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HODGE THEORY LEFSCHETZ PENCILS 1. More on Lefschetz
HODGE THEORY LEFSCHETZ PENCILS JACOB KELLER 1. More on Lefschetz Pencils When studying the vanishing homology of a hyperplane section, we want to view the hyperplane as one element of a Lefschetz pencil. This is because associated to each singular divisor in the pencil, there is an element of the homology of our hyperplane that is not detected by the Lefschetz hyperplane theorem. In order to do this we should know more about Lefschetz pencils, and that’s how we will start this lecture. We want to be able to produce Lefschetz pencils containing a given hyperplane section. So we should get a better characterization of a Lefschetz pencil that is easier to compute. Let X be a projective variety embedded in PN that is not contained in a hyperplane. We consider the variety N∗ Z = {(x, H)|x ∈ X ∩ H is a singular point} ⊆ X × P To better understand the geometry of Z, we consider it’s projection π1 : Z → X. For any x ∈ X and N hyperplane H ⊆ P , the tangent plane to H ∩ X at x is the intersection of H and TxX. Thus X ∩ H will −1 be singular at x if and only if H contains TxX. Thus for x ∈ X, π1 (Z) is the set of hyperplanes that contain TxX. These hyperplanes form an N − n − 1 dimensional projective space, and thus π1 exhibits Z as a PN − n − 1-bundle over X, and we deduce that Z is smooth. N∗ Now we consider the projection π2 : Z → P . The image of this map is denoted DX and is called the discriminant of X. -
Homology Theory on Algebraic Varieties
HOMOLOGY THEORYon ALGEBRAIC VARIETIES HOMOLOGY THEORY ON ALGEBRAIC VARIETIES by ANDREW H. WALLACE Assistant Professor of Mathematics University of Toronto PERGAMON PRESS LONDON NEW YORK PARIS LOS ANGELES 1958 . PERGAMON PRESS LTD. 4 and 5 Fitzroy Square, London W.I. PERGAMON PRESS INC. 122 East 55th Street, New York, N.Y. 10638 South Wilton Place, Los Angeles 47, California PERGAMON PRESS S.A.R.L. 24 Rue des Ecoles, Paris V" Copyright 0 1958 A. H. Wallace Library of Congress Card Number 57-14497 Printed in Northern Ireland at The Universities Press, Be fast CONTENTS INTRODUCTION vu 1. -LINEAR SECTIONS OF AN ALGEBRAIC VARIETY 1. Hyperplane sections of a non-singular variety 1 2. A family of linear sections of W 2 3. The fibring of a variety defined over the complex numbers 7. 4. Homology groups related to V(K) 17 II. THE SINGULAR SECTIONS 1. Statement of the results 23 2. Proof of Theorem 11 25 III. A PENCIL OF HYPERPLANE SECTIONS 1. The choice of a pencil 34 2. Notation 37 3. Reduction to local theorems 38. IV. LEFSCHETZ'S FIRST AND SECOND THEOREMS 1. Lefschetz's first main theorem 43 2. Statement of Lefschetz's second main theorem 49 3. Sketch proof of Theorem 19 49- .4. Some immediate consequences _ 84 V. PROOF OF LEFSCHETZ'S SECOND THEOREM 1. Deformation theorems 56 2. Some remarks dh Theorem 19 80 3. Formal verification of Theorem 19; the vanishing cycle62 4. Proof of Theorem 19, parts (1) and (2) 64 5. Proof of Theorem 19, part (3) 87 v Vi CONTENTS VI. -
Homology Stratifications and Intersection Homology 1 Introduction
ISSN 1464-8997 (on line) 1464-8989 (printed) 455 Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest Pages 455–472 Homology stratifications and intersection homology Colin Rourke Brian Sanderson Abstract A homology stratification is a filtered space with local ho- mology groups constant on strata. