The Riemann-Roch Theorem
|
|
- Rodney Owens
- 5 years ago
- Views:
Transcription
1 The Riemann-Roch Theorem Paul Baum Penn State Texas A&M University College Station, Texas, USA April 4, 2014
2 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch theorem The minicourse is based on joint work with Ron Douglas and is dedicated to Ron Douglas.
3 THE RIEMANN-ROCH THEOREM Topics in this talk : 1. Classical Riemann-Roch 2. Hirzebruch-Riemann-Roch (HRR) 3. Grothendieck-Riemann-Roch (GRR) 4. RR for possibly singular varieties (Baum-Fulton-MacPherson)
4 CLASSSICAL RIEMANN - ROCH M compact connected Riemann surface genus of M = # of holes = 1 2 [rankh 1(M; Z)]
5 D a divisor of M D consists of a finite set of points of M p 1, p 2,..., p l and an integer assigned to each point n 1, n 2,..., n l Equivalently D is a function D : M Z with finite support Support(D) = {p M D(p) 0} Support(D) is a finite subset of M
6 D a divisor on M deg(d) := p M D(p) Remark D 1, D 2 two divisors D 1 D 2 iff p M, D 1 (p) D 2 (p) Remark D a divisor, D is ( D)(p) = D(p)
7 Example Let f : M C { } be a meromorphic function. Define a divisor δ(f) by: 0 if p is neither a zero nor a pole of f δ(f)(p) = order of the zero if f(p) = 0 (order of the pole) if p is a pole of f
8 Example Let ω be a meromorphic 1-form on M. Locally ω is f(z)dz where f is a (locally defined) meromorphic function. Define a divisor δ(ω) by: 0 if p is neither a zero nor a pole of ω δ(ω)(p) = order of the zero if ω(p) = 0 (order of the pole) if p is a pole of ω
9 D a divisor on M { } meromorphic functions H 0 (M, D) := δ(f) D f : M C { } { } meromorphic 1-forms H 1 (M, D) := ω on M δ(ω) D Lemma H 0 (M, D) and H 1 (M, D) are finite dimensional C vector spaces dim C H 0 (M, D) < dim C H 1 (M, D) <
10 Theorem (RR) Let M be a compact connected Riemann surface and let D be a divisor on M. Then: dim C H 0 (M, D) dim C H 1 (M, D) = d g + 1 d = degree (D) g = genus (M)
11 HIRZEBRUCH-RIEMANN-ROCH M non-singular projective algebraic variety / C E an algebraic vector bundle on M E = sheaf of germs of algebraic sections of E H j (M, E) := j-th cohomology of M using E, j = 0, 1, 2, 3,...
12 LEMMA For all j = 0, 1, 2,... dim C H j (M, E) <. For all j > dim C (M), H j (M, E) = 0. χ(m, E) := n = dim C (M) n ( 1) j dim C H j (M, E) j=0 THEOREM[HRR] Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]
13 Hirzebruch-Riemann-Roch Theorem (HRR) Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]
14 EXAMPLE. Let M be a compact complex-analytic manifold. Set Ω p,q = C (M, Λ p,q T M) Ω p,q is the C vector space of all C differential forms of type (p, q) Dolbeault complex 0 Ω 0,0 Ω 0,1 Ω 0,2 Ω 0,n 0 The Dirac operator (of the underlying Spin c manifold) is the assembled Dolbeault complex + : j Ω 0, 2j j 0, 2j+1 Ω The index of this operator is the arithmetic genus of M i.e. is the Euler number of the Dolbeault complex.
15 K-theory and K-homology in algebraic geometry Let X be a (possibly singular) projective algebraic variety / C. Grothendieck defined two abelian groups: Kalg 0 (X) = Grothendieck group of algebraic vector bundles on X. K alg 0 (X) = Grothendieck group of coherent algebraic sheaves on X. Kalg 0 (X) = the algebraic geometry K-theory of X contravariant. K alg 0 (X) = the algebraic geometry K-homology of X covariant.
