By Viktor S. Kulikov, P. F. Kurchanov, V. V. Shokurov (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

ISBN-10: 3642081185

ISBN-13: 9783642081187

ISBN-10: 3662036622

ISBN-13: 9783662036624

The first contribution of this EMS quantity almost about complicated algebraic geometry touches upon a number of the critical difficulties during this massive and extremely energetic zone of present learn. whereas it's a lot too brief to supply entire assurance of this topic, it presents a succinct precis of the parts it covers, whereas offering in-depth insurance of yes extremely important fields - a few examples of the fields taken care of in higher aspect are theorems of Torelli sort, K3 surfaces, version of Hodge constructions and degenerations of algebraic varieties.

the second one half presents a quick and lucid creation to the new paintings at the interactions among the classical quarter of the geometry of complicated algebraic curves and their Jacobian types, and partial differential equations of mathematical physics. The paper discusses the paintings of Mumford, Novikov, Krichever, and Shiota, and will be an exceptional better half to the older classics at the topic through Mumford.

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**Additional info for Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians**

**Example text**

S. Kulikov, P. F. Kurchanov 28 §5. Connections on Holomorphic Vector Bundles One of the central concepts of differential geometry is that of an affine connection, which makes it possible to define the concept of parallel translation on vector bundles. In this section we extend the concept of an affine connection into the complex setting. 1. The generalization of the concept of a vector bundle to the complex setting is the concept of a holomorphic vector bundle. Definition. A holomorphic mapping vector bundle of rank n if 1r : E -+ X is called a holomorphic 1) There exists an open cover {Ua} of the manifold X and biholomorphic mappings cPa en X Ua -+ 7r- 1 (Ua), such that the following diagram commutes: en X Ua _ _ _¢_"'_ _ _ 7r- 1 (Ua) ~/.

Let (u 0 : ... : un) be homogeneous coordinates in lPn. Consider the differential (1, 1) form n1 = i88log t k=O I ~k 2 1 J in the neighborhood { u 1 =1- 0}. Since in the open set {u 1 =f- 0, u 1 =f- 0} 88log I~: 1= o, 2 the forms nj and nl coincide in that neighborhood and thus they define a form n globally on lPn. This form is closed since d88 = 8 2 8 - 882 = 0. Let z1 = ~~, 0}. Then j = 1, ... , n be nonhomogeneous coordinates in U0 = {u 0 =f- where H = log(1 + L lz1l 2 ). Evidently, (wr,s) is a Hermitian matrix.

Now, suppose the line bundle E is positive. Then, by Kodaira's theorem, E®n = [V], where V is the divisor of a hyperplane section of X under some inclusion X'-+ JIPN. Let {}be the form associated to the Kahler metric on X, and let e be the curvature form of the metric connection. We can define an operator by setting and we have the operator = {} 1\ 'Tl ® s, );rL. L(ry ® s) e= Vik. S. Kulikov, P. F. Kurchanov 58 Let D = D' + 8 be the metric connection on E. Then the operator 8 can be interpreted as 81] = D 2 1J.

### Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians by Viktor S. Kulikov, P. F. Kurchanov, V. V. Shokurov (auth.), A. N. Parshin, I. R. Shafarevich (eds.)

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