hodge theory and complex algebraic geometry i volume 1 cambridge studies in advanced mathematics

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Hodge Theory And Complex Algebraic Geometry I

Author : Claire Voisin
ISBN : 1139437690
Genre : Mathematics
File Size : 32. 15 MB
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The first of two volumes offering a modern introduction to Kaehlerian geometry and Hodge structure. The book starts with basic material on complex variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory, the latter being treated in a more theoretical way than is usual in geometry. The author then proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The book culminates with the Hodge decomposition theorem. The meanings of these results are investigated in several directions. Completely self-contained, the book is ideal for students, while its content gives an account of Hodge theory and complex algebraic geometry as has been developed by P. Griffiths and his school, by P. Deligne, and by S. Bloch. The text is complemented by exercises which provide useful results in complex algebraic geometry.

Hodge Theory And Complex Algebraic Geometry

Author : Voisin
ISBN : 052117032X
Genre : Hodge theory
File Size : 23. 25 MB
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The second of two volumes offering a modern account of Kaehlerian geometry and Hodge theory for researchers in algebraic and differential geometry.

Introduction To Hodge Theory

Author : José Bertin
ISBN : 0821820400
Genre : Mathematics
File Size : 66. 76 MB
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Hodge theory is a powerful tool in analytic and algebraic geometry. This book consists of expositions of aspects of modern Hodge theory, with the purpose of providing the nonexpert reader with a clear idea of the current state of the subject. The three main topics are: $L^2$ Hodge theory and vanishing theorems; Hodge theory in characteristic $p$; and variations of Hodge structures and mirror symmetry. Each section has a detailed introduction and numerous references. Many open problems are also included. The reader should have some familiarity with differential and algebraic geometry, with other prerequisites varying by chapter. The book is suitable as an accompaniment to a second course in algebraic geometry.

Period Mappings And Period Domains

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ISBN : 9781108422628
Genre :
File Size : 85. 13 MB
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3264 And All That

Author : David Eisenbud
ISBN : 9781107017085
Genre : Mathematics
File Size : 74. 79 MB
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This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincaré's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.

Positivity In Algebraic Geometry I

Author : R.K. Lazarsfeld
ISBN : 3540225331
Genre : Mathematics
File Size : 37. 83 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. Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II.

Hodge Theory And Complex Algebraic Geometry Ii

Author : Claire Voisin
ISBN : 1139437704
Genre : Mathematics
File Size : 49. 46 MB
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The 2003 second volume of this account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. Proofs of the Lefschetz theorem on hyperplane sections, the Picard–Lefschetz study of Lefschetz pencils, and Deligne theorems on the degeneration of the Leray spectral sequence and the global invariant cycles follow. The main results of the second part are the generalized Noether–Lefschetz theorems, the generic triviality of the Abel–Jacobi maps, and most importantly Nori's connectivity theorem, which generalizes the above. The last part of the book is devoted to the relationships between Hodge theory and algebraic cycles. The book concludes with the example of cycles on abelian varieties, where some results of Bloch and Beauville, for example, are expounded. The text is complemented by exercises giving useful results in complex algebraic geometry. It will be welcomed by researchers in both algebraic and differential geometry.

Complex Manifolds And Deformation Of Complex Structures

Author : K. Kodaira
ISBN : 9781461385905
Genre : Mathematics
File Size : 81. 54 MB
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Algebraic Geometry From Algebraic Varieties To Schemes

Author : Kenji Ueno
ISBN : 9780821808627
Genre : Mathematics
File Size : 46. 62 MB
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Beginning algebraic geometers are well served by Uneno's inviting introduction to the language of schemes. Grothendieck's schemes and Zariski's emphasis on algebra and rigor are primary sources for this introduction to a rich mathematical subject. Ueno's book is a self-contained text suitable for an introductory course on algebraic geometry.

Rational Curves On Algebraic Varieties

Author : Janos Kollar
ISBN : 9783662032763
Genre : Mathematics
File Size : 29. 80 MB
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The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.

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