Moduli Spaces of Curves, Mapping Class Groups and Field Theory

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This is a collection of articles that grew out of a workshop organized to discuss deep links among various topics that were previously considered unrelated. Rather than a typical workshop, this gathering was unique as it was structured more like a course for advanced graduate students and research mathematicians.

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This book is the first to present a new area of mathematical research that combines topology, geometry, and logic. Shmuel Weinberger seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity. He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow.

The main sort of algorithmic problem that arises is recognition: is the presented object equivalent to some standard one? If it is difficult to determine whether the problem is solvable, then the original object has doppelgängers--that is, other objects that are extremely difficult to distinguish from it.

Many new questions emerge about the algorithmic nature of known geometric theorems, about "dichotomy problems," and about the metric entropy of moduli space. Weinberger studies them using tools from group theory, computability, differential geometry, and topology, all of which he explains before use. Since several examples are worked out, the overarching principles are set in a clear relief that goes beyond the details of any one problem.

Period Spaces for p-divisible Groups (AM-141)

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Period Spaces for p-divisible Groups (AM-141)
Michael Rapoport , and Thomas Zink
Manufacturer: Princeton University Press
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In this monograph p-adic period domains are associated to arbitrary reductive groups. Using the concept of rigid-analytic period maps the relation of p-adic period domains to moduli space of p-divisible groups is investigated. In addition, non-archimedean uniformization theorems for general Shimura varieties are established.

The exposition includes background material on Grothendieck's "mysterious functor" (Fontaine theory), on moduli problems of p-divisible groups, on rigid analytic spaces, and on the theory of Shimura varieties, as well as an exposition of some aspects of Drinfelds' original construction. In addition, the material is illustrated throughout the book with numerous examples.

Complex Manifolds and Deformation of Complex Structures (Classics in Mathematics)

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A Japanese mathematician
Superb

Complex Manifolds and Deformation of Complex Structures (Classics in Mathematics)
Kunihiko Kodaira
Manufacturer: Springer
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Complex Geometry: An Introduction (Universitext)
Complex Manifolds (AMS Chelsea Publishing)
Differential Analysis on Complex Manifolds (Graduate Texts in Mathematics, Vol 65)
Differential Forms in Algebraic Topology (Graduate Texts in Mathematics)
Algebraic Geometry (Graduate Texts in Mathematics)

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Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algebraic Cobordism (Springer Monographs in Mathematics)
Introduction to Singularities and Deformations (Springer Monographs in Mathematics)

ASIN: 3540226141

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From the reviews:

"The author, [...], has written a book which will be of service to all who are interested in this by now vast subject. [...] This is a book of many virtues: mathematical, historical, and pedagogical. Parts of it could be used for a graduate complex manifolds course.
J.A. Carlson in Mathematical Reviews, 1987

"There are many mathematicians, or even physicists, who would find this book useful and accessible, but its distinctive attribute is the insight it gives into a brilliant mathematician's work. [...] It is intriguing to sense between the lines Spencer's optimism, Kodaira's scepticism or the shadow of Grauert with his very different methods, as it is to hear of the surprises and ironies which appeared on the way. Most of all it is a piece of work which shows mathematics as lying somewhere between discovery and invention, a fact which all mathematicians know, but most inexplicably conceal in their work."
N.J. Hitchin in the Bulletin of the London Mathematical Society, 1987

Customer Reviews:

A Japanese mathematician.......2006-03-01

Kunihiko Kodaira is one of the greatest japanese mathematicians.

He is one of the last students of Teiji Takagi who is famous
for class field theory.

Kunihiko Kodaira died at Kohu, Yamanashi prefecture, Japan in 1997.

He also wrote essays.

Superb.......2005-09-12

Of importance to applications such as superstring theories in high-energy physics, the theory of complex manifolds and the deformation of complex structures are explained in great detail in this book by one of the major contributors to the subject. One of the valuable features of the book that is actually rare in more recent books on mathematics is that the author tries (and succeeds) to give motivation for the subject. This feature is actually quite common in older books on mathematics, for with few exceptions writers at that time believed that a proper understanding of mathematics can only come with explanations that are given outside the deductive structures that are created in the process of doing mathematics. These explanations frequently involve the use of diagrams, pictures, intuitive arguments, and historical analogies, and so are not held to be rigorous from a mathematical standpoint. They are however extremely valuable to students of mathematics and those who are interested in applying it, like physicists and engineers. There seems to be an inverse relationship between rigor and understanding of mathematics, and given the emphasis on the former in modern works of mathematics, one can expect students to have more trouble learning a particular branch of mathematics than those students of a few decades ago.

