Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems
01 Mar 1970-Bulletin of the American Mathematical Society (American Mathematical Society)-Vol. 76, Iss: 2, pp 228-296
TL;DR: In this paper, the geometric situation giving rise to variation of Hodge structure is discussed and a number of results about holomorphic and locally constant cohomology classes are given. But these results are restricted to local monodromy of homology.
Abstract: 0. Introduction 229 Par t I. Summary of main results 231 1. The geometric situation giving rise to variation of Hodge structure. . . . 231 2. Data given by the variation of Hodge structure 232 3. Theorems about monodromy of homology 235 4. Theorems about Picard-Fuchs equations (Gauss-Manin connex ion) . . . . 237 5. Global theorems about holomorphic and locally constant cohomology classes 242 6. Global results on variation of Hodge structure 246 Par t I I . Problems and conjectures 247 7. Problems on Torelli-type theorems 247 8. Problems on local monodromy and variation of Hodge structure 248 9. Questions on compactification and the behavior of periods at infinity. . 251
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TL;DR: In this article, the authors present a table of table of tables of this article : Table of Table 1.3.1.1-3.2.0.1]
Abstract: Table of
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TL;DR: In this paper, it is proved that an algebraic surface of type K3 is uniquely determined by prescribing the integrals of its holomorphic differential forms with respect to a basis of cycles of the two-dimensional homology group, if the homology class of a hyperplane section is distinguished.
Abstract: In this paper it is proved that an algebraic surface of type K3 is uniquely determined by prescribing the integrals of its holomorphic differential forms with respect to a basis of cycles of the two-dimensional homology group, if the homology class of a hyperplane section is distinguished.
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Book•
26 Sep 2016TL;DR: Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular and each chapter ends with questions and open problems.
Abstract: K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.
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01 Jan 1968
TL;DR: The Singular Points of Complex Hypersurfaces (AM-61) as mentioned in this paper is a seminal work in the area of complex hypersurfaces, and is based on as mentioned in this paper.
Abstract: The description for this book, Singular Points of Complex Hypersurfaces. (AM-61), will be forthcoming.
2,676 citations
"Periods of integrals on algebraic m..." refers background in this paper
..., and reported on in Milnor's book [8], where also numerous examples are given....
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...transformation) obtained by transporting cycles around the branch of S—S [3], [7], [8], [l9J....
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TL;DR: The notion of analytic fiber spaces and their equivalences was introduced in this paper, where it was shown that a fiber space with a sheaf (3 of structure groups) is an analytic fiber space.
Abstract: THEOREM 11.1. The family 9(,{, G) consists of all analytic fibre spaces Bi, '2C H'(A, f2(B#)). The notion of fibre spaces and their equivalences depends on the sheaf of structure groups.' By a (63-fibre space we mean a fibre space with a sheaf (3 of structure groups, and we say that two fibre spaces are (X-equivalent if they are equivalent as (63-fibre spaces. The fibre space Bi may be considered as an analytic fibre space, as an f2(BO)-fibre space, or as an f2(Bl)-fibre space. The cohomology class C) C H'(Ay, f(BO)) represents the f2(Bl)-equivalence class of Be. Let C& denote the fibre of BI over u c A. It is clear that
1,011 citations
690 citations
"Periods of integrals on algebraic m..." refers background in this paper
...[10], [24] wh ere further references are given)....
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...The Satake compactification is obtained by letting D* be the disjoint union of D with the rational boundary components of D in D [10], [25]....
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TL;DR: In this article, the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).
Abstract: © Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
631 citations
"Periods of integrals on algebraic m..." refers background in this paper
...Thus Ùx/s is a graded, differential sheaf and so the Leray hypercohomology sheaves [5] R/*(Ùx/s) are defined and are coherent algebraic sheaves on S....
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...By the main theorem in [5] we see that the analytic coherent sheaf associated to R/*(Ùx/s) is just Rf* (C)®©^....
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...Among these applications are: (i) Grothendieck's algebraic deRham theorem [5]; (ii) the local monodromy theorem stated in (3....
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TL;DR: In this article, the authors implique l'accord avec les conditions generales d'utilisation (http://www.numdam.org/legal.php).
Abstract: © Publications mathematiques de l’I.H.E.S., 1968, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.
376 citations