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## Abelian Varieties

*D. Huybrechts*

### in Fourier-Mukai Transforms in Algebraic Geometry

- 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 ... More

## Fourier-Mukai Transforms in Algebraic Geometry

*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 ... More

## Hodge Structures with Complex Multiplication

*Mark Green, Phillip Griffiths, and Matt Kerr*

### in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

- 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 ... More

## Structures of Semi-Abelian Schemes

*Kai-Wen Lan*

### in Arithmetic Compactifications of PEL-Type Shimura Varieties

- 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 ... More

## Theory of Degeneration for Polarized Abelian Schemes

*Kai-Wen Lan*

### in Arithmetic Compactifications of PEL-Type Shimura Varieties

- 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 ... More

## Mordell-Weil Groups and Generating Series

*Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang*

### in The Gross-Zagier Formula on Shimura Curves

- 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 ... More

## Arithmetic Compactifications of PEL-Type Shimura Varieties

*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 ... More

##
On the Chow ring of *K3* surfaces and hyper-Kahler manifolds

*Claire Voisin*

### in Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

- 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 ... More

## Introduction and Statement of Main Results

*Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang*

### in The Gross-Zagier Formula on Shimura Curves

- 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 ... More

## Stable Betti Numbers of (Partial) Toroidal Compactifications of the Moduli Space of Abelian Varieties

*Samuel Grushevsky, Klaus Hulek, Orsola Tommasi, and Mathieu Dutour Sikirić*

### in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

- 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 ... More

## Trace of the Generating Series

*Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang*

### in The Gross-Zagier Formula on Shimura Curves

- 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 ... More

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