By Vasile Brinzanescu

ISBN-10: 3540498451

ISBN-13: 9783540498452

ISBN-10: 3540610189

ISBN-13: 9783540610182

The goal of this booklet is to give the on hand (sometimes basically partial) ideas to the 2 basic difficulties: the life challenge and the type challenge for holomorphic buildings in a given topological vector package deal over a compact complicated floor. specific gains of the nonalgebraic surfaces case, like irreducible vector bundles and balance with recognize to a Gauduchon metric, are thought of. The reader calls for a grounding in geometry at graduate scholar point. The e-book might be of curiosity to graduate scholars and researchers in complicated, algebraic and differential geometry.

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**Additional info for Holomorphic Vector Bundles over Compact Complex Surfaces**

**Example text**

A compact, reduced, connected curve C on a nonsingular surface X is called exceptional, if there is a bimeromorphic map 7r : X --+ Y, such that there is an open neighbourhood U of C in X , a point y E I/, and a neighbourhood V of y in Y with the property that 7r : U \ C --+ V \ {y} is biholomorphie and 7r(C) = y. We shall say also in this case that C is contracted to y. CN) is negative definite. e. 3 Classification and examples of surfaces 39 frequently called (-1)-curves. In fact, the contraction of a (-1)-curve is the inverse construction of a a-process.

E. F is a free subgroup of C 2 of rank 4). 3 Let F be defined by the vectors Vl, v2, v3, v 4 linearly independent over IR, F = 7/Vl + 7/v2 + 7/v3 + 77v4 and let X = C2/F. The complex surface X is homeomorphic to (IR/7/) 4 and H2(X, 77) _~ 776 where the group H2(X, 77) is generated by the cycles Sij , 1 < i < j <_ 4, obtained as the images of the planes ]Rvi + IRvj by the natural projection (~2 _+ X. Let C C X be a curve on X. Taking a triangulation of the normalization of C we can consider the curve C in a natural way as a singular cycle on X.

3), the existence of a holomorphic structure in a given topological vector bundle is equivalent to the existence of an algebraic structure. Let us consider the case dim X = 3. If E is topological vector bundle of rank r > 3, then the Chern classes (Cl(E), c2(E), c3(E)) determine up to an isomorphism the bundle E. This fact follows by the above mentioned results and it was firstly proved in [Bc]. If r = rk(E) > 3, then E ~ E ' @ (r - 3)11, where E ' is a vector bundle of rank 3, uniquely determined (up to an isomorphism) by the isomorphism class of E.

### Holomorphic Vector Bundles over Compact Complex Surfaces by Vasile Brinzanescu

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