## 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

## Derived Categories of Coherent Sheaves

*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/9780199296866.003.0003
- Subject:
- Mathematics, Geometry / Topology

The discussion of the previous chapter is applied to the derived category of the abelian category of coherent sheaves. The Serre functor is introduced, and particular spanning classes are ... More

## Where to Go from Here

*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/9780199296866.003.0013
- Subject:
- Mathematics, Geometry / Topology

This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the ... More

## Convolution of Perverse Sheaves

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0003
- Subject:
- Mathematics, Number Theory

Let k be a finite field, q its cardinality, p its characteristic, 𝓁 ≠ p a prime number, and 𝓖/k a smooth commutative group scheme which over k¯ becomes isomorphic to 𝔾ₘ/k¯. This chapter is concerned ... More

## Appendix: Deligne’s Fibre Functor

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0031
- Subject:
- Mathematics, Number Theory

This chapter takes up the proof of Theorem 3.1. It shows that N ↦ ω(N) := H⁰(𝔸¹/k¯, j0!N) is a fiber functor on the Tannakian category Ƿsubscript geom of those perverse sheaves on 𝔾ₘ/k¯ satisfying ... More

## Structures on Manifolds

*Charles P. Boyer and Krzysztof Galicki*

### in Sasakian Geometry

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter begins by introducing various geometries that play important roles in the way they relate to Sasakian structures. It espouses the point of view that a geometric structure is best ... More

## Kähler–Einstein Metrics

*Charles P. Boyer and Krzysztof Galicki*

### in Sasakian Geometry

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0006
- Subject:
- Mathematics, Geometry / Topology

This chapter is a second trip into the realm of Kähler geometry, focusing on Kähler-Einstein metrics, in particular positive scalar curvature Kähler-Einstein metrics on compact Fano orbifolds, which ... More

## Extension theorems and the existence of flips

*Christopher D. Hacon and James McKernan*

### in Flips for 3-folds and 4-folds

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198570615
- eISBN:
- 9780191717703
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570615.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter provides a detailed and self-contained exposition of Hacon and McKernan's construction of pl flips in dimension n assuming minimal models with scaling in dimension n-1. The construction ... More

## UNIVERSES IN TOPOSES

*Thomas Streicher*

### in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566519
- eISBN:
- 9780191713927
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566519.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter discusses a notion of universe in toposes, which from a logical point of view gives rise to an extension of Higher Order Intuitionistic Arithmetic (HAH). In this way, one can construct ... More

##
*GL*(*n*) × *GL*(*n*) × … × *GL*(*n*) Examples

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0021
- Subject:
- Mathematics, Number Theory

This chapter investigates the question which begings as follows. Suppose we have a geometrically irreducible middle extension sheaf G on 𝔾ₘ/k which is pure of weight zero, such that the object N := ... More

##
*G*_{2} Examples: the Overall Strategy

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0026
- Subject:
- Mathematics, Number Theory

This fixes, for each prime p, a prime 𝓁 ≠ p and a choice of nontrivial ℚ¯ℓX-valued additive character ψ of the prime field 𝔽ₚ. Given a finite extension field k/𝔽ₚ, it takes as nontrivial additive ... More

##
*G*_{2} Examples: Construction in Characteristic Two

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0027
- Subject:
- Mathematics, Number Theory

This chapter treats the case of characteristic two separately because it is somewhat simpler than the case of odd characteristic. Recall from the first paragraph of Chapter 25 that for k a finite ... More

##
*G*_{2} Examples: Construction in Odd Characteristic

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0028
- Subject:
- Mathematics, Number Theory

This chapter treats the case of the odd characteristic. The situation in odd characteristic is complicated by the quadratic character χ₂: the ! hypergeometric sheaf of type (7, 1), though lisse on ... More

## Hodge Theory (MN-49)

*Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161341
- eISBN:
- 9781400851478
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161341.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The ... More

## From Sheaf Cohomology to the Algebraic de Rham Theorem

*Fouad El Zein and Loring W. Tu*

*Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)*

### in Hodge Theory (MN-49)

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161341
- eISBN:
- 9781400851478
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161341.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter proves Grothendieck's algebraic de Rham theorem. It first proves Grothendieck's algebraic de Rham theorem more or less from scratch for a smooth complex projective variety X, namely, ... More

## Divisors, line bundles and Jacobians

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0012
- Subject:
- Mathematics, Geometry / Topology, Analysis

This chapter first examines cohomology and line bundles, covering sheaves and cohomology, line bundles and projective embeddings, and divisors and unique factorisation. It then turns to Jacobians of ... More

##
The Cohomology of GL_{n}

*Günter Harder and A. Raghuram*

### in Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions: (AMS-203)

- Published in print:
- 2019
- Published Online:
- September 2020
- ISBN:
- 9780691197890
- eISBN:
- 9780691197937
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691197890.003.0002
- Subject:
- Mathematics, Number Theory

This chapter addresses the cohomology of GLn. It first discusses the adèlic locally symmetric space. Next, the chapter turns to the highest weight modules 𝓜λ and the sheaves 𝓜̃λ. From there, the ... More

## The First Piano Sonata

*Kyle Gann*

### in Charles Ives's Concord: Essays after a Sonata

- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780252040856
- eISBN:
- 9780252099366
- Item type:
- chapter

- Publisher:
- University of Illinois Press
- DOI:
- 10.5406/illinois/9780252040856.003.0013
- Subject:
- Music, History, American

Ives’s First Piano Sonata (1901-1917, written concurrently with the Concord) is analyzed here to show differences between its formal design and the Concord’s. Particularly evident is its greater ... More

## Diamonds

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter investigates the notion of a diamond. The idea is that there should be a functor which “forgets the structure morphism to Zp.” The desired quotient in the example provided in the chapter ... More

## Examples of diamonds

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0015
- Subject:
- Mathematics, Geometry / Topology

This chapter assesses some interesting examples of diamonds. So far, the only example encountered is the self-product of copies of Spd Qp. The chapter first studies this self-product. It is useful to ... More