By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

ISBN-10: 3540519955

ISBN-13: 9783540519959

"... To sum up, this booklet is helping to profit algebraic geometry very quickly, its concrete type is agreeable for college kids and divulges the great thing about mathematics." --Acta Scientiarum Mathematicarum

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**Additional resources for Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes**

**Example text**

The classical de Rham decomposition of a Riemannian manifold into irreducible factors takes a much more precise form in the special case of a Ricci-flat compact Kiihler manifold. In fact,[7]: If X is a compact Ricci-flat Kiihler manifold, then it has a finite, unramified covering X = T x Ml x . . x Mj x Nl x . . x Nk, where T is a flat, complex torus, the Mi are compact, simply connected irreducible hyperKghIer manifolds, and the Ni are compact, simply connected irreducible Kfihler manifolds which have no holomorphic forms except for the form trivializing the canonical line bundle.

In the decomposition H'(M, O(End(e))) ~_ Hi(M, O(EndÂ°(C))) + HI(M, O) of the tangent space at D" E He(E, h), the holomorphic sectional curvature vanishes in the direction of Hi(M, 0). The question remains whether it is positive in the direction of Hl( M, O( EndÂ°( ED") )). Although we can compute the curvature of 2~I(E, h) explicitly using the Gauss equation for CR-submersions [8], the fact that the holomorphic sectional curvature is nonnegative can be understood intuitively from the general principle that the holomorphic curvature decreases with a holomorphic subbundle and increases with a quotient bundle.

Intrinsic Distances, Measures and Geometric Function Theory. Bull. Am. Math. Soc. 82, 357-416 (1976) - - On Moduli of Vector Bundles Shoshichi K o b a y a s h i D e p a r t m e n t of Mathematics, University of California, Berkeley 1. I n t r o d u c t i o n . We wish to discuss here moduli spaces of simple vector bundles on a compact K~hler manifold M from differential geometric viewpoints, placing emphasis on the case where M is symplectic Kg~hler. In order to explain the construction of such moduli spaces, we consider first an analogous construction of the moduli space of complex structures on a differentiable manifold M.

### Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

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