16 K-theory in algebraic geometry Vect alg X = set of isomorphism classes of algebraic vector bundles on X. A(Vect alg X) = free abelian group with one generator for each element [E] Vect alg X. For each short exact sequence ξ 0 E E E 0 of algebraic vector bundles on X, let r(ξ) A(Vect alg X) be r(ξ) := [E ] + [E ] [E]
17 K-theory in algebraic geometry R A(Vect alg (X)) is the subgroup of A(Vect alg X) generated by all r(ξ) A(Vect alg X). DEFINITION. K 0 alg (X) := A(Vect algx)/r Let X, Y be (possibly singular) projective algebraic varieties /C. Let f : X Y be a morphism of algebraic varieties. Then have the map of abelian groups f : K 0 alg (X) K0 alg (Y ) [f E] [E] Vector bundles pull back. f E is the pull-back via f of E.
18 K-homology in algebraic geometry S alg X = set of isomorphism classes of coherent algebraic sheaves on X. A(S alg X) = free abelian group with one generator for each element [E] S alg X. For each short exact sequence ξ 0 E E E 0 of coherent algebraic sheaves on X, let r(ξ) A(S alg X) be r(ξ) := [E ] + [E ] [E]
19 K-homology in algebraic geometry R A(S alg (X)) is the subgroup of A(S alg X) generated by all r(ξ) A(S alg X). DEFINITION. K alg 0 (X) := A(S alg X)/R Let X, Y be (possibly singular) projective algebraic varieties /C. Let f : X Y be a morphism of algebraic varieties. Then have the map of abelian groups f : K alg 0 (X) K alg 0 (Y ) [E] Σ j ( 1) j [(R j f)e]
20 f : X Y morphism of algebraic varieties E coherent algebraic sheaf on X For j 0, define a presheaf (W j f)e on Y by U H j (f 1 U; E f 1 U) U an open subset of Y Then (R j f)e := the sheafification of (W j f)e
21 f : X Y morphism of algebraic varieties f : K alg 0 (X) K alg 0 (Y ) [E] Σ j ( 1) j [(R j f)e]
22 SPECIAL CASE of f : K alg 0 (X) K alg 0 (Y ) Y is a point. Y = ɛ: X is the map of X to a point. Kalg 0 ( ) = Kalg 0 ( ) = Z ɛ : K alg 0 (X) K alg 0 ( ) = Z ɛ (E) = χ(x; E) = Σ j ( 1) j dim C H j (X; E)
23 X non-singular = K 0 alg (X) = K alg 0 (X) Let X be non-singular. Let E be an algebraic vector bundle on X. E denotes the sheaf of germs of algebraic sections of E. Then E E is an isomorphism of abelian groups Kalg 0 (X) Kalg 0 (X) This is Poincaré duality within the context of algebraic geometry K-theory&K-homology.
24 X non-singular = K 0 alg (X) = K alg 0 (X) Let X be non-singular. The inverse map is defined as follows. K alg 0 (X) K 0 alg (X) Let F be a coherent algebraic sheaf on X. Since X is non-singular, F has a finite resolution by algebraic vector bundles.
25 X non-singular = K 0 alg (X) = K alg 0 (X) F has a finite resolution by algebraic vector bundles. i.e. algebraic vector bundles on X E r, E r 1,..., E 0 and an exact sequence of coherent algebraic sheaves 0 E r E r 1... E 0 F 0 Then K alg 0 (X) Kalg 0 (X) is F Σ j ( 1) j E j
26 Grothendieck-Riemann-Roch Theorem (GRR) Let X, Y be non-singular projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) ch( ) T d(x) ch( ) T d(y ) H (X; Q) H (Y ; Q)
27 WARNING!!! The horizontal arrows in the GRR commutative diagram K 0 alg (X) K0 alg (Y ) ch( ) T d(x) ch( ) T d(y ) are wrong-way (i.e. Gysin) maps. H (X; Q) H (Y ; Q) K 0 alg (X) = K alg 0 (X) f K alg 0 (Y ) = K 0 alg (Y ) H (X; Q) = H (X; Q) f H (Y ; Q) = H (Y ; Q) Poincaré duality Poincaré duality
28 K-homology is the dual theory to K-theory. How can K-homology be taken from algebraic geometry to topology? There are three ways in which this has been done: Homotopy Theory K-homology is the homology theory determined by the Bott spectrum. K-Cycles K-homology is the group of K-cycles. C* algebras K-homology is the Atiyah-BDF-Kasparov group KK (A, C).
