Geometric Invariant Theory

Author: David Mumford
Publisher: Springer Science & Business Media
ISBN: 9783540569633
File Size: 22,65 MB
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"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants.

Geometric Invariant Theory

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Number Theory

Author: David V. Chudnovsky
Publisher: Springer Science & Business Media
ISBN: 1475741588
File Size: 53,44 MB
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New York Number Theory Seminar started its regular meeting in January, 1982. The Seminar has been meeting on a regular basis weekly during the academic year since then. The meeting place of the seminar is in midtown Manhattan at the Graduate School and University Center of the City Uni versity of New York. This central location allows number-theorists in the New York metropolitan area and vistors an easy access. Four volumes of the Seminar proceedings, containing expanded texts of Seminar's lectures had been published in the Springer's Lecture Notes in Mathematics series as volumes 1052 (1984), 1135 (1985), 1240 (1987), and 1383 (1989). Seminar co chairmen are pleased that some of the contributions to the Seminar opened new avenues of research in Number Theory and related areas. On a histori cal note, one of such contributions proved to be a contribution by P. Landweber. In addition to classical and modern Number Theory, this Semi nar encourages Computational Number Theory. This book presents a selection of invited lectures presented at the New York Number Theory Seminar during 1989-1990. These papers cover wide areas of Number Theory, particularly modular functions, Aigebraic and Diophantine Geometry, and Computational Number Theory. The review of C-L. Chai presents a broad view of the moduli of Abelian varieties based on recent work of the author and many other prominent experts. This provides the reader interested in Diophantine Analysis with access to state of the art research. The paper of D. V. and G. V.

Symmetry And Spaces

Author: H.E.A. Eddy Campbell
Publisher: Springer Science & Business Media
ISBN: 0817648755
File Size: 46,18 MB
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This volume includes articles that are a sampling of modern day algebraic geometry with associated group actions from its leading experts. There are three papers examining various aspects of modular invariant theory (Broer, Elmer and Fleischmann, Shank and Wehlau), and seven papers concentrating on characteristic 0 (Brion, Daigle and Freudenberg, Greb and Heinzner, Helminck, Kostant, Kraft and Wallach, Traves).

The Geometry Of Some Special Arithmetic Quotients

Author: Bruce Hunt
Publisher: Springer
ISBN: 354069997X
File Size: 51,95 MB
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The book discusses a series of higher-dimensional moduli spaces, of abelian varieties, cubic and K3 surfaces, which have embeddings in projective spaces as very special algebraic varieties. Many of these were known classically, but in the last chapter a new such variety, a quintic fourfold, is introduced and studied. The text will be of interest to all involved in the study of moduli spaces with symmetries, and contains in addition a wealth of material which has been only accessible in very old sources, including a detailed presentation of the solution of the equation of 27th degree for the lines on a cubic surface.

Compact Complex Surfaces

Author: W. Barth
Publisher: Springer
ISBN: 3642577393
File Size: 40,85 MB
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In the 19 years which passed since the first edition was published, several important developments have taken place in the theory of surfaces. The most sensational one concerns the differentiable structure of surfaces. Twenty years ago very little was known about differentiable structures on 4-manifolds, but in the meantime Donaldson on the one hand and Seiberg and Witten on the other hand, have found, inspired by gauge theory, totally new invariants. Strikingly, together with the theory explained in this book these invariants yield a wealth of new results about the differentiable structure of algebraic surfaces. Other developments include the systematic use of nef-divisors (in ac cordance with the progress made in the classification of higher dimensional algebraic varieties), a better understanding of Kahler structures on surfaces, and Reider's new approach to adjoint mappings. All these developments have been incorporated in the present edition, though the Donaldson and Seiberg-Witten theory only by way of examples. Of course we use the opportunity to correct some minor mistakes, which we ether have discovered ourselves or which were communicated to us by careful readers to whom we are much obliged.

L2 Invariants Theory And Applications To Geometry And K Theory

Author: Wolfgang Lück
Publisher: Springer Science & Business Media
ISBN: 3662046873
File Size: 12,42 MB
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In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. The book, written in an accessible manner, presents a comprehensive introduction to this area of research, as well as its most recent results and developments.

Cohomology Theory Of Topological Transformation Groups

Author: W.Y. Hsiang
Publisher: Springer Science & Business Media
ISBN: 3642660525
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Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of L. E. 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of P. A. Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tl by more general compact groups.

Quasi Projective Moduli For Polarized Manifolds

Author: Eckart Viehweg
Publisher: Springer Science & Business Media
ISBN: 3642797458
File Size: 69,26 MB
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The concept of moduli goes back to B. Riemann, who shows in [68] that the isomorphism class of a Riemann surface of genus 9 ~ 2 depends on 3g - 3 parameters, which he proposes to name "moduli". A precise formulation of global moduli problems in algebraic geometry, the definition of moduli schemes or of algebraic moduli spaces for curves and for certain higher dimensional manifolds have only been given recently (A. Grothendieck, D. Mumford, see [59]), as well as solutions in some cases. It is the aim of this monograph to present methods which allow over a field of characteristic zero to construct certain moduli schemes together with an ample sheaf. Our main source of inspiration is D. Mumford's "Geometric In variant Theory". We will recall the necessary tools from his book [59] and prove the "Hilbert-Mumford Criterion" and some modified version for the stability of points under group actions. As in [78], a careful study of positivity proper ties of direct image sheaves allows to use this criterion to construct moduli as quasi-projective schemes for canonically polarized manifolds and for polarized manifolds with a semi-ample canonical sheaf.

Positivity In Algebraic Geometry 2

Author: R.K. Lazarsfeld
Publisher: Springer Science & Business Media
ISBN: 9783540225348
File Size: 63,23 MB
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This two volume work on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Whereas Volume I is more elementary, the present Volume II is more at the research level and somewhat more specialized. Both volumes are also available as hardcover edition as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".