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersection ho- mology, homology stratifications do not appear to have been studied in any detail and their properties remain obscure. Here we use them to present a simplified version of the Goresky–MacPherson proof valid for PL spaces, and we ask a number of questions. The proof uses a new technique, homology general position, which sheds light on the (open) problem of defining generalised intersection homology. AMS Classification 55N33, 57Q25, 57Q65; 18G35, 18G60, 54E20, 55N10, 57N80, 57P05 Keywords Permutation homology, intersection homology, homology stratification, homology general position Rob Kirby has been a great source of encouragement. His help in founding the new electronic journal Geometry & Topology has been invaluable. It is a great pleasure to dedicate this paper to him. 1 Introduction Homology stratifications are filtered spaces with local homology groups constant on strata; they include stratified sets as special cases. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersec- tion homology, they do not appear to have been studied in any detail and their properties remain obscure. It is the purpose of this paper is to publicise these neglected but powerful tools. The main result is that the intersection homology groups of a PL homology stratification are given by singular cycles meeting the strata with appropriate dimension restrictions. -
Generalized Rudin–Shapiro Sequences
ACTA ARITHMETICA LX.1 (1991) Generalized Rudin–Shapiro sequences by Jean-Paul Allouche (Talence) and Pierre Liardet* (Marseille) 1. Introduction 1.1. The Rudin–Shapiro sequence was introduced independently by these two authors ([21] and [24]) and can be defined by u(n) εn = (−1) , where u(n) counts the number of 11’s in the binary expansion of the integer n (see [5]). This sequence has the following property: X 2iπnθ 1/2 (1) ∀ N ≥ 0, sup εne ≤ CN , θ∈R n<N where one can take C = 2 + 21/2 (see [22] for improvements of this value). The order of magnitude of the left hand term in (1), as N goes to infin- 1/2 ity, is exactly N ; indeed, for each sequence (an) with values ±1 one has 1/2 X 2iπn(·) X 2iπn(·) N = ane ≤ ane , 2 ∞ n<N n<N where k k2 denotes the quadratic norm and k k∞ the supremum norm. Note that for almost every√ sequence (an) of ±1’s the supremum norm of the above sum is bounded by N Log N (see [23]). The inequality (1) has been generalized in [2] (see also [3]): X 2iπxu(n) 0 α(x) (2) sup f(n)e ≤ C N , f∈M2 n<N where M2 is the set of 2-multiplicative sequences with modulus 1. The * Research partially supported by the D.R.E.T. under contract 901636/A000/DRET/ DS/SR 2 J.-P. Allouche and P. Liardet exponent α(x) is explicitly given and satisfies ∀ x 1/2 ≤ α(x) ≤ 1 , ∀ x 6∈ Z α(x) < 1 , ∀ x ∈ Z + 1/2 α(x) = 1/2 . -
Fibonacci, Kronecker and Hilbert NKS 2007
Fibonacci, Kronecker and Hilbert NKS 2007 Klaus Sutner Carnegie Mellon University www.cs.cmu.edu/∼sutner NKS’07 1 Overview • Fibonacci, Kronecker and Hilbert ??? • Logic and Decidability • Additive Cellular Automata • A Knuth Question • Some Questions NKS’07 2 Hilbert NKS’07 3 Entscheidungsproblem The Entscheidungsproblem is solved when one knows a procedure by which one can decide in a finite number of operations whether a given logical expression is generally valid or is satisfiable. The solution of the Entscheidungsproblem is of fundamental importance for the theory of all fields, the theorems of which are at all capable of logical development from finitely many axioms. D. Hilbert, W. Ackermann Grundzuge¨ der theoretischen Logik, 1928 NKS’07 4 Model Checking The Entscheidungsproblem for the 21. Century. Shift to computer science, even commercial applications. Fix some suitable logic L and collection of structures A. Find efficient algorithms to determine A |= ϕ for any structure A ∈ A and sentence ϕ in L. Variants: fix ϕ, fix A. NKS’07 5 CA as Structures Discrete dynamical systems, minimalist description: Aρ = hC, i where C ⊆ ΣZ is the space of configurations of the system and is the “next configuration” relation induced by the local map ρ. Use standard first order logic (either relational or functional) to describe properties of the system. NKS’07 6 Some Formulae ∀ x ∃ y (y x) ∀ x, y, z (x z ∧ y z ⇒ x = y) ∀ x ∃ y, z (y x ∧ z x ∧ ∀ u (u x ⇒ u = y ∨ u = z)) There is no computability requirement for configurations, in x y both x and y may be complicated. -