Luckily though the author of this book has given the reader valuable insights into the nature of complex manifolds and what is means to deform a complex structure. Complex manifolds are different from real manifolds due to the notion of holomorphicity, but are similar in the sense that they are constructed from domains that are "glued together". In complex manifolds, the "glue" is provided by biholomorphic maps between the domains, the latter of which are open sets called `polydisks'. A `deformation' of the complex manifold is then considered to be a glueing of the same polydisks but via a different identification. For an n-dimensional complex manifold, the maps could thus be dependent on say m parameters, which are labeled as "t" by the author. This dependence on t would result in a differentiable family of complex manifolds. One thus expects the complex manifold to be dependent on t, but the author discusses a counterexample that indicates that one must not be cavalier about this approach.

The definition that is arrived at involves letting t be an element of a domain B in m-dimensional Euclidean space, and considering a collection of compact complex n-dimensional manifolds that depends on t. This collection will be a `differentiable family' if: 1. There exists a differentiable manifold M and a C-infinity map W from M onto B such that the rank of the Jacobian matrix of W is equal to m at every point of M. 2. M(t), the inverse image of t under W is a compact connected subset of M, and in fact is equal to a member of the collection. 3. M has a locally finite open covering along with smooth coordinate functions on the covering that have non-empty intersection with each member of the covering. Beginning with an initial element of B, each member of the inverse image of t under W is viewed as a deformation of the initial member. The crucial point made by the author is that the restricting the domain of the parameter t to a sufficiently small interval allows the representation of the member M(t) as a union of polydisks that are independent of t. Therefore only the coordinate transformations depend on t, and thus only the way of glueing the polydisks depends on t.

To show that these constructions are meaningful, namely that the complex structure of M(t) actually depends on t, the author studies the case of m = 1. In the process he constructs the infinitesimal deformation of M(t), and interprets it as the derivative of the complex structure of M(t) with respect to t. He also shows that the infinitesimal deformation does not depend on the choice of systems of local coordinates, and that the infinitesimal deformation vanishes when M(t) does not vary with t. The author then defines, using a notion of equivalence between two differentiable families, a differentiable family (M, B, W) to be `trivial' if it is equivalent to a product (M x B, B, P). Restricting this triviality to a subdomain gives a notion of `local triviality', which implies immediately that each M(t) will be biholomorphically equivalent to a fixed M. He then shows that if the infinitesimal deformation vanishes then the differentiable family is locally trivial. A more substantial statement of this result is encapsulated in the Frolicher-Nijenhuis theorem, which follows from the results that the author proves in the book. These results involve the theory of strongly elliptic differential operators and considerations of the first cohomology group of M(t) with coefficients in the sheaf of germs of holomorphic vector fields over M(t).

The case of a complex analytic family of compact complex manifolds entails that B will be domain in complex n-space. The author shows that a complex analytic family will be trivial if it is trivial as a differentiable family. As expected, because of the nature of analyticity learned from the theory of complex variables, the proof of these results involves the theory of harmonic differential forms. The author gives these proofs in detail in the book. He also considers the question whether if given an element b of the first cohomology group of a compact complex manifold Mwith coefficients in the sheaf of germs of holomorphic vector fields over M, one can find a complex analytic family that takes M as its initial element and the derivative equal to b. This question, as expected, involves the use of obstruction theory, which the author develops in great detail. In these considerations, the reader will see the and origin and role of the moduli of complex structures. These are essentially the number of parameters m, as long as the complex analytic family is `complete.'

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Snowbird Lectures on String Geometry
AMS-IMS-SIAM JOINT SUMMER RESEARCH CONFE , and Katrin Becker
Manufacturer: American Mathematical Society
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The interaction and cross-fertilization of mathematics and physics is ubiquitous in the history of both disciplines. In particular, the recent developments of string theory have led to some relatively new areas of common interest among mathematicians and physicists, some of which are explored in the papers in this volume. These papers provide a reasonably comprehensive sampling of the potential for fruitful interaction between mathematicians and physicists that exists as a result of string theory.

