Download PDF by Pogorelov, A.V.: Analytical geometry

By Pogorelov, A.V.

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The case n = 2 can be treated in a similar way. Part ii) also follows from Theorem 8 remarking that any two smooth hyperquadricS in pn are projectively isomorphic. D. 31 Using Proposition 3 and the same method as in the proof of Theorem 8, part i) of the abov~ corollary can be generalized in the following way. Theorem ~. Le t n, q l " ' ' ' q i be natural numbers such that n>/2 and qj-! ~i. If n - 2 assume moreover that char(k) = o. l~,%l, .... qi ) is rigid. n+l times 1 1 Theorem io (Char(k)= o).

Z . p(R*) } q ~Z and let 0p(R, ) (i) be the t a u t o l o g i c a l the notations: tion q* T[=q"'(TI) ,T~=~*(T2) (R~)~Op(R,)(1) gives invertible and y'=q'* (y). sheaf on p(R*), we The canonical use surjec- a morphism 0p (R)(-1)-q*(R)--Ext,,(Iy~q,r~), hence a global section ~ in Ext,, Let now V be an affine =X×V. We have open the i s o m o r p h i s m (Iy,@T{,T2@P'*(Op(R,)(1))). subset in M(d,r)~p(R*) and V'=p'-I(v) = 42 Ext I ( V , ; . . )) (v';Iy,®T~,T~Op'" (0p (R') (I)) be m e n t of [I V" The e l e m e n t ~ gives an e x t e n s i o n the c o r r e s p o n d i n g ele- on V' O-T'~p'*(02 p(R*) (l))-F-Iy, ~ T[-O where the sheaf F is o b v i o u s l y flat over V.

T o p o l o g i s c h triviale h o l o m o r m h e V e k t o r b ~ n d e l auf ~n, P r e p r i n t INCREST, (1982 J. reine angew. Math. 344 (1983) lo2-119. : V a r i a t i o n der g l o b a l e n Ext in D e f o r m a t i o n e n k o m p a k t e r k o m p l e x e r Raume, M a t h . A n n . 2 5 0 ( 1 9 8 0 ) . : T o p o l o g i c a l ~ t r i v i a l a l g e b r a i c 2-vector bundles on ruled surfaces I, P r e p r i n t INCREST (1981); (to appear in Rev. R o u m . M a t h . ). : On the moduli space for stable rank-2 vector bundles on p2.

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Analytical geometry by Pogorelov, A.V.

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