Download Algebraic Geometry III: Complex Algebraic Varieties by Viktor S. Kulikov, P. F. Kurchanov, V. V. Shokurov (auth.), PDF

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 almost about complicated algebraic geometry touches upon some of the critical difficulties during this monstrous and extremely energetic sector of present examine. whereas it really is a lot too brief to supply whole insurance of this topic, it offers a succinct precis of the components it covers, whereas offering in-depth insurance of sure vitally important fields - a few examples of the fields taken care of in higher aspect are theorems of Torelli kind, K3 surfaces, version of Hodge buildings and degenerations of algebraic varieties.
the second one half presents a short and lucid advent to the new paintings at the interactions among the classical zone of the geometry of advanced 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 superb better half to the older classics at the topic by means of Mumford.

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This ebook through Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
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Additional info for Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians

Example text

Indeed, since D" = 8, then D" 2 = 0, and hence 8°• 2 = 0. But with respect to a unitary basis {e }, the matrix Be is skew-Hermitian, hence e =dO- 01\0 is also skew-Hermitian. Therefore, 8 2,0 = _t8 o,2 = o. Let D be the metric connection on an Hermitian bundle E ---* X. Jc(E*)), over an open set U in X. In particular, if {ei} is a basis of the bundle E over U and {ei} is the dual basis of E*, and 0 and 0* are the corresponding connection matrices, then 0 = d(ei, ej) = Oii hence 0 =-to*. + Oji, (5) Periods of Integrals and Hodge Structures 33 §6.

In order to do this, consider a discrete subgroup (lattice) r of en generated by 2n vectors linearly independent over JR. Let T = en I r be the quotient complex torus. The complex structure on en induces a complex structure on T and gives it the structure of a Kahler manifold, whose Kahler metric is induced by an arbitrary hermitian metric hijdZi ® d:Zj with constant coefficients on en . Conversely, given a Kahler metric ds 2 = "'£ hij (z )dzi ® tlzj on the torus T, we can integrate the coefficients of this metric over T to obtain a Kahler metric with constant coefficients "'£ hijdZi ® tlzj, where where dV is a translation-invariant volume form, rescaled so that the volume of T is 1.

Theorem. If L = [D] for some divisor D on a compact complex manifold X, then c1(L) = 7rD· In the exact sequence (14) the morphism j : H 2 (X, Z)--+ H 2 (X, Ox) can be represented as a composition If X is a compact Kahler manifold, it can be shown that the morphism a coincides with the projection II0 ,2 onto the space of harmonic (0, 2)-forms, and hence the kernel of a contains the cocycles of H'f 1 (Z) C H 2 (X, Z) which can be represented by closed (1, 1)-forms. Since the' Chern classes c1(L) E H 2 (X, C) are represented by (1, 1)-forms, by exactness of the sequence (14) we get Theorem.

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