By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

ISBN-10: 0821834762

ISBN-13: 9780821834763

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**Extra resources for Affine Algebraic Geometry**

**Example text**

4 If c 1 < o, then H 0 (M,T~) = o for all p> q, and every holomorphic tensor field of covariant and contravariant degree p > o is parallel. In particular, the global sections of smne ® Km and Smn Q 1 ® K-m are parallel. 3 Let X be a (not necessarily compact) Kahler manifold of dimension d. e. the group which acts on Ta:X by parallel translation along closed loops which are homotopic to zero. e. its curvature is parallel) or b) ~ 0 is one of the following groups: (i) U(d), (ii) SU(d), (iii) Sp(~).

7 Thus every manifold covered by the non-compact dual D of Qn admits a natural quadric structure. 8 In dimension 2 the noncompact dual Dis isomorphic to the bidisc; thus for instance every product of two compact Riemann surfaces of genus >1 carries a quadric structure. 9 M carries a quadric structure if and only if there is an immersion ~ : ~ ~ Qn of the universal covering such that every covering transformation y satisfies ~Y with some XY E G. 11 the normal bundly of every simply connected complex submanifold of M is generated by its global sections.

Note however that an affine structure need not be quadric in general: the Hopf surface a: 2 */(az 1 ,az 2 +z 1 ) does not even admit a conformal structure (see [12]). 2 We will now introduce an infinitesimal concept corresponding to quadric structures. 1 Consider the symmetric tensor field of covariant degree 2 (as usual we omit the tensor symbol) on a:n+ 2 ' {o}. Evidently w is invariant under the action of G. 3 form given by the matrix Q. •. ,~n+1)Ea:n+2,{o}. 2 the (ga) define a global section g in 48 ~1 0n ® ~1 °n ® 0 JPn+1 (2)1Q n • Actually, g€H 0 (Qn,s 2 ~6 ®OJP (2) 1 0 ), as Q is synunetric.

### Affine Algebraic Geometry by Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

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