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Moduli Theory
Shigeru Mukai , and W.M. Oxbury
Manufacturer: Cambridge University Press
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Fourier-Mukai Transforms in Algebraic Geometry (Oxford Mathematical Monographs)

ASIN: 0521809061

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Incorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, its influence is not confined there; for example, the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. Researchers and graduate students working in areas ranging from Donaldson or Seiberg-Witten invariants to more concrete problems such as vector bundles on curves will find this to be a valuable resource. Among other things this volume includes an improved presentation of the classical foundations of invariant theory that, in addition to geometers, would be useful to those studying representation theory. This translation gives an accurate account of Mukai's influential Japanese texts.

Moduli of Curves (Graduate Texts in Mathematics)

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Superb

Moduli of Curves (Graduate Texts in Mathematics)
Joe Harris , and Ian Morrison
Manufacturer: Springer
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Principles of Algebraic Geometry
Intersection Theory
Period Mappings and Period Domains (Cambridge Studies in Advanced Mathematics)
Algebraic Geometry: A First Course (Graduate Texts in Mathematics)
The Geometry of Schemes

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Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algebraic Cobordism (Springer Monographs in Mathematics)
Introduction to Singularities and Deformations (Springer Monographs in Mathematics)

ASIN: 0387984291

Book Description

This book provides a guide to a rich and fascinating subject: algebraic curves and how they vary in families. The aim has been to provide a broad but compact overview of the field, which will be accessible to readers with a modest background in algebraic geometry. Many techniques including Hilbert schemes, deformation theory, stable reduction, intersection theory, and geometric invariant theory are developed, with a focus on examples and applications arising in the study of moduli of curves. From such foundations, the book goes on to show how moduli spaces of curves are constructed, to illustrate typical applications with the proofs of the Brill-Noether and Gieseker -Petri theorems via limit linear series, and to survey the most important results about their geometry ranging from irreducibility and complete subvarieties to ample divisors and Kodaira dimension. With over 180 exercises and 70 figures, the book also provides a concise introduction to the main results and open problems about important

Customer Reviews:

Superb.......2001-09-02

The canonical strategy of modern mathematics when studying an object is to put this object into a collection, and see what properties they have in common. Most commonly, the objects depend on some parameter(s), and the goal is to find out how the objects vary with these parameters. The authors of this book take this approach to studying algebraic curves, with the parametrization being called the moduli space, and it enables one to gain information about the geometry of a family of objects from the moduli space and vice versa. The objects are typically schemes, sheaves, or morphisms parametrized by a scheme called the base. Putting an equivalence relation on the families gives a functor, called the moduli functor, which acts on the category of schemes to the category of sets. The functor is representable in the category of schemes if there is an isomorphism between the functor and the functor of points of a scheme. This particular scheme is called the fine moduli space for the functor, as distinguished from the coarse moduli space, where the functor is not representable, i.e. only a natural transformation, and not an isomorphism exists.

The authors clarify the distinction between a moduli space and a parameter space, the former used for problems that involve intrinsic data, the latter for problems involving extrinsic data. An example of the latter, the Hilbert scheme, is discussed in detail in the first chapter, and an example due to Mumford of a component of a Hilbert scheme of space curves that is everywhere nonreduced is given to illustrate the pathologies that can arise in the extrinsic case, and to motivate the use of intrinsic moduli spaces to eliminate these difficulties. Severi varieties are discussed as objects that are more well-behaved than Hilbert schemes but still do not permit a scheme structure to be defined on them so that they represent the functor of families of plane curves with the correct geometric properties.

The second chapter gives a general overview of the approaches taken in the construction of moduli spaces of curves. The authors first study the case of genus 1 (elliptic) curves to illustrate the difficulties involved in constructing fine moduli spaces. The role of automorphisms on the curves as an obstruction to the existence of fine moduli spaces is outlined, as well as approaches to deal with these automorphims, particularly the role of marked points. The authors briefly discuss the role of algebraic spaces and algebraic stacks in the moduli problem. They explain also the various approaches to the construction of the moduli space of smooth curves of genus g, namely the Teichmuller, Hodge-theoretic, and geometric invariant theoretic approaches. The local properties of the moduli space are outlined, along with a discussion of to what extent the moduli space deviates from being a projective or affine variety. The rational cohomology ring of the moduli space is also treated, in low dimensions via the Harer stability theorem, and for high dimensions via the Mumford conjecture. Most interestingly, Witten's conjectures and the Kontsevich formulas are introduced, as a theory of moduli spaces of stable maps. The famous Gromov-Witten invariants of a projective scheme and the quantum cohomology ring are briefly discussed. These have generated an enormous amount of research, the results of which show the power of viewing mathematical constructions from a "quantum" point of view.

