DOCSLIB.ORG

Mixed Hodge Structures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mixed Hodge Structures Mixed Hodge structures 8.1 Definition of a mixed Hodge structure · Definition 1. A mixed Hodge structure V = (VZ;W·;F ) is a triple, where: (i) VZ is a free Z-module of finite rank. (ii) W· (the weight filtration) is a finite, increasing, saturated filtration of VZ (i.e. Wk=Wk−1 is torsion free for all k). · (iii) F (the Hodge filtration) is a finite decreasing filtration of VC = VZ ⊗Z C, such that the induced filtration on (Wk=Wk−1)C = (Wk=Wk−1)⊗ZC defines a Hodge structure of weight k on Wk=Wk−1. Here the induced filtration is p p F (Wk=Wk−1)C = (F + Wk−1 ⊗Z C) \ (Wk ⊗Z C): Mixed Hodge structures over Q or R (Q-mixed Hodge structures or R-mixed Hodge structures) are defined in the obvious way. A weight k Hodge structure V is in particular a mixed Hodge structure: we say that such mixed Hodge structures are pure of weight k. The main motivation is the following theorem: Theorem 2 (Deligne). If X is a quasiprojective variety over C, or more generally any separated scheme of finite type over Spec C, then there is a · mixed Hodge structure on H (X; Z) mod torsion, and it is functorial for k morphisms of schemes over Spec C. Finally, W`H (X; C) = 0 for k < 0 k k and W`H (X; C) = H (X; C) for ` ≥ 2k. More generally, a topological structure definable by algebraic geometry has a mixed Hodge structure: relative cohomology, punctured neighbor- hoods, the link of a singularity, . Two extreme cases of the theorem are compact but possibly singular varieties or smooth but not necessarily compact varieties. The mixed Hodge structures look somewhat dual to each other in these two cases. In the first k k k case, the weight filtration on H (X; C) satisfies: WkH (X; C) = H (X; C), k k k and hence W`H (X; C) = H (X; C) for all ` ≥ k. Also, W`H (X; C) = 0 if ` < 2k − 2n. Of course, this is only interesting if k > n. In case X k is smooth but not necessarily compact, W`H (X; C) = 0 for all ` < k. k k k k Moreoer, W2nH (X; C) = H (X; C), and hence W`H (X; C) = H (X; C) for all ` ≥ 2n; again, this is only interesting if k > n. 1 We describe the mixed Hodge structure of a curve C in more detail: Nodal curves: Suppose that, for simplicity, C is an irreducible curve with at worst nodes as singularities. (It is not hard to modify the above discussion to the case where C is not necessarily irreducible.) Let ν : Ce ! C be the −1 normalization map. For each node xi of C, write ν (xi) = fpi; qig. Then we have an exact sequence 1 1 1 0 ! W0H (C; Z) ! H (C; Z) ! H (C; Z) ! 0; 1 ∗ 1 1 where W0H (C; Z) = Kerfν : H (C; Z) ! H (C; Z)g. In terms of hyper- cohomology, let C· be the complex 1 OC ! ν∗Ω : Ce · 1 Then C is a subcomplex of ν∗Ω , which is quasi-isomorphic to ν∗ , and we Ce C have an exact sequence · 1 0 !C ! ν∗Ω ! ν∗O =OC ! 0: Ce Ce Here ν O =O is a skyscraper sheaf with stalk at each of the double ∗ Ce C C points. Taking hypercohomology gives 0 0 d 1 1 H (C; C) ! H (Ce; C) ! C ! H (C; C) ! H (Ce; C) ! 0: 0 0 Since C is irreducible, and hence Ce is connected, H (C; C) ! H (Ce; C) is 1 1 1 ∼ an isomorphism, and hence W0H (C; C) = KerfH (C; C) ! H (Ce; C)g = d C . Of course, a similar statement holds over Z, and it is easy to identify 1 1 W0H (C; Z) with H1(C; Z)/ν∗(H (Ce; Z)); a set of representatives for this quotient is given by paths γi on Ce with γi(0) = pi and γi(1) = qi. Open smooth curves: Suppose that C = C − S, where C is a smooth projective curve and S is a finite set. Let S = fx1; : : : ; xsg. Then we have an exact sequence 1 1 s−1 0 ! H (C; Z) ! H (C; Z) ! Z ! 0; s−1 where we can identify Z with the subgroup of the free abelian group Z[S] Ps P consisting of formal sums i=1 ni · xi such that i ni = 0. In this case, we 1 1 set W1 = H (C; Z) and W2 = H (C; Z). The quotient W2=W1 is dual to the subspace of H1(C; Z) generated by small loops in C around the xi (which are 1 homologous to 0 in C). As we shall show, H (C; C) is the hypercohomology of the complex Ω· (log S) on C defined by C 1 OC ! ΩC (x1 + ··· + xs): 2 The inclusion Ω· ! Ω· (log S) gives an exact sequence C C s · · M 0 ! ΩC ! ΩC (log S) ! Cxi [−1] ! 