Math 6280 - Class 27
MATH 6280 - CLASS 27 Contents 1. Reduced and relative homology and cohomology 1 2. Eilenberg-Steenrod Axioms 2 2.1. Axioms for unreduced homology 2 2.2. Axioms for reduced homology 4 2.3. Axioms for cohomology 5 These notes are based on • Algebraic Topology from a Homotopical Viewpoint, M. Aguilar, S. Gitler, C. Prieto • A Concise Course in Algebraic Topology, J. Peter May • More Concise Algebraic Topology, J. Peter May and Kate Ponto • Algebraic Topology, A. Hatcher 1. Reduced and relative homology and cohomology Definition 1.1. For A ⊂ X a sub-complex, let Cn(A) ⊂ Cn(X) as the cells of A are a subset of the cells of X. • Let C∗(X; A) = C∗(X)=C∗(A) and H∗(X; A) = H∗(C∗(X; A)). • Let ∗ ∗ ∗ C (X; A) = ker(C (X) ! C (A)) = Hom(C∗(X; A); Z) and H∗(X) = H∗(C∗(X; A)). We can give similar definitions for H∗(X; A; M) and H∗(X; A; M): Definition 1.2. For X a based CW-complex with based point ∗ a zero cell, 1 2 MATH 6280 - CLASS 27 • Let Ce∗(X) = C∗(X)=C∗(∗) and He∗(X) = H∗(Ce∗(X)). • Let ∗ ∗ ∗ Ce (X) = ker(C (X) ! C (∗)) = Hom(Ce∗(X); Z) and He ∗(X) = H∗(Ce∗(X)). ∼ Remark 1.3. Note that C∗(X; A) = C∗(X)=C∗(A) = Ce∗(X=A). Indeed, X=A has a CW-structure all cells in A identified to the base point and one cell for each cell not in A. Therefore, we have a natural isomorphism ∼ H∗(X; A) = He∗(X=A): Similarly, H∗(X; A) =∼ He ∗(X=A) Unreduced cohomology can be though of as a functor from the homotopy category of pairs of topological spaces to abelian groups: H∗(−; −; M): hCWpairs ! Ab where H∗(X; M) = H∗(X; ;; M): 2. -
Formal Groups, Witt Vectors and Free Probability
Formal Groups, Witt vectors and Free Probability Roland Friedrich and John McKay December 6, 2019 Abstract We establish a link between free probability theory and Witt vectors, via the theory of formal groups. We derive an exponential isomorphism which expresses Voiculescu's free multiplicative convolution as a function of the free additive convolution . Subsequently we continue our previous discussion of the relation between complex cobordism and free probability. We show that the generic nth free cumulant corresponds to the cobordism class of the (n 1)-dimensional complex projective space. This permits us to relate several probability distributions− from random matrix theory to known genera, and to build a dictionary. Finally, we discuss aspects of free probability and the asymptotic representation theory of the symmetric group from a conformal field theoretic perspective and show that every distribution with mean zero is embeddable into the Universal Grassmannian of Sato-Segal-Wilson. MSC 2010: 46L54, 55N22, 57R77, Keywords: Free probability and free operator algebras, formal group laws, Witt vectors, complex cobordism (U- and SU-cobordism), non-crossing partitions, conformal field theory. Contents 1 Introduction 2 2 Free probability and formal group laws4 2.1 Generalised free probability . .4 arXiv:1204.6522v3 [math.OA] 5 Dec 2019 2.2 The Lie group Aut( )....................................5 O 2.3 Formal group laws . .6 2.4 The R- and S-transform . .7 2.5 Free cumulants . 11 3 Witt vectors and Hopf algebras 12 3.1 The affine group schemes . 12 3.2 Witt vectors, λ-rings and necklace algebras . 15 3.3 The induced ring structure . -
Lefschetz Section Theorems for Tropical Hypersurfaces
