Equivariant Algebraic K-Theory
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The Extension Problem in Complex Analysis II; Embeddings with Positive Normal Bundle Author(S): Phillip A
The Extension Problem in Complex Analysis II; Embeddings with Positive Normal Bundle Author(s): Phillip A. Griffiths Source: American Journal of Mathematics, Vol. 88, No. 2 (Apr., 1966), pp. 366-446 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/2373200 Accessed: 25/10/2010 12:00 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=jhup. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. http://www.jstor.org THE EXTENSION PROBLEM IN COMPLEX ANALYSIS II; EMBEDDINGS WITH POSITIVE NORMAL BUNDLE. -
1.10 Partitions of Unity and Whitney Embedding 1300Y Geometry and Topology
1.10 Partitions of unity and Whitney embedding 1300Y Geometry and Topology Theorem 1.49 (noncompact Whitney embedding in R2n+1). Any smooth n-manifold may be embedded in R2n+1 (or immersed in R2n). Proof. We saw that any manifold may be written as a countable union of increasing compact sets M = [Ki, and that a regular covering f(Ui;k ⊃ Vi;k;'i;k)g of M can be chosen so that for fixed i, fVi;kgk is a finite ◦ ◦ cover of Ki+1nKi and each Ui;k is contained in Ki+2nKi−1. This means that we can express M as the union of 3 open sets W0;W1;W2, where [ Wj = ([kUi;k): i≡j(mod3) 2n+1 Each of the sets Ri = [kUi;k may be injectively immersed in R by the argument for compact manifolds, 2n+1 since they have a finite regular cover. Call these injective immersions Φi : Ri −! R . The image Φi(Ri) is bounded since all the charts are, by some radius ri. The open sets Ri; i ≡ j(mod3) for fixed j are disjoint, and by translating each Φi; i ≡ j(mod3) by an appropriate constant, we can ensure that their images in R2n+1 are disjoint as well. 0 −! 0 2n+1 Let Φi = Φi + (2(ri−1 + ri−2 + ··· ) + ri) e 1. Then Ψj = [i≡j(mod3)Φi : Wj −! R is an embedding. 2n+1 Now that we have injective immersions Ψ0; Ψ1; Ψ2 of W0;W1;W2 in R , we may use the original argument for compact manifolds: Take the partition of unity subordinate to Ui;k and resum it, obtaining a P P 3-element partition of unity ff1; f2; f3g, with fj = i≡j(mod3) k fi;k. -
Characteristic Classes and K-Theory Oscar Randal-Williams
Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. -
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020 Lectures by R. Inanç˙ Baykur University of Massachusetts Amherst Disclaimer These notes were taken during lecture using the vimtex package of the editor neovim. Any errors are mine and not the instructor’s. In addition, my notes are picture-free (but will include commutative diagrams) and are a mix of my mathematical style (omit lengthy computations, use category theory) and that of the instructor. If you find any errors, please contact me at [email protected]. Contents Contents • 2 1 January 21 • 5 1.1 Course Description • 5 1.2 Organization • 5 1.2.1 Notational conventions•5 1.3 Basic Notions • 5 1.4 Symplectic Linear Algebra • 6 2 January 23 • 8 2.1 More Basic Linear Algebra • 8 2.2 Compatible Complex Structures and Inner Products • 9 3 January 28 • 10 3.1 A Big Theorem • 10 3.2 More Compatibility • 11 4 January 30 • 13 4.1 Homework Exercises • 13 4.2 Subspaces of Symplectic Vector Spaces • 14 5 February 4 • 16 5.1 Linear Algebra, Conclusion • 16 5.2 Symplectic Vector Bundles • 16 6 February 6 • 18 6.1 Proof of Theorem 5.11 • 18 6.2 Vector Bundles, Continued • 18 6.3 Compatible Triples on Manifolds • 19 7 February 11 • 21 7.1 Obtaining Compatible Triples • 21 7.2 Complex Structures • 21 8 February 13 • 23 2 3 8.1 Kähler Forms Continued • 23 8.2 Some Algebraic Geometry • 24 8.3 Stein Manifolds • 25 9 February 20 • 26 9.1 Stein Manifolds Continued • 26 9.2 Topological Properties of Kähler Manifolds • 26 9.3 Complex and Symplectic Structures on 4-Manifolds • 27 10 February 25 -
NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. -
Arxiv:1307.5568V2 [Math.AG]