29 Riemann-Roch for possibly singular complex projective algebraic varieties Let X be a (possibly singular) projective algebraic variety / C Then (Baum-Fulton-MacPherson) there are functorial maps α X : K 0 alg (X) K0 top(x) K-theory contravariant natural transformation of contravariant functors β X : K alg 0 (X) K top 0 (X) K-homology covariant natural transformation of covariant functors Everything is natural. No wrong-way (i.e. Gysin) maps are used.
30 α X : K 0 alg (X) K0 top(x) is the forgetful map which sends an algebraic vector bundle E to the underlying topological vector bundle of E. α X (E) := E topological
31 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) α X α Y K 0 top(x) K 0 top(y ) i.e. natural transformation of contravariant functors
32 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) α X α Y K 0 top(x) K 0 top(y ) ch ch H (X; Q) H (Y ; Q)
33 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K alg 0 (X) K alg 0 (Y ) β X β Y K top 0 (X) Ktop 0 (Y ) i.e. natural transformation of covariant functors Notation. K top is K-cycle K-homology.
34 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) α X α Y K 0 top(x) K 0 top(y ) ch ch H (X; Q) H (Y ; Q)
35 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K alg 0 (X) K alg 0 (Y ) β X β Y K top 0 (X) Ktop 0 (Y ) ch ch H (X; Q) H (Y ; Q)
36 Definition of β X : K alg 0 (X) K top 0 (X) Let F be a coherent algebraic sheaf on X. Choose an embedding of projective algebraic varieties where W is non-singular. ι: X W ι F is the push forward (i.e. extend by zero) of F. ι F is a coherent algebraic sheaf on W.
37 ι F is a coherent algebraic sheaf on W. Since W is non-singular, ι F has a finite resolution by algebraic vector bundles. Consider 0 E r E r 1... E 0 ι F 0 0 E r E r 1... E 0 0 These are algebraic vector bundles on W and maps of algebraic vector bundles such that for each p W ι(x) the sequence of finite dimensional C vector spaces is exact. 0 (E r ) p (E r 1 ) p... (E 0 ) p 0
38 Choose Hermitian structures for E r, E r 1,..., E 0 Then for each vector bundle map there is the adjoint map σ : E j E j 1 σ : E j E j 1 σ σ : j E 2j j E 2j+1 is a map of topological vector bundles which is an isomorphism on W ι(x).
39 Let Ω be an open set in W with smooth boundary Ω such that Ω = Ω Ω is a compact manifold with boundary which retracts onto ι(x). Ω ι(x). Set M = Ω Ω Ω M is a closed Spin c manifold which maps to X by: ϕ: M = Ω Ω Ω Ω ι(x) = X
40 On M = Ω Ω Ω let E be the topological vector bundle E 2j (σ σ ) E 2j+1 E = j Then β X : K alg 0 (X) K top 0 (X) is : F (M, E, ϕ) j
41 Module structure Kalg 0 (X) is a ring and Kalg 0 (X) is a module over this ring. α X : K 0 alg (X) K0 top(x) is a homomorphism of rings. β X : K alg 0 (X) K top 0 (X) respects the module structures.
42 Todd class Set td(x) = ch (β X (O X )) td(x) H (X; Q) If X is non-singular, then td(x) = Todd(X) [X]. With X possibly singular and E an algebraic vector bundle on X χ(x, E) = ɛ (ch(e) td(x)) ɛ: X is the map of X to a point. ɛ : H (X; Q) H ( ; Q) = Q
The Riemann-Roch Theorem
The Riemann-Roch Theorem Paul Baum Penn State TIFR Mumbai, India 20 February, 2013 THE RIEMANN-ROCH THEOREM Topics in this talk : 1. Classical Riemann-Roch 2. Hirzebruch-Riemann-Roch (HRR) 3. Grothendieck-Riemann-Roch
More informationThe Riemann-Roch Theorem
The Riemann-Roch Theorem TIFR Mumbai, India Paul Baum Penn State 7 August, 2015 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch
More informationAtiyah-Singer Revisited
Atiyah-Singer Revisited Paul Baum Penn State Texas A&M Universty College Station, Texas, USA April 1, 2014 From E 1, E 2,..., E n obtain : 1) The Dirac operator of R n D = n j=1 E j x j 2) The Bott generator
More informationWHAT IS K-HOMOLOGY? Paul Baum Penn State. Texas A&M University College Station, Texas, USA. April 2, 2014
WHAT IS K-HOMOLOGY? Paul Baum Penn State Texas A&M University College Station, Texas, USA April 2, 2014 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, 2014 1 / 56 Let X be a compact C manifold without
More informationDirac Operator. Texas A&M University College Station, Texas, USA. Paul Baum Penn State. March 31, 2014
Dirac Operator Paul Baum Penn State Texas A&M University College Station, Texas, USA March 31, 2014 Miniseries of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond
More informationDirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017
Dirac Operator Göttingen Mathematical Institute Paul Baum Penn State 6 February, 2017 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. The Riemann-Roch theorem 5. K-theory