0; i=1 where the map Ω1 (log S) ! Ls [1] is given by taking the residue C p i=1 Cxi −1 at xi (up to a factor of (2π −1) if we want to make sure the integral 1 1 Ls structure is correct). Here, the image of H (C; C) in H (C; i=1 Cxi [−1]) = 0 Ls P H (C; i=1 Cxi ) consists of s-tuples (t1; : : : ; ts) such that i ti = 0, by the residue theorem: the exact sequence s 0 1 0 1 M 1 1 H (C;ΩC ) ! H (C;ΩC (x1 + ··· + xs)) ! Cxi ! H (C;ΩC ) ! 0 i=1 implies that the image of H0(C;Ω1 (x + ··· + x )) in [S] has codimension C 1 s C one and is otherwise constrained by the residue theorem. 8.2 The category of mixed Hodge structures Definition 3. Let V1 and V2 be two mixed Hodge structures. A morphism of mixed Hodge structures is a homomorphism ':(V1)Z ! (V2)Z such that ' preserves the weight filtrations (i.e. '(WkV1) ⊆ WkV2 for all k), and the extension of ' to a linear map (V1)C ! (V2)C preserves the Hodge filtrations. k In particular, the induced maps on grW are morphisms of Hodge structures for all k. Sub-mixed Hodge structures are then defined in the obvious way: V 0 ⊆ V is a sub-mixed Hodge structure of V if it carries a mixed Hodge structure and the inclusion is a morphism of mixed Hodge structures. As we will see, this implies that the mixed Hodge structure on V 0 is the one 0 0 induced from the mixed Hodge structure on V , i.e. WkV = WkV \ V and p 0 p 0 F V = F V \ V . For example, WkV is a sub-mixed Hodge structure of V for all k. One could also define morphisms of degree r, generalizing the above (morphisms of degree 0), by requiring that '(WkV1) ⊆ Wk+2rV2 for all k and p p+r k '(F V1) ⊆ F V2 for all p, so that the induced maps on grW are morphisms of Hodge structures of type (r; r). However, as in the case of morphisms of Hodge structures, we can always reduce to morphisms of weight 0 via Tate twists, as we describe below. Convention: we will write V for VZ, VQ, VR, or VC, relying on the context to make precise which we mean, unless we have to avoid confusion. We 3 can also speak of Q-morphisms or R-morphisms of mixed Hodge structures. p Finally, we shall often write Wk for either WkV or WkVC, and similarly F for F pV , when V is clear from the context. Given the definition of morphisms, mixed Hodge structures form a cat- egory, and we can carry out the usual linear algebra operations. For exam- ple, the direct sum V1 ⊕ V2 is defined in the obvious way: Wk(V1 ⊕ V2) = WkV1 ⊕ WkV2, and similarly for the Hodge filtration. Likewise, tensor prod- uct: X Wk(V1 ⊗ V2) = WaV1 ⊗ WbV2; a+b≤k p X a b F (V1 ⊗ V2) = F V1 ⊗ F V2: a+b≥p Example 4. If Z(k) is the Tate Hodge structure of weight −2, then V (k) = V ⊗ Z(k) is the Tate twist of V . For Hom, we define: Ws Hom(V1;V2) = f' 2 Hom(V1;V2): '(WkV1) ⊆ Wk+sV2g; r p p+r F Hom(V1;V2) = f' 2 Hom(V1;V2): '(F V1) ⊆ F V2g: 0 In particular, W0 Hom(V1;V2)Z \ F Hom(V1;V2) is the group of morphisms of mixed Hodge structures from V1 to V2. Theorem 5. If ': V1 ! V2 is a morphism, or R-morphism, of mixed Hodge structures, then ' is strict with respect to both the weight and Hodge filtra- tions, i.e. Im ' \ WkV2 = '(WkV1); p p Im ' \ F V2 = '(F V1): To prove the theorem, we use the existence of a Deligne decomposition: Theorem 6. Let V be an R-mixed Hodge structure. Then there exist vector p;q subspaces I ⊆ VC such that: L p;q L p;q (i) WkVC = p+q≤k I , and in particular VC = p;q I . p L r;q (ii) F V = r≥p I . Hence p;q q;p I = I mod Wp+q−1: 4 (iii) The construction is functorial with respect to morphisms of mixed Hodge structures. Proof. Define 0 1 p;q p q X q−i+1 I = (F \ Wp+q) \ @F \ Wp+q + F \ Wp+q−iA : i≥2 p;q p;q p By definition, I ⊆ Wp+q, I ⊆ F , and p;q p q I = F \ F \ Wp+q mod Wp+q−2: The construction is also clearly functorial with respect to morphisms of mixed Hodge structures.