The next chapter gives a very specialized overview of the techniques used to study moduli spaces. The authors are very meticulous in their explanations of where the names of the concepts come from, and this is an enormous help to those seeking an in-depth understanding of the topics. One of the first is the dualizing sheaf of a nodal curve, which is the analogue of the canonical bundle of a smooth curve. The authors then describe, by taking a point as the base, the scheme-theoretic automorphism group of a stable curve, and show that it is finite and reduced. Deformation theory is introduced first as over smooth varieties. Readers will appreciate the discussion more if they have a background in the deformation theory of compact, complex manifolds. The authors then tackle the stable reduction problem, and give several beautiful examples, with lots of diagrams, to illustrate the concepts. This is one of the best discussions I have seen in print on these topics. After a brief interlude on the properties of the moduli stack, the authors treat the generalization of the Riemann-Roch formula due to Grothendieck. This section is very important to physicists working in superstring theory. The Porteous formula is also stated and applied to the determination of the class in the rational Picard group of hyperelliptic curves. The determination of the class of the locus of hyperelliptic stable curves of genus 3 is continued in two more sections using the method of test curves and admissible covers.

The actual construction of the moduli space is the subject of chapter 4, from the viewpoint of geometric invariant theory. A nice example of this approach is given for the case of the set of smooth curves of genus 1. The numerical criterion for stability is discussed in detail, with Giesecker's criterion given the main focus. The case of the moduli space of curves with genus greater than two is tackled via the potential stability theorem.

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This book presents a complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space. Most results are new and have never been published before. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables. The book also provides a new approach to Siegel modular forms. This work should serve as a valuable reference source for researchers and graduate students interested in algebraic geometry, Shimura varieties, or diophantine geometry.

Geometric Invariant Theory (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 2 Folge)

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Geometric Invariant Theory (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 2 Folge)
David Mumford , John Fogarty , and Frances Clare Kirwan
Manufacturer: Springer
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Lectures on Curves on an Algebraic Surface. (AM-59) (Annals of Mathematics Studies)
Compact Complex Surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)
Hodge Theory and Complex Algebraic Geometry I (Cambridge Studies in Advanced Mathematics)
Deformations of Algebraic Schemes (Grundlehren der mathematischen Wissenschaften)
Moduli Theory

ASIN: 0387569634

Book Description

"Geometric Invariant Theory" by Mumford/Fogarty (the first edition was published in 1965, a second, enlarged editon appeared in 1982) is the standard reference on applications of invariant theory to the construction of moduli spaces. This third, revised edition has been long awaited for by the mathematical community. It is now appearing in a completely updated and enlarged version with an additional chapter on the moment map by Prof. Frances Kirwan (Oxford) and a fully updated bibliography of work in this area. The book deals firstly with actions of algebraic groups on algebraic varieties, separating orbits by invariants and construction quotient spaces; and secondly with applications of this theory to the construction of moduli spaces. It is a systematic exposition of the geometric aspects of the classical theory of polynomial invariants.

Selected Papers: On the Classification of Varieties and Moduli Spaces

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Selected Papers: On the Classification of Varieties and Moduli Spaces
David Mumford
Manufacturer: Springer
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Fundamental Algebraic Geometry (Mathematical Surveys & Monographs)
Lectures on Curves on an Algebraic Surface. (AM-59) (Annals of Mathematics Studies)
Algebraic Geometry: A First Course (Graduate Texts in Mathematics)
Birational Geometry of Algebraic Varieties (Cambridge Tracts in Mathematics)

ASIN: 038721092X

Book Description

These 30 articles span the years from 1961-1980 while David Mumford was an active researcher in the area of algebraic geometry. In addition, each of the three sections is introduced with never before published commentary by David Gieseker, Eckart Viehweg, and George Kempf and Herbert Lange.

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