More informationBEYOND ELLIPTICITY. Paul Baum Penn State. Fields Institute Toronto, Canada. June 20, 2013
BEYOND ELLIPTICITY Paul Baum Penn State Fields Institute Toronto, Canada June 20, 2013 Paul Baum (Penn State) Beyond Ellipticity June 20, 2013 1 / 47 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer
More informationRecall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,
Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n
More informationHochschild homology and Grothendieck Duality
Hochschild homology and Grothendieck Duality Leovigildo Alonso Tarrío Universidade de Santiago de Compostela Purdue University July, 1, 2009 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality
More informationAPPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY
APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then
More informationThe Canonical Sheaf. Stefano Filipazzi. September 14, 2015
The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationAlgebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism
Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism UWO January 25, 2005 Marc Levine Prelude: From homotopy theory to A 1 -homotopy theory A basic object in homotopy theory is a generalized
More information3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).
3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)
More informationK-Homology, Assembly and Rigidity Theorems for Relative Eta Invariants
K-Homology, Assembly and Rigidity Theorems for Relative Eta Invariants Department of Mathematics Pennsylvania State University Potsdam, May 16, 2008 Outline K-homology, elliptic operators and C*-algebras.
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula
20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More informationTopology of Toric Varieties, Part II
Topology of Toric Varieties, Part II Daniel Chupin April 2, 2018 Abstract Notes for a talk leading up to a discussion of the Hirzebruch-Riemann-Roch (HRR) theorem for toric varieties, and some consequences
More informationInvariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem
PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents
More informationNOTES ON DIVISORS AND RIEMANN-ROCH
NOTES ON DIVISORS AND RIEMANN-ROCH NILAY KUMAR Recall that due to the maximum principle, there are no nonconstant holomorphic functions on a compact complex manifold. The next best objects to study, as
More informationCategories and functors
Lecture 1 Categories and functors Definition 1.1 A category A consists of a collection ob(a) (whose elements are called the objects of A) for each A, B ob(a), a collection A(A, B) (whose elements are called
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More informationHODGE NUMBERS OF COMPLETE INTERSECTIONS
HODGE NUMBERS OF COMPLETE INTERSECTIONS LIVIU I. NICOLAESCU 1. Holomorphic Euler characteristics Suppose X is a compact Kähler manifold of dimension n and E is a holomorphic vector bundle. For every p
More informationTopics in Algebraic Geometry
Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes
More informationAlgebraic Curves and Riemann Surfaces
Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex
More informationCelebrating One Hundred Fifty Years of. Topology. ARBEITSTAGUNG Bonn, May 22, 2013
Celebrating One Hundred Fifty Years of Topology John Milnor Institute for Mathematical Sciences Stony Brook University (www.math.sunysb.edu) ARBEITSTAGUNG Bonn, May 22, 2013 Algebra & Number Theory 3 4
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 4
COMPLEX ALGEBRAIC SURFACES CLASS 4 RAVI VAKIL CONTENTS 1. Serre duality and Riemann-Roch; back to curves 2 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion
More informationALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES
ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open
More informationAn overview of D-modules: holonomic D-modules, b-functions, and V -filtrations
An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
More informationChern numbers and Hilbert Modular Varieties
Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point
More informationAlgebraic Cobordism. 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006.
Algebraic Cobordism 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006 Marc Levine Outline: Describe the setting of oriented cohomology over a
More informationGK-SEMINAR SS2015: SHEAF COHOMOLOGY
GK-SEMINAR SS2015: SHEAF COHOMOLOGY FLORIAN BECK, JENS EBERHARDT, NATALIE PETERNELL Contents 1. Introduction 1 2. Talks 1 2.1. Introduction: Jordan curve theorem 1 2.2. Derived categories 2 2.3. Derived
More informationHODGE GENERA OF ALGEBRAIC VARIETIES, II.