Load more

Recommended publications
  • Mixed Hodge Structure of Affine Hypersurfaces
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Hossein MOVASATI Mixed Hodge structure of affine hypersurfaces Tome 57, no 3 (2007), p. 775-801. <http://aif.cedram.org/item?id=AIF_2007__57_3_775_0> © Association des Annales de l’institut Fourier, 2007, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 57, 3 (2007) 775-801 MIXED HODGE STRUCTURE OF AFFINE HYPERSURFACES by Hossein MOVASATI Abstract. — In this article we give an algorithm which produces a basis of the n-th de Rham cohomology of the affine smooth hypersurface f −1(t) compatible with the mixed Hodge structure, where f is a polynomial in n + 1 variables and satisfies a certain regularity condition at infinity (and hence has isolated singulari- ties). As an application we show that the notion of a Hodge cycle in regular fibers of f is given in terms of the vanishing of integrals of certain polynomial n-forms n+1 in C over topological n-cycles on the fibers of f.
    [Show full text]
  • Lectures on Motivic Aspects of Hodge Theory: the Hodge Characteristic
    Lectures on Motivic Aspects of Hodge Theory: The Hodge Characteristic. Summer School Istanbul 2014 Chris Peters Universit´ede Grenoble I and TUE Eindhoven June 2014 Introduction These notes reflect the lectures I have given in the summer school Algebraic Geometry and Number Theory, 2-13 June 2014, Galatasaray University Is- tanbul, Turkey. They are based on [P]. The main goal was to explain why the topological Euler characteristic (for cohomology with compact support) for complex algebraic varieties can be refined to a characteristic using mixed Hodge theory. The intended audi- ence had almost no previous experience with Hodge theory and so I decided to start the lectures with a crash course in Hodge theory. A full treatment of mixed Hodge theory would require a detailed ex- position of Deligne's theory [Del71, Del74]. Time did not allow for that. Instead I illustrated the theory with the simplest non-trivial examples: the cohomology of quasi-projective curves and of singular curves. For degenerations there is also such a characteristic as explained in [P-S07] and [P, Ch. 7{9] but this requires the theory of limit mixed Hodge structures as developed by Steenbrink and Schmid. For simplicity I shall only consider Q{mixed Hodge structures; those coming from geometry carry a Z{structure, but this iwill be neglected here. So, whenever I speak of a (mixed) Hodge structure I will always mean a Q{Hodge structure. 1 A Crash Course in Hodge Theory For background in this section, see for instance [C-M-P, Chapter 2]. 1 1.1 Cohomology Recall from Loring Tu's lectures that for a sheaf F of rings on a topolog- ical space X, cohomology groups Hk(X; F), q ≥ 0 have been defined.
    [Show full text]
  • Transcendental Hodge Algebra
    M. Verbitsky Transcendental Hodge algebra Transcendental Hodge algebra Misha Verbitsky1 Abstract The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in p-th cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natu- ral algebraic structure, and compute this algebra explicitly for a hyperk¨ahler manifold. As an application, we obtain a theorem about dimension of a compact torus T admit- ting a holomorphic symplectic embedding to a hyperk¨ahler manifold M. If M is generic in a d-dimensional family of deformations, then dim T > 2[(d+1)/2]. Contents 1 Introduction 2 1.1 Mumford-TategroupandHodgegroup. 2 1.2 Hyperk¨ahler manifolds: an introduction . 3 1.3 Trianalytic and holomorphic symplectic subvarieties . 3 2 Hodge structures 4 2.1 Hodgestructures:thedefinition. 4 2.2 SpecialMumford-Tategroup . 5 2.3 Special Mumford-Tate group and the Aut(C/Q)-action. .. .. 6 3 Transcendental Hodge algebra 7 3.1 TranscendentalHodgelattice . 7 3.2 TranscendentalHodgealgebra: thedefinition . 7 4 Zarhin’sresultsaboutHodgestructuresofK3type 8 4.1 NumberfieldsandHodgestructuresofK3type . 8 4.2 Special Mumford-Tate group for Hodge structures of K3 type .. 9 arXiv:1512.01011v3 [math.AG] 2 Aug 2017 5 Transcendental Hodge algebra for hyperk¨ahler manifolds 10 5.1 Irreducible representations of SO(V )................ 10 5.2 Transcendental Hodge algebra for hyperk¨ahler manifolds . .. 11 1Misha Verbitsky is partially supported by the Russian Academic Excellence Project ’5- 100’. Keywords: hyperk¨ahler manifold, Hodge structure, transcendental Hodge lattice, bira- tional invariance 2010 Mathematics Subject Classification: 53C26, –1– version 3.0, 11.07.2017 M.
