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Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians: Encyclopaedia of Mathematical Sciences, cartea 36

Editat de A.N. Parshin Traducere de I. Rivin Contribuţii de V.S. Kulikov Editat de I.R. Shafarevich Contribuţii de P.F. Kurchanov, V.V. Shokurov
en Limba Engleză Hardback – 8 dec 1997
Starting with the end of the seventeenth century, one of the most interesting directions in mathematics (attracting the attention as J. Bernoulli, Euler, Jacobi, Legendre, Abel, among others) has been the study of integrals of the form r dz l Aw(T) = -, TO W where w is an algebraic function of z. Such integrals are now called abelian. Let us examine the simplest instance of an abelian integral, one where w is defined by the polynomial equation (1) where the polynomial on the right hand side has no multiple roots. In this case the function Aw is called an elliptic integral. The value of Aw is determined up to mv + nv , where v and v are complex numbers, and m and n are 1 2 1 2 integers. The set of linear combinations mv+ nv forms a lattice H C C, and 1 2 so to each elliptic integral Aw we can associate the torus C/ H. 2 On the other hand, equation (1) defines a curve in the affine plane C = 2 2 {(z,w)}. Let us complete C2 to the projective plane lP' = lP' (C) by the addition of the "line at infinity", and let us also complete the curve defined 2 by equation (1). The result will be a nonsingular closed curve E C lP' (which can also be viewed as a Riemann surface). Such a curve is called an elliptic curve.
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Specificații

ISBN-13: 9783540546818
ISBN-10: 3540546812
Pagini: 270
Ilustrații: VIII, 270 p.
Dimensiuni: 155 x 235 x 19 mm
Greutate: 0.58 kg
Ediția:1998
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Encyclopaedia of Mathematical Sciences

Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

Cuprins

I. Complex Algebraic Varieties: Periods of Integrals and Hodge Structures.- II. Algebraic Curves and Their Jacobians.

Textul de pe ultima copertă

The first contribution of this EMS volume on the subject of complex algebraic geometry touches upon many of the central problems in this vast and very active area of current research. While it is much too short to provide complete coverage of this subject, it provides a succinct summary of the areas it covers, while providing in-depth coverage of certain very important fields - some examples of the fields treated in greater detail are theorems of Torelli type, K3 surfaces, variation of Hodge structures and degenerations of algebraic varieties.
The second part provides a brief and lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties, and partial differential equations of mathematical physics. The paper discusses the work of Mumford, Novikov, Krichever, and Shiota, and would be an excellent companion to the older classics on the subject by Mumford.