D. Huybrechts
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0009
- Subject:
- Mathematics, Geometry / Topology
Historically, Mukai's equivalence with the Poincare bundle on the product of an abelian variety and its dual as kernel was the fist Fourier-Mukai transform. The first section in this chapter ...
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Historically, Mukai's equivalence with the Poincare bundle on the product of an abelian variety and its dual as kernel was the fist Fourier-Mukai transform. The first section in this chapter functions as a reminder of the basic facts from the rich theory of abelian varieties, and the case of principally polarized abelian varieties is studied. A general investigation of derived equivalences between abelian varieties and derived autoequivalences of a single abelian variety is included.Less
Historically, Mukai's equivalence with the Poincare bundle on the product of an abelian variety and its dual as kernel was the fist Fourier-Mukai transform. The first section in this chapter functions as a reminder of the basic facts from the rich theory of abelian varieties, and the case of principally polarized abelian varieties is studied. A general investigation of derived equivalences between abelian varieties and derived autoequivalences of a single abelian variety is included.
D. Huybrechts
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this ...
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This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.Less
This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.
Mark Green, Phillip Griffiths, and Matt Kerr
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691154244
- eISBN:
- 9781400842735
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691154244.003.0006
- Subject:
- Mathematics, Analysis
This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining ...
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This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining an n-orientation of a totally imaginary number field and constructs a precise correspondence between these and certain important kinds of CM Hodge structures. In the classical case of weight n = 1, the abelian variety associated to a CM type is recovered. The notion of the Kubota rank and reflex field associated to a CM Hodge structure is then generalized to the totally imaginary number field setting. When the Kubota rank is maximal, the CM Hodge structure is non-degenerate. The discussion also covers oriented imaginary number fields, Hodge structures with special endomorphisms, polarization and Mumford-Tate groups, and the Mumford-Tate group in the Galois case.Less
This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining an n-orientation of a totally imaginary number field and constructs a precise correspondence between these and certain important kinds of CM Hodge structures. In the classical case of weight n = 1, the abelian variety associated to a CM type is recovered. The notion of the Kubota rank and reflex field associated to a CM Hodge structure is then generalized to the totally imaginary number field setting. When the Kubota rank is maximal, the CM Hodge structure is non-degenerate. The discussion also covers oriented imaginary number fields, Hodge structures with special endomorphisms, polarization and Mumford-Tate groups, and the Mumford-Tate group in the Galois case.
Kai-Wen Lan
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691156545
- eISBN:
- 9781400846016
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156545.003.0003
- Subject:
- Mathematics, Geometry / Topology
This chapter explains well-known notions important for the study of semi-abelian schemes. It first studies groups of multiplicative type and the torsors under them. A fundamental property of groups ...
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This chapter explains well-known notions important for the study of semi-abelian schemes. It first studies groups of multiplicative type and the torsors under them. A fundamental property of groups of multiplicative type is that they are rigid in the sense that they cannot be deformed. The chapter then turns to biextensions, cubical structures, semi-abelian schemes, Raynaud extensions, and certain dual objects for the last two notions extending the notion of dual abelian varieties. Such notions are, as this chapter shows, of fundamental importance in the study of the degeneration of abelian varieties. The main objective here is to understand the statement and the proof of the theory of degeneration data.Less
This chapter explains well-known notions important for the study of semi-abelian schemes. It first studies groups of multiplicative type and the torsors under them. A fundamental property of groups of multiplicative type is that they are rigid in the sense that they cannot be deformed. The chapter then turns to biextensions, cubical structures, semi-abelian schemes, Raynaud extensions, and certain dual objects for the last two notions extending the notion of dual abelian varieties. Such notions are, as this chapter shows, of fundamental importance in the study of the degeneration of abelian varieties. The main objective here is to understand the statement and the proof of the theory of degeneration data.
Kai-Wen Lan
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691156545
- eISBN:
- 9781400846016
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156545.003.0004
- Subject:
- Mathematics, Geometry / Topology
This chapter reproduces the theory of degeneration data for polarized abelian varieties, following D. Mumford's 1972 monograph, An analytic construction of degenerating abelian varieties over ...