PARTIAL POSITIVITY: GEOMETRY AND COHOMOLOGY OF q-AMPLE LINE BUNDLES DANIEL GREB AND ALEX KURONYA¨ To Rob Lazarsfeld on the occasion of his 60th birthday Abstract. We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs. Contents 1. Introduction 1 2. Overview of the theory of q-ample line bundles 4 2.1. Vanishing of cohomology groups and partial ampleness 4 2.2. Basic properties of q-ampleness 7 2.3. Sommese’s geometric q-ampleness 15 2.4. Ample subschemes, and a Lefschetz hyperplane theorem for q-ample divisors 17 3. q-Kodaira vanishing for Du Bois divisors and log canonical pairs 19 References 23 1. Introduction Ampleness is one of the central notions of algebraic geometry, possessing the extremely useful feature that it has geometric, numerical, and cohomological characterizations. Here we will concentrate on its cohomological side. The fundamental result in this direction is the theorem of Cartan–Serre–Grothendieck (see [Laz04, Theorem 1.2.6]): for a complete arXiv:1307.5568v2 [math.AG] 23 Jan 2014 projective scheme X, and a line bundle L on X, the following are equivalent to L being ample: ⊗m (1) There exists a positive integer m0 = m0(X, L) such that L is very ample for all m ≥ m0. (2) For every coherent sheaf F on X, there exists a positive integer m1 = m1(X, F, L) ⊗m for which F ⊗ L is globally generated for all m ≥ m1. -
Topological K-Theory
TOPOLOGICAL K-THEORY ZACHARY KIRSCHE Abstract. The goal of this paper is to introduce some of the basic ideas sur- rounding the theory of vector bundles and topological K-theory. To motivate this, we will use K-theoretic methods to prove Adams' theorem about the non- existence of maps of Hopf invariant one in dimensions other than n = 1; 2; 4; 8. We will begin by developing some of the basics of the theory of vector bundles in order to properly explain the Hopf invariant one problem and its implica- tions. We will then introduce the theory of characteristic classes and see how Stiefel-Whitney classes can be used as a sort of partial solution to the problem. We will then develop the methods of topological K-theory in order to provide a full solution by constructing the Adams operations. Contents 1. Introduction and background 1 1.1. Preliminaries 1 1.2. Vector bundles 2 1.3. Classifying Spaces 4 2. The Hopf invariant 6 2.1. H-spaces, division algebras, and tangent bundles of spheres 6 3. Characteristic classes 8 3.1. Definition and construction 9 3.2. Real projective space 10 4. K-theory 11 4.1. The ring K(X) 11 4.2. Clutching functions and Bott periodicity 14 4.3. Adams operations 17 Acknowledgments 19 References 19 1. Introduction and background 1.1. Preliminaries. This paper will presume a fair understanding of the basic ideas of algebraic topology, including homotopy theory and (co)homology. As with most writings on algebraic topology, all maps are assumed to be continuous unless otherwise stated. -
4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module. -
D-Modules and the Riemann-Hilbert Correspondence
D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE JOSE´ SIMENTAL Abstract. These are notes for a talk given at the MIT/NEU Seminar on Hodge modules, Fall 2016. We give a quick introduction to the theory of D-modules, following mostly [BCEY, HTT, K]. 1. Differential operators 1.1. Setup. Let X be a smooth algebraic variety, and let dX := dim X. We remark that all algebraic varieties here are always assumed to be quasi-projective. By OX we will denote the structure sheaf of X, and VectX denotes the sheaf of vector fields on X, that is, the sheaf of regular sections of TX. Consider the sheaf EndCX (OX ), its sections on an open set U are HomCX (OU ; OU ). Note that we have embeddings OX ,!EndCX (OX ) and VectX ,!EndCX (OX ). Definition 1.1. The sheaf of differential operators DX is the sheaf of algebras that is generated by OX ; VectX inside of EndCX (OX ). Let us remark that DX is actually a coherent sheaf of algebras on X. This follows from the corresponding result for OX and results of Subsection 1.2 below, see Corollary 1.4. We also remark that DX is a sheaf of noncommutative algebras. Indeed, if ξ 2 VectX ; f 2 OX , then [ξ; f] = ξ(f), which is not necessarily zero. In local coordinates x1; : : : ; xn; @x1; : : : ; @xn, and element P 2 DX may be written as X α P = aα(x)@x n α2Z≥0 α1+···+αn n α @ where aα = 0 for all but a finite number of α 2 Z≥0, and @x = α1 αn . -
Some Very Basic Deformation Theory