HODGE GENERA OF ALGEBRAIC VARIETIES, II. SYLVAIN E. CAPPELL, ANATOLY LIBGOBER, LAURENTIU MAXIM, AND JULIUS L. SHANESON Abstract. We study the behavior of Hodge-theoretic genera under morphisms of complex
More informationOverview of Atiyah-Singer Index Theory
Overview of Atiyah-Singer Index Theory Nikolai Nowaczyk December 4, 2014 Abstract. The aim of this text is to give an overview of the Index Theorems by Atiyah and Singer. Our primary motivation is to understand
More informationAlgebraic v.s. Analytic Point of View
Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationTHE GROTHENDIECK-RIEMANN-ROCH THEOREM FOR VARIETIES
THE GROTHENDIECK-RIEMANN-ROCH THEOREM FOR VARIETIES PETER XU ABSTRACT. We give an exposition of the Grothendieck-Riemann-Roch theorem for algebraic varieties. Our proof follows Borel and Serre [3] and
More informationDerived categories, perverse sheaves and intermediate extension functor
Derived categories, perverse sheaves and intermediate extension functor Riccardo Grandi July 26, 2013 Contents 1 Derived categories 1 2 The category of sheaves 5 3 t-structures 7 4 Perverse sheaves 8 1
More informationIntroduction to Index Theory. Elmar Schrohe Institut für Analysis
Introduction to Index Theory Elmar Schrohe Institut für Analysis Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a
More informationAn introduction to calculus of functors
An introduction to calculus of functors Ismar Volić Wellesley College International University of Sarajevo May 28, 2012 Plan of talk Main point: One can use calculus of functors to answer questions about
More informationCoherent sheaves on elliptic curves.
Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of
More informationFROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS
FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS ZHIQIN LU. Introduction It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer.
More informationSERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN
SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank
More informationContents. Chapter 3. Local Rings and Varieties Rings of Germs of Holomorphic Functions Hilbert s Basis Theorem 39.
Preface xiii Chapter 1. Selected Problems in One Complex Variable 1 1.1. Preliminaries 2 1.2. A Simple Problem 2 1.3. Partitions of Unity 4 1.4. The Cauchy-Riemann Equations 7 1.5. The Proof of Proposition
More informationKähler manifolds and variations of Hodge structures
Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic
More informationx X p K i (k(x)) K i 1 (M p+1 )
5. Higher Chow Groups and Beilinson s Conjectures 5.1. Bloch s formula. One interesting application of Quillen s techniques involving the Q construction is the following theorem of Quillen, extending work
More informationA users guide to K-theory
A users guide to K-theory K-theory Alexander Kahle alexander.kahle@rub.de Mathematics Department, Ruhr-Universtät Bochum Bonn-Cologne Intensive Week: Tools of Topology for Quantum Matter, July 2014 Outline
More informationDEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE
DEFINITION OF ABELIAN VARIETIES AND THE THEOREM OF THE CUBE ANGELA ORTEGA (NOTES BY B. BAKKER) Throughout k is a field (not necessarily closed), and all varieties are over k. For a variety X/k, by a basepoint
More informationOperator Algebras II, Homological functors, derived functors, Adams spectral sequence, assembly map and BC for compact quantum groups.
II, functors, derived functors, Adams spectral, assembly map and BC for compact quantum groups. University of Copenhagen 25th July 2016 Let C be an abelian category We call a covariant functor F : T C
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More informationKR-theory. Jean-Louis Tu. Lyon, septembre Université de Lorraine France. IECL, UMR 7502 du CNRS
Jean-Louis Tu Université de Lorraine France Lyon, 11-13 septembre 2013 Complex K -theory Basic definition Definition Let M be a compact manifold. K (M) = {[E] [F] E, F vector bundles } [E] [F] [E ] [F
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More information= Spec(Rf ) R p. 2 R m f gives a section a of the stalk bundle over X f as follows. For any [p] 2 X f (f /2 p), let a([p]) = ([p], a) wherea =
LECTURES ON ALGEBRAIC GEOMETRY MATH 202A 41 5. Affine schemes The definition of an a ne scheme is very abstract. We will bring it down to Earth. However, we will concentrate on the definitions. Properties
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More informationSome remarks on symmetric correspondences
Some remarks on symmetric correspondences H. Lange Mathematisches Institut Universitat Erlangen-Nurnberg Bismarckstr. 1 1 2 D-91054 Erlangen (Germany) E. Sernesi Dipartimento di Matematica Università Roma
More informationAPPENDIX 3: AN OVERVIEW OF CHOW GROUPS
APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss
More informationThe Theorem of Gauß-Bonnet in Complex Analysis 1
The Theorem of Gauß-Bonnet in Complex Analysis 1 Otto Forster Abstract. The theorem of Gauß-Bonnet is interpreted within the framework of Complex Analysis of one and several variables. Geodesic triangles
More informationTheta divisors and the Frobenius morphism
Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following
More informationContributors. Preface
Contents Contributors Preface v xv 1 Kähler Manifolds by E. Cattani 1 1.1 Complex Manifolds........................... 2 1.1.1 Definition and Examples.................... 2 1.1.2 Holomorphic Vector Bundles..................
