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Derived Categories of Coherent Sheaves

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

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

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 constructed. The usual geometric functors, direct and inverse image, tensor product, and global sections, are derived and extended to functors between derived categories. The compatibilities between them are reviewed. The final section focuses on the Grothendieck-Verdier duality.Less

Where to Go from Here

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

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

This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the equivariant-derived category of a variety endowed with the action of a finite group and the derived category of a crepant resolution of the quotient. This chapter gives the results from Bridgeland, King, and Reid for a special crepant resolution provided by Hilbert schemes and of Bezrukavnikov and Kaledin for symplectic vector spaces. A brief discussion of Kontsevich's homological mirror symmetry is included, as well as a discussion of stability conditions on triangulated categories. Twisted sheaves and their derived categories can be dealt with in a similar way, and some of the results in particular for K3 surfaces are presented.Less

Convolution of Perverse Sheaves

Nicholas M. Katz

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

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

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 with perverse sheaves on 𝓖/k and on 𝓖/k¯.Less

Appendix: Deligne’s Fibre Functor

Nicholas M. Katz

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

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

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 Ƿ, under middle convolution.Less

Structures on Manifolds

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

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

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 described as a G-structure which may or may not be (partially) integrable. Some selected topics include: Riemannian metrics, complex structures, symplectic structures, contact structures, quaternionic structures, group actions, pseudogroups, sheaves, bundles, connections, holonomy, curvature and integrability.Less

Kähler–Einstein Metrics

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

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

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 gives rise to the famous Monge-Amfipere equation. Some basic techniques such as the continuity method, Tian's invariant, and multipliers ideal sheaves are introduced. These provide for proving various existence results concerning orbifold Kähler-Einstein metrics. The Matsushima-Lichnerowicz theorem and Futaki invariant are briefly discussed in the section on obstructions.Less

Extension theorems and the existence of flips

Christopher D. Hacon and James McKernan

in Flips for 3-folds and 4-folds

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

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 is based on the key notion of an ‘adjoint algebra’. The chapter contains an introduction to multiplier ideals, and the celebrated lifting lemma is developed from first principles. Key ideas of the minimal model program for real pairs are also developed from the ground up.Less

UNIVERSES IN TOPOSES

Thomas Streicher

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

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

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 families of types in the universe by structural recursion and quantify over such families. Further, it shows that (hierarchies of) such universes do exist in all sheaf and realizability toposes. They do not exist instead either in the free topos or in the Vω+ω model of Zermelo set theory. Though universes in the category Set are necessarily of strongly inaccessible cardinality, it remains an open question as to whether toposes with a universe allow one to construct internal models of Intuitionistic Zermelo Fraenkel set theory (IZF).Less

GL(n) × GL(n) × … × GL(n) Examples

Nicholas M. Katz

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

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

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 := G(1/2)[1] ɛ Garith is “dimension” n and has Gsubscript geom,N = Garith,N = GL(n). Suppose in addition we are given s ≤ 2 distinct characters χ‎ᵢ of kˣ.Less

G₂ Examples: the Overall Strategy

Nicholas M. Katz

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

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

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 character of k the composition k := ψ‎ ○ Trk/𝔽p, whenever a nontrivial additive character of k is (implicitly or explicitly) called for (for instance in the definition of a Kloosterman sheaf, or of a hypergeometric sheaf, on 𝔾ₘ/k).Less

G₂ Examples: Construction in Characteristic Two

Nicholas M. Katz

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

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

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 field of characteristic 2, and any character χ‎ of kË£, the Tate-twisted Kloosterman sheaf of rank seven has Ggeom = Garith = Gâ‚‚. The first task is to express its stalk at a fixed point a É› kË£ as the finite field Mellin transform of the desired object N(a; k).Less

G₂ Examples: Construction in Odd Characteristic

Nicholas M. Katz

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

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

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 𝔾ₘ is not pure of weight zero, nor is its Ggeom the group G₂.Less

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)

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

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, that there is an isomorphism H*(Xₐₙ,ℂ) ≃ H*X,Ω‎subscript alg superscript bullet) between the complex singular cohomology of Xan and the hypercohomology of the complex Ω‎subscript alg superscript bullet of sheaves of algebraic differential forms on X. The proof necessitates a discussion of sheaf cohomology, coherent sheaves, and hypercohomology. The chapter then develops more machinery, mainly the Čech cohomology of a sheaf and the Čech cohomology of a complex of sheaves, as tools for computing hypercohomology. The chapter thus proves that the general case of Grothendieck's theorem is equivalent to the affine case.Less

Divisors, line bundles and Jacobians

Simon Donaldson

in Riemann Surfaces

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

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 Riemann surfaces, covering the Abel-Jacobi Theorem; abstract theory; and the geometry of symmetric products. It concludes by examining the special case when Σ has genus 2, so that Jac(Σ) is a complex surface and there is a surjective map j : Sym2( Σ) → Jac.Less

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)

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

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 chapter illustrates the cohomology of the sheaves 𝓜̃λ‎. A fundamental problem at the heart of this monograph is to understand the arithmetic information contained in the sheaf-theoretically defined cohomology groups H⦁(SGKf,𝓜̃λ‎,E). Finally, the chapter briefly discusses how to refine many of the foregoing considerations to talk about integral sheaves and their cohomology and why this is interesting. This aspect is not so relevant for the results in this book, but it will become relevant when applying certain refinements of the results to arithmetic questions.Less

The First Piano Sonata

Kyle Gann

in Charles Ives's Concord: Essays after a Sonata

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

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 reliance on ragtime and quoted hymn tunes, including its jazzy rendition of “Bringing in the Sheaves.”Less

Diamonds

Peter Scholze and Jared Weinstein

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

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

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 exists in a category of sheaves on the site of perfectoid spaces with pro-étale covers. The chapter then defines pro-étale morphisms between perfectoid spaces. A morphism of perfectoid spaces is pro-étale if it is locally (on the source and target) affinoid pro-étale. The intuitive definition of diamonds involved the tilting functor in case of perfectoid spaces of characteristic 0. For this reason, diamonds are defined as certain pro-étale sheaves on the category of perfectoid spaces of characteristic p.Less

Examples of diamonds

Peter Scholze and Jared Weinstein

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

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

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 keep in mind that a diamond can have multiple “incarnations.” Another important class of diamonds, which in fact were one of the primary motivations for their definition, is the category of Banach-Colmez spaces. Recently, le Bras has reworked their theory in terms of perfectoid spaces. The category of Banach-Colmez spaces over C is the thick abelian subcategory of the category of pro-étale sheaves of Qp-modules. This is similar to a category considered by Milne in characteristic p.Less