LEFSCHETZ SECTION THEOREMS FOR TROPICAL HYPERSURFACES CHARLES ARNAL, ARTHUR RENAUDINEAU, AND KRISTIN SHAW Abstract. We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersur- faces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which n are compact or contained in R are torsion free. We prove a relation- ship between the coefficients of the χy genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellu- lar chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropi- cal hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge-Deligne numbers of their complex counterparts. Contents 1. Introduction 1 Acknowledgement 5 2. Preliminaries 6 3. Tropical Lefschetz hyperplane section theorem 16 4. The tropical homology of hypersurfaces is torsion free 24 5. Betti numbers of tropical homology and Hodge numbers 27 References 32 1. Introduction Tropical homology is a homology theory with non-constant coefficients for polyhedral spaces. Itenberg, Katzarkov, Mikhalkin, and Zharkov, show that under suitable conditions, the Q-tropical Betti numbers of the tropical limit arXiv:1907.06420v1 [math.AG] 15 Jul 2019 of a family of complex projective varieties are equal to the corresponding Hodge numbers of a generic member of the family [IKMZ16]. This explains the particular interest of these homology groups in tropical and complex algebraic geometry. -
The Derived Category of Sheaves and the Poincare-Verdier Duality
THE DERIVED CATEGORY OF SHEAVES AND THE POINCARE-VERDIER´ DUALITY LIVIU I. NICOLAESCU ABSTRACT. I think these are powerful techniques. CONTENTS 1. Derived categories and derived functors: a short introduction2 2. Basic operations on sheaves 16 3. The derived functor Rf! 31 4. Limits 36 5. Cohomologically constructible sheaves 47 6. Duality with coefficients in a field. The absolute case 51 7. The general Poincare-Verdier´ duality 58 8. Some basic properties of f ! 63 9. Alternate descriptions of f ! 67 10. Duality and constructibility 70 References 72 Date: Last revised: April, 2005. Notes for myself and whoever else is reading this footnote. 1 2 LIVIU I. NICOLAESCU 1. DERIVED CATEGORIES AND DERIVED FUNCTORS: A SHORT INTRODUCTION For a detailed presentation of this subject we refer to [4,6,7,8]. Suppose A is an Abelian category. We can form the Abelian category CpAq consisting of complexes of objects in A. We denote the objects in CpAq by A or pA ; dq. The homology of such a complex will be denoted by H pA q. P p q A morphism s HomCpAq A ;B is called a quasi-isomorphism (qis brevity) if it induces an isomorphism in co-homology. We will indicate qis-s by using the notation A ùs B : Define a new additive category KpAq whose objects coincide with the objects of CpAq, i.e. are complexes, but the morphisms are the homotopy classes of morphisms in CpAq, i.e. p q p q{ HomKpAq A ;B : HomCpAq A ;B ; where denotes the homotopy relation. Often we will use the notation r s p q A ;B : HomKpAq A ;B : The derived category of A will be a category DpAq with the same objects as KpAq but with a q much larger class of morphisms. -
Agnieszka Bodzenta
June 12, 2019 HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY AGNIESZKA BODZENTA Contents 1. Categories, functors, natural transformations 2 1.1. Direct product, coproduct, fiber and cofiber product 4 1.2. Adjoint functors 5 1.3. Limits and colimits 5 1.4. Localisation in categories 5 2. Abelian categories 8 2.1. Additive and abelian categories 8 2.2. The category of modules over a quiver 9 2.3. Cohomology of a complex 9 2.4. Left and right exact functors 10 2.5. The category of sheaves 10 2.6. The long exact sequence of Ext-groups 11 2.7. Exact categories 13 2.8. Serre subcategory and quotient 14 3. Triangulated categories 16 3.1. Stable category of an exact category with enough injectives 16 3.2. Triangulated categories 22 3.3. Localization of triangulated categories 25 3.4. Derived category as a quotient by acyclic complexes 28 4. t-structures 30 4.1. The motivating example 30 4.2. Definition and first properties 34 4.3. Semi-orthogonal decompositions and recollements 40 4.4. Gluing of t-structures 42 4.5. Intermediate extension 43 5. Perverse sheaves 44 5.1. Derived functors 44 5.2. The six functors formalism 46 5.3. Recollement for a closed subset 50 1 2 AGNIESZKA BODZENTA 5.4. Perverse sheaves 52 5.5. Gluing of perverse sheaves 56 5.6. Perverse sheaves on hyperplane arrangements 59 6. Derived categories of coherent sheaves 60 6.1. Crash course on spectral sequences 60 6.2. Preliminaries 61 6.3. Hom and Hom 64 6.4. -
