By Michael Atiyah (auth.), Vinicio Villani (eds.)

The quantity comprises the texts of the most talks introduced on the overseas Symposium on advanced Geometry and research held in Pisa, may possibly 23-27, 1988. The Symposium used to be geared up at the celebration of the 60th birthday of Edoardo Vesentini. the purpose of the lectures used to be to explain the current scenario, the new advancements and study developments for numerous correct subject matters within the box. The contributions are through unusual mathematicians who've actively collaborated with the mathematical college in Pisa over the last thirty years.

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**Extra info for Complex Geometry and Analysis: Proceedings of the International Symposium in honour of Edoardo Vesentini held in Pisa (Italy), May 23–27, 1988**

**Sample text**

In particular, we have n <_ 2n - dim~°(S) < d i m H l ( M , O ( E n d ( S ) ) ) . The bundle $ is said to be semihomogeneous if for every x E M there is a holomorphic line bundle £ such that Clearly, S is semihomogeneous if and only if dim~°(S) = n. From the inequa~ties above it follows that if d i m H l ( M , O(End(S))) = n, then S is semihomogeneous. An extensive study of semihomogeneous vector bundles has been made by Mukai. 1). For a simple holomorphic vector bundle S on an abelian variety M, of dimension n, the following conditions are mutually equivalent.

We first assume that the deformation is real analytic. 4, restricted to X , , and Y~ be the subbundle of those tangent vectors along the fibers. Let Ot resp. q~ be the sheaves of holomorphic cross sections in ~-, resp. g,. th. , sra(~). ,I define a O-cochain {U~} E C°(U, q t ) in the Cech complex defined by the sections of the sheaf k9, and the covering /d. ~} E Zl(ld, 6);) which, by passing to the limit, yields a cohomology class p,(~) E H i ( X , , 6)~). This is uniquely determined and p, : TB,, -* H~(X,, 6);) is a real linear map.

They occur also as moduli spaces of simple or stable holomorphic vector bundles over a compact holomorphic symplectic manifolds as we see in this section. In order to discuss such moduli spaces, we consider infinite dimensional holomorphic symplectic manifolds. Let V be a complex Banach manifold. A holomorphic symplectic structure on V is given by a holomorphic 2-form wy which is closed and non-degenerate in the following sense: ~v(X,Y)=O V}" E T x V :a X =O. 52 This condition means that the linear map T , V -~ T*V defined by wv is injective (but not necessarily bijective).