More informationAn Introduction to Spectral Sequences
An Introduction to Spectral Sequences Matt Booth December 4, 2016 This is the second half of a joint talk with Tim Weelinck. Tim introduced the concept of spectral sequences, and did some informal computations,
More informationCATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.
CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists
More informationIND-COHERENT SHEAVES AND SERRE DUALITY II. 1. Introduction
IND-COHERENT SHEAVES AND SERRE DUALITY II 1. Introduction Let X be a smooth projective variety over a field k of dimension n. Let V be a vector bundle on X. In this case, we have an isomorphism H i (X,
More informationHodge Theory of Maps
Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar
More informationRIEMANN S INEQUALITY AND RIEMANN-ROCH
RIEMANN S INEQUALITY AND RIEMANN-ROCH DONU ARAPURA Fix a compact connected Riemann surface X of genus g. Riemann s inequality gives a sufficient condition to construct meromorphic functions with prescribed
More informationAlgebraic Topology II Notes Week 12
Algebraic Topology II Notes Week 12 1 Cohomology Theory (Continued) 1.1 More Applications of Poincaré Duality Proposition 1.1. Any homotopy equivalence CP 2n f CP 2n preserves orientation (n 1). In other
More informationRiemann Surfaces and Algebraic Curves
Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1
More informationA First Lecture on Sheaf Cohomology
A First Lecture on Sheaf Cohomology Elizabeth Gasparim Departamento de Matemática, Universidade Federal de Pernambuco Cidade Universitária, Recife, PE, BRASIL, 50670-901 gasparim@dmat.ufpe.br I. THE DEFINITION
More informationTHE ARITHMETIC GROTHENDIECK-RIEMANN-ROCH THEOREM FOR GENERAL PROJECTIVE MORPHISMS
THE ARITHMETIC GROTHENDIECK-RIEMANN-ROCH THEOREM FOR GENERAL PROJECTIVE MORPHISMS JOSÉ IGNACIO BURGOS GIL, GERARD FREIXAS I MONTPLET, AND RĂZVAN LIŢCANU Abstract. In this paper we generalize the arithmetic
More informationFourier Mukai transforms II Orlov s criterion
Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and
More informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationThe Dirac-Ramond operator and vertex algebras
The Dirac-Ramond operator and vertex algebras Westfälische Wilhelms-Universität Münster cvoigt@math.uni-muenster.de http://wwwmath.uni-muenster.de/reine/u/cvoigt/ Vanderbilt May 11, 2011 Kasparov theory
More informationPatrick Iglesias-Zemmour
Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries
More informationAlgebraic cycle complexes
Algebraic cycle complexes June 2, 2008 Outline Algebraic cycles and algebraic K-theory The Beilinson-Lichtenbaum conjectures Bloch s cycle complexes Suslin s cycle complexes Algebraic cycles and algebraic
More informationDirect Limits. Mathematics 683, Fall 2013
Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry
More informationAFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES
AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.
More informationUseful theorems in complex geometry
Useful theorems in complex geometry Diego Matessi April 30, 2003 Abstract This is a list of main theorems in complex geometry that I will use throughout the course on Calabi-Yau manifolds and Mirror Symmetry.