Characteristic Classes of Mixed Hodge Modules
Topology of Stratified Spaces MSRI Publications Volume 58, 2011 Characteristic classes of mixed Hodge modules JORG¨ SCHURMANN¨ ABSTRACT. This paper is an extended version of an expository talk given at the workshop “Topology of Stratified Spaces” at MSRI in September 2008. It gives an introduction and overview about recent developments on the in- teraction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito’s deep theory of mixed Hodge modules as a black box, thinking about them as “constructible or perverse sheaves of Hodge struc- tures”, having the same functorial calculus of Grothendieck functors. For the “constant Hodge sheaf”, one gets the “motivic characteristic classes” of Brasselet, Schurmann,¨ and Yokura, whereas the classes of the “intersection homology Hodge sheaf” were studied by Cappell, Maxim, and Shaneson. The classes associated to “good” variation of mixed Hodge structures where studied in connection with understanding the monodromy action by these three authors together with Libgober, and also by the author. There are two versions of these characteristic classes. The K-theoretical classes capture information about the graded pieces of the filtered de Rham complex of the filtered D-module underlying a mixed Hodge module. Appli- cation of a suitable Todd class transformation then gives classes in homology. These classes are functorial for proper pushdown and exterior products, to- gether with some other properties one would expect for a satisfactory theory of characteristic classes for singular spaces. For “good” variation of mixed Hodge structures they have an explicit clas- sical description in terms of “logarithmic de Rham complexes”. -
Intersection Homology Duality and Pairings: Singular, PL, and Sheaf
Intersection homology duality and pairings: singular, PL, and sheaf-theoretic Greg Friedman and James E. McClure July 22, 2020 Contents 1 Introduction 2 2 Conventions and some background 11 2.1 Pseudomanifolds and intersection homology . .. 11 3 Hypercohomology in the derived category 13 4 Some particular sheaf complexes 16 4.1 The Verdier dualizing complex and Verdier duality . 16 4.2 TheDelignesheaf................................. 17 4.3 The sheaf complex of intersection chains . .. 19 4.4 Thesheafofintersectioncochains . .. 20 ∗ 4.4.1 Ip¯C is quasi-isomorphic to PDp¯ ..................... 22 5 Sheafification of the cup product 24 5.1 Applications: A multiplicative de Rham theorem for perverse differential forms and compatibility with the blown-up cup product . 26 arXiv:1812.10585v3 [math.GT] 22 Jul 2020 5.1.1 Perversedifferentialforms . 26 5.1.2 Blown-up intersection cohomology . 31 6 Classical duality via sheaf maps 32 6.1 Orientationsandcanonicalproducts. ... 33 6.2 ProofofTheorem6.1............................... 35 7 Compatibility of cup, intersection, and sheaf products 50 8 Classical duality and Verdier duality 55 A Co-cohomology 59 1 B A meditation on signs 60 2000 Mathematics Subject Classification: Primary: 55N33, 55N45 Secondary: 55N30, 57Q99 Keywords: intersection homology, intersection cohomology, pseudomanifold, cup product, cap product, intersection product, Poincar´e duality, Verdier dual- ity, sheaf theory Abstract We compare the sheaf-theoretic and singular chain versions of Poincar´eduality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the sin- gular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky- MacPherson PL intersection product.