains no compact real analytic Levi flat hypersurfaces if n 【greater than or equal】 2. It was extended in a joint work with K.Matsumoto by studying the geometry of Levi flat hypersurfaces in complex tori. Unlike the case of projective spaces, tori contain infinitely many compact Levi flat hypersurfaces. We determined all such jypersurfaces under the assumption of real analyticity when the dimension of tori is 2.