Some very basic deformation theory Carlo Pagano W We warm ourselves up with some infinitesimal deformation tasks, to get a feeling of the type of question we are dealing with: 1 Infinitesimal Deformation of a sub-scheme of a scheme Take k a field, X a scheme defined over k, Y a closed subscheme of X. An 0 infinitesimal deformation of Y in X, for us, is any closed subscheme Y ⊆ X ×spec(k) 0 spec(k[]), flat over k[], such that Y ×Spec(k[]) Spec(k) = Y . We want to get a description of these deformations. Let's do it locally, working the affine case, and then we'll get the global one by patching. Translating the above conditions you get: classify ideals I0 ⊆ B0 := Bj], such that I0 reduces to I modulo and B0=I0 is flat over k[]. Flatness over k[] can be tested just looking at the sequence 0 ! () ! k[] ! k ! 0(observe that the first and the third are isomorphic as k[]-modules), being so equivalent to the exactness of the sequence 0 ! B=I ! B0=I0 ! B=I ! 0. So in this situation you see that if you have such an I0 then every element y 2 I lift uniquely to an element of B=I, giving 2 rise to a map of B − modules, 'I : I=I ! B=I, proceeding backwards from 0 0 ' 2 HomB(I; B=I), you gain an ideal I of the type wanted(to check this use again the particularly easy test of flatness over k[]). So we find that our set of 2 deformation is naturally parametrized by HomB(I=I ; B=I). -
(Quasi)Coherent Sheaves on Projective Spaces
18.725 Algebraic Geometry I Lecture 14 Lecture 14: (Quasi)coherent sheaves on Projective Spaces First an abstract lemma. Let L : C1 !C2; R : C2 !C1 be an adjoint pair; if L is fully faithful and R is conservative, then they are inverses. The unit is Id −!Lu ◦ R and the counit is Id −!" R ◦ L. Additionally, we "()R R(u) have R −−−!R ◦ L ◦ R −−−!R = Id. Example 1. C1 = C2 = Vect. Let V be a finite dimensional vector space. Let R : U ! V ⊗ U, L : U ! E⊗δ7! E(δ) V ∗ ⊗ U = Hom(V; U). Then the operation above becomes VVVV−δ−7!−I−d−⊗!δ ⊗ ∗ ⊗ −−−−−−−! V . L is fully faithful implies Id =∼ R ◦ L. What about L ◦ R =∼ Id? It suffices to use R ! R ◦ L ◦ R ! R. Last time we showed that the set of affine maps between X and Y is the same as the set of quasicoherent sheaves of OX -algebras (which are locally finitely generated and reduced). ∼ n −1 ∼ n Definition 1. X ! Y is a vector bundle if locally = A × Y , i.e. there exists a covering f (Ui) = A × Ui n and agree on the intersection, i.e. the two copies of A × (Ui \ Uj) are glued together using GLn(k[Ui \ Uj]). The equivalence between the category of locally free sheaves and the category of vector bundles is given i by E 7! Spec(⊕iSym (E)), which is a contravariant functor. The opposite maps are from a vector bundle to the sheaf of sections of the dual bundle. Note that the total space is given by Tot(E) = Spec(Sym(E_)) where E_ = Hom(E; O). -
Constructible Sheaves on Affine Grassmannians and Geometry of the Dual Nilpotent Cone
CONSTRUCTIBLE SHEAVES ON AFFINE GRASSMANNIANS AND GEOMETRY OF THE DUAL NILPOTENT CONE PRAMOD N. ACHAR AND SIMON RICHE Abstract. In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group Gˇ, contructible with respect to the ˇ stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov{ Bezrukavnikov{Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of the Langlands dual re- ductive group G. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians. 1. Introduction 1.1. Let Gˇ be a complex connected reductive group, and let ˇ ˇ GrGˇ := G(K)=G(O) be the associated affine Grassmannian (where K = C((x)) and O = C[[x]]). The Satake equivalence is an equivalence of tensor categories ∼ SG : Rep(G) −! PGˇ(O)-eq(GrGˇ ) ˇ between the category PGˇ(O)-eq(GrGˇ ) of G(O)-equivariant perverse sheaves on GrGˇ (endowed with the convolution product) and the category Rep(G) of finite-dimen- sional G-modules (endowed with the tensor product), where G is the complex re- ductive group which is Langlands dual to Gˇ. This equivalence is \functorial with respect to restriction to a Levi" in the sense that, if L ⊂ G is a Levi subgroup and Lˇ ⊂ Gˇ a dual Levi subgroup, the restriction functor Rep(G) ! Rep(L) can be realized geometrically as a (renormalized) hyperbolic localization functor G RL : PGˇ(O)-eq(GrGˇ ) ! PLˇ(O)-eq(GrLˇ ) in the sense of Braden [Bra].