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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


G2 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


G2 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


G2 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 GLn

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


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