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This chapter reproduces the theory of degeneration data for polarized abelian varieties, following D. Mumford's 1972 monograph, An analytic construction of degenerating abelian varieties over complete rings, as well as the first three chapters of the Degeneration of abelian varieties (1990), by G. Faltings and C.-L. Chai. Although there is essentially nothing new in this chapter, some modifications have been introduced to make the statements compatible with a certain understanding of the proofs. Moreover, since Mumford and Faltings and Chai have supplied full details only in the completely degenerate case, this chapter balances the literature by avoiding the special case and treating all cases equally.Less
This chapter reproduces the theory of degeneration data for polarized abelian varieties, following D. Mumford's 1972 monograph, An analytic construction of degenerating abelian varieties over complete rings, as well as the first three chapters of the Degeneration of abelian varieties (1990), by G. Faltings and C.-L. Chai. Although there is essentially nothing new in this chapter, some modifications have been introduced to make the statements compatible with a certain understanding of the proofs. Moreover, since Mumford and Faltings and Chai have supplied full details only in the completely degenerate case, this chapter balances the literature by avoiding the special case and treating all cases equally.
Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.003.0003
- Subject:
- Mathematics, Number Theory
This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before ...
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This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before introducing a theorem, which is an identity between the analytic kernel and the geometric kernel. It then defines the generating series and uses it to describe the geometric kernel. It also presents a theorem, which is an identity formulated in terms of projectors, and reviews some basic notations and results on Shimura curves. Other topics covered include the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves, normalization of the geometric kernel, and the analytic kernel function. The chapter concludes with an analysis of the kernel identity implied in the first theorem.Less
This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before introducing a theorem, which is an identity between the analytic kernel and the geometric kernel. It then defines the generating series and uses it to describe the geometric kernel. It also presents a theorem, which is an identity formulated in terms of projectors, and reviews some basic notations and results on Shimura curves. Other topics covered include the Eichler–Shimura theory for abelian varieties parametrized by Shimura curves, normalization of the geometric kernel, and the analytic kernel function. The chapter concludes with an analysis of the kernel identity implied in the first theorem.
Kai-Wen Lan
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691156545
- eISBN:
- 9781400846016
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156545.001.0001
- Subject:
- Mathematics, Geometry / Topology
By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical ...
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By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary, which this book explains in detail. Through the discussion, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.Less
By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary, which this book explains in detail. Through the discussion, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties.
Claire Voisin
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160504
- eISBN:
- 9781400850532
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160504.003.0005
- Subject:
- Mathematics, Geometry / Topology
This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was ...
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This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π : X → B of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.Less
This chapter considers varieties whose Chow ring has special properties. This includes abelian varieties, K3 surfaces, and Calabi–Yau hypersurfaces in projective space. For K3 surfaces S, it was discovered that they have a canonical 0-cycle o of degree 1 with the property that the product of two divisors of S is a multiple of o in CH₀(S). This result would later be extended to Calabi–Yau hypersurfaces in projective space. The chapter also considers a decomposition in CH(X × X × X)ℚ of the small diagonal Δ ⊂ X × X × X that was established for K3 surfaces, and is partially extended to Calabi–Yau hypersurfaces. Finally, the chapter uses this decomposition and the spreading principle to show that for families π : X → B of smooth projective K3 surfaces, there is a decomposition isomorphism that is multiplicative over a nonempty Zariski dense open set of B.
Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.003.0001
- Subject:
- Mathematics, Number Theory
This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves ...
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This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.Less
This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.
Samuel Grushevsky, Klaus Hulek, Orsola Tommasi, and Mathieu Dutour Sikirić
- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.003.0024
- Subject:
- Mathematics, Geometry / Topology
This chapter presents an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of Ag, satisfying ...
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This chapter presents an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of Ag, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular, the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree.Less
This chapter presents an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of Ag, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular, the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree.
Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.003.0004
- Subject:
- Mathematics, Number Theory
This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the ...
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This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.Less
This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, the chapter considers the new space of Schwartz functions and constructs theta series and Eisenstein series from such functions. It proceeds by discussing discrete series at infinite places, modularity of the generating series, degree of the generating series, and the trace identity. It also presents the pull-back formula for the compact and non-compact cases. In particular, it describes CM cycles on the Shimura curve, pull-back as cycles, degree of the pull-back, and some coset identities.
