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

The first contribution of this EMS quantity with reference to advanced algebraic geometry touches upon the various crucial difficulties during this huge and intensely lively zone of present learn. whereas it truly is a lot too brief to supply whole insurance of this topic, it offers a succinct precis of the parts it covers, whereas supplying in-depth insurance of yes extremely important fields - a few examples of the fields taken care of in better element are theorems of Torelli style, K3 surfaces, version of Hodge constructions and degenerations of algebraic varieties.

the second one half presents a quick and lucid advent to the hot paintings at the interactions among the classical sector of the geometry of complicated algebraic curves and their Jacobian forms, and partial differential equations of mathematical physics. The paper discusses the paintings of Mumford, Novikov, Krichever, and Shiota, and will be a very good spouse to the older classics at the topic by way of Mumford.

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

**Sample text**

F. e differ by a co boundary. X). X) corresponds to tensor product of line bundles. X) is called the Picard group of the manifold X, and denoted by Pic X. 2. The analogue to the concept of a Euclidean vector bundle is that of a Hermitian vector bundle. Definition A holomorphic vector bundle 1r : E -+ X is called a Hermitian vector bundle, if each fiber Ex is equipped with a Hermitian scalar product, which depends smoothly on x E X. Smoothness of the scalar product means that if we choose a basis {e; ( x)}, over an open set U C X, smoothly depending on x E U (in other words we choose a trivialization ¢u : -+ 7r- 1 (U)), then the functions h;j(x) = (e;(x), ej(x)) are of class coo.

For any real closed (1, 1)-form w of class c1 (E) E H 2 (X,C) there exists a metric connection on the line bundle E with curvature form = yCI 21rw. e Periods of Integrals and Hodge Structures 55 Indeed, let isl 2 be a metric on E. As we saw in section 8, if¢: U x C--+ Eu is the trivialization of E over an open set U, then the metric lsl 2 is given by a positive function au : lsl 2 = aususu, while the curvature form and Chern class are given by formulas e = 88logau, c1(E) = [ ~8] E H 2(X,C). For another metric is'l 2 onE with the curvature form 8' we have ef, where f is a real C 00 function.

If X= ]pm and V = IP'n- 1 is a hyperplane, then a direct computation (see, Griffiths-Harris [1978]) that the Chern class c1 ([IP'n- 1 ]) coincides with the cohomology class of the (1, 1)-form associated with the Fubini-Study metric. Let V be a hypersurface of X. The linear functional fv ¢on H 2n- 2(X, Z) defines a homology class (V) E H2n-2(X, Z). The Poincare dual class 1rv E H 2 (X, C) is called the fundamental class of the hypersurface V. Define the fundamental class 1r D E H 2 (X, C) of a divisor D = E r i Vi as Using Stokes' theorem it is not too hard to obtain the following (see GriffithsBarris [1978]).