By V. Villani (Ed.)
ISBN-10: 3540524347
ISBN-13: 9783540524342
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Let M be a compact complex manifold with a holomorphic symplectic fOrTl~ OdM and E a C °° complex vector bundle over M. Then ¢dM induces, in a natural way, a holomorphic symplectic form on the nonsingular part of ](A(E). This theorem was first proved by Mukai [12] when M is an abelian surface or a K3 surface. 2, we apply the reduction theorem to Y = L2(7)"(E)), G = L~+I(GL(E))/C*, g = L~+I(gl(E))/C, where L~ and Lk+l, 2 (k > dimM), denote the Sobolev completion. We define a holomorphic symplectic form wy on V by wV(a,/3)= / M t r ( a A ~ ) A w ~ l A ~ 4 - 1 a, Z6TD,,(V), where ce and ~ axe considered as elements of L~(AO,I(End(E))) the complex dimension of M.
A uP), we would get only a p-form arising from the fibering A t ( E ) -~ Pic°(M). 3. O n C u r v a t u r e o f M o d u l i S p a c e s o f B u n d l e s o v e r C u r v e s . [1,3,5,8,9] Let M be a compact Riemann surface of genus g, and let E be a C ~° complex vector bundle of rank r over M. 1 that the moduli space A~i(E) of simple holomorphic structures on E is a (possibly non-Hausdorff) nonsingular complex manifold of dimension r2(g - 1) + 1. For a fixed line bundle £, the moduli space A~4(E, £) of holomorphic structures with prescribed determinant bundle L: has dimension r2(g - 1) + 1 - g = (r 2 - 1)(g - 1).
The analytic continuation of I , ( r , s) for s m o o t h ~ is based on the following two lemmas. Ind~(I LEMMA 1. Let ~o E 7r C Ip ® ~) be smooth. ( I I~ ~ o is a representation of P via m n ~-* lamlaa(hm)). Then I~(r, s) has a meromorphic continuation to all of C with possible poles at s =- i p - k for k = 0 , 1 , 2 , - . . l f p ~ 0 these poles are simple. I~(T, S) Can have a pole at : k p - k only if T is a constituent of a ® R ~ where R is the standard representation of H -- SO(r) on R". The point of this l e m m a is t h a t if T hes in an irreducible representation lr, then formulas of Wallach [6] and Casselman [1] give explicit asymptotics for W~(m) as a,~ --* 0.
Complex Geometry and Analysis by V. Villani (Ed.)
by Richard
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