Complex Geometry and Analysis: Proceedings of the by Michael Atiyah (auth.), Vinicio Villani (eds.)

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

The quantity includes the texts of the most talks introduced on the foreign Symposium on complicated Geometry and research held in Pisa, might 23-27, 1988. The Symposium was once geared up at the get together of the 60th birthday of Edoardo Vesentini. the purpose of the lectures was once to explain the current scenario, the hot advancements and learn developments for a number of proper subject matters within the box. The contributions are by way of individual mathematicians who've actively collaborated with the mathematical tuition in Pisa during the last thirty years.

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Example text

This can be proved like in [KS,II], but we also can get this immediately by using a general t h e o r e m stated in [F1]. 3 If ~r : X --* B is semiuniversal in t E B for holomorphic deformations, then it there also is semiuniversal for real analytic deformations. , x l . th. , s ~ . th. ,~lu = X o & and d& is uniquely determined. By restricting a = &luna we have Zlcrnc = X o a . 0 and ~ to a the m a p do is uniquely determined. f. [Abi] or see below). We need some preparations: We put Y ' = Y \ { P 1 , .

Desired properties. So o u r d e f o r m a t i o n t~ : Z ---+ G h a s all W e n o w c a n p r o v e t h e following: 4 . 1). , s k ) -+ B' ) depend real analytically on B ' . , s ' ) depends real analytically on the fiber parameter ~ C B I . Theorem Proof. L e t t E B t b e a r b i t r a r y . 2 t h e r e e x i s t s a h o l o m o r p h i c s e m i u n i v e r s a l d e f o r m a t i o n 7r : X -+ B C (P~ w i t h X0 = X~ a n d n = dirn~ HI(Xo, ®~). 3 we k n o w t h a t lr : X -+ B is also s e m i u n i v e r s a l for r e a l a n a l y t i c deformations.

H o l o m o r p h i c S y m p l e c t i c S t r u c t u r e s . [9,12] We consider now K~ihler manifolds with holomorphic symplectic structure or hyperKiihler manifolds. While Atiyah's article in this volume treats hyper-K~ihler manifolds more from the real or quaternionic viewpoint, we shall emphasize the complex analytic viewpoint. 2, the reader is referred to Atiyah's article and references therein. A hyper-Kiihler manifold is a Kiihler manifold with a nongenerate holomorphic, parallel 2-form.

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