More informationCrystalline Cohomology and Frobenius
Crystalline Cohomology and Frobenius Drew Moore References: Berthelot s Notes on Crystalline Cohomology, discussions with Matt Motivation Let X 0 be a proper, smooth variety over F p. Grothendieck s etale
More informationMT845: ALGEBRAIC CURVES
MT845: ALGEBRAIC CURVES DAWEI CHEN Contents 1. Sheaves and cohomology 1 2. Vector bundles, line bundles and divisors 9 3. Preliminaries on curves 16 4. Geometry of Weierstrass points 27 5. Hilbert scheme
More information6. Lecture cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not
6. Lecture 6 6.1. cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not only motivic cohomology, but also to Morel-Voevodsky
More informationINTERSECTION THEORY CLASS 19
INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed
More informationCohomology jump loci of local systems
Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to
More informationLECTURE 3: RELATIVE SINGULAR HOMOLOGY
LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following
More informationAn introduction to derived and triangulated categories. Jon Woolf
An introduction to derived and triangulated categories Jon Woolf PSSL, Glasgow, 6 7th May 2006 Abelian categories and complexes Derived categories and functors arise because 1. we want to work with complexes
More informationConstructible Derived Category
Constructible Derived Category Dongkwan Kim September 29, 2015 1 Category of Sheaves In this talk we mainly deal with sheaves of C-vector spaces. For a topological space X, we denote by Sh(X) the abelian
More informationSTEENROD OPERATIONS IN ALGEBRAIC GEOMETRY
STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY ALEXANDER MERKURJEV 1. Introduction Let p be a prime integer. For a pair of topological spaces A X we write H i (X, A; Z/pZ) for the i-th singular cohomology group
More informationEQUIVARIANT CHARACTERISTIC CLASSES OF SINGULAR COMPLEX ALGEBRAIC VARIETIES
EQUIVARIANT CHARACTERISTIC CLASSES OF SINGULAR COMPLEX ALGEBRAIC VARIETIES SYLVAIN E. CAPPELL, LAURENTIU MAXIM, JÖRG SCHÜRMANN, AND JULIUS L. SHANESON Abstract. Homology Hirzebruch characteristic classes
More informationAbelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005)
Abelian Varieties and the Fourier Mukai transformations (Foschungsseminar 2005) U. Bunke April 27, 2005 Contents 1 Abelian varieties 2 1.1 Basic definitions................................. 2 1.2 Examples
More informationwhich is a group homomorphism, such that if W V U, then
4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV
More informationDAVE ANDERSON AND SAM PAYNE
OPERATIONAL K-THEORY DAVE ANDERSON AND SAM PAYNE Abstract. We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has
More informationGeometry of Conformal Field Theory
Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU) Contents 0. Introduction
More informationRTG Mini-Course Perspectives in Geometry Series
RTG Mini-Course Perspectives in Geometry Series Jacob Lurie Lecture IV: Applications and Examples (1/29/2009) Let Σ be a Riemann surface of genus g, then we can consider BDiff(Σ), the classifying space
More informationMath 231b Lecture 16. G. Quick
Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector
More informationThe diagonal property for abelian varieties
The diagonal property for abelian varieties Olivier Debarre Dedicated to Roy Smith on his 65th birthday. Abstract. We study complex abelian varieties of dimension g that have a vector bundle of rank g
More informationLECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES
LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain
More informationBERTRAND GUILLOU. s G q+r
STABLE A 1 -HOMOTOPY THEORY BERTRAND GUILLOU 1. Introduction Recall from the previous talk that we have our category pointed A 1 -homotopy category Ho A 1, (k) over a field k. We will often refer to an
More informationOn the geometric Langlands duality
On the geometric Langlands duality Peter Fiebig Emmy Noether Zentrum Universität Erlangen Nürnberg Schwerpunkttagung Bad Honnef April 2010 Outline This lecture will give an overview on the following topics:
More informationThe Ordinary RO(C 2 )-graded Cohomology of a Point
The Ordinary RO(C 2 )-graded Cohomology of a Point Tiago uerreiro May 27, 2015 Abstract This paper consists of an extended abstract of the Master Thesis of the author. Here, we outline the most important
More informationA p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationOperator algebras and topology
Operator algebras and topology Thomas Schick 1 Last compiled November 29, 2001; last edited November 29, 2001 or later 1 e-mail: schick@uni-math.gwdg.de www: http://uni-math.gwdg.de/schick Fax: ++49-251/83
More information