    [Show full text]
  • Weak Positivity Via Mixed Hodge Modules
    Weak positivity via mixed Hodge modules Christian Schnell Dedicated to Herb Clemens Abstract. We prove that the lowest nonzero piece in the Hodge filtration of a mixed Hodge module is always weakly positive in the sense of Viehweg. Foreword \Once upon a time, there was a group of seven or eight beginning graduate students who decided they should learn algebraic geometry. They were going to do it the traditional way, by reading Hartshorne's book and solving some of the exercises there; but one of them, who was a bit more experienced than the others, said to his friends: `I heard that a famous professor in algebraic geometry is coming here soon; why don't we ask him for advice.' Well, the famous professor turned out to be a very nice person, and offered to help them with their reading course. In the end, four out of the seven became his graduate students . and they are very grateful for the time that the famous professor spent with them!" 1. Introduction 1.1. Weak positivity. The purpose of this article is to give a short proof of the Hodge-theoretic part of Viehweg's weak positivity theorem. Theorem 1.1 (Viehweg). Let f : X ! Y be an algebraic fiber space, meaning a surjective morphism with connected fibers between two smooth complex projective ν algebraic varieties. Then for any ν 2 N, the sheaf f∗!X=Y is weakly positive. The notion of weak positivity was introduced by Viehweg, as a kind of birational version of being nef. We begin by recalling the { somewhat cumbersome { definition.
    [Show full text]
  • Hodge Polynomials of Singular Hypersurfaces
    HODGE POLYNOMIALS OF SINGULAR HYPERSURFACES ANATOLY LIBGOBER AND LAURENTIU MAXIM Abstract. We express the difference between the Hodge polynomials of the singular and resp. generic member of a pencil of hypersurfaces in a projective manifold by using a stratification of the singular member described in terms of the data of the pencil. In particular, if we assume trivial monodromy along all strata in the singular member of the pencil, our formula computes this difference as a weighted sum of Hodge polynomials of singular strata, the weights depending only on the Hodge-type information in the normal direction to the strata. This extends previous results (cf. [19]) which related the Euler characteristics of the generic and singular members only of generic pencils, and yields explicit formulas for the Hodge χy-polynomials of singular hypersurfaces. 1. Introduction and Statements of Results Let X be an n-dimensional non-singular projective variety and L be a bundle on X. Let L ⊂ P(H0(X; L)) be a line in the projectivization of the space of sections of L, i.e., a pencil of hypersurfaces in X. Assume that the generic element Lt in L is non-singular and that L0 is a singular element of L. The purpose of this note is to relate the Hodge polynomials of the singular and resp. generic member of the pencil, i.e., to understand the difference χy(L0) − χy(Lt) in terms of invariants of the singularities of L0. A special case of this situation was considered by Parusi´nskiand Pragacz ([19]), where the Euler characteristic is studied for pencils satisfying the assumptions that the generic element Lt of the pencil L is transversal to the strata of a Whitney stratification of L0.
    [Show full text]
  • University of Alberta
    University of Alberta FILTRATIONS ON HIGHER CHOW GROUPS AND ARITHMETIC NORMAL FUNCTIONS by Jos´eJaime Hern´andezCastillo A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematical and Statistical Sciences c Jos´eJaime Hern´andezCastillo Spring 2012 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission. Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-78000-8 Our file Notre référence ISBN: 978-0-494-78000-8 NOTICE: AVIS: The author has granted a non- L'auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distrbute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats.
    [Show full text]
  • Hodge Numbers of Landau-Ginzburg Models
    HODGE NUMBERS OF LANDAU–GINZBURG MODELS ANDREW HARDER Abstract. We study the Hodge numbers f p,q of Landau–Ginzburg models as defined by Katzarkov, Kont- sevich, and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau–Ginzburg mirrors of Fano three- folds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold X there is a natural LG mirror (Y,w) so that hp,q(X) = f 3−q,p(Y,w). 1. Introduction The goal of this paper is to study Hodge theoretic invariants associated to the class of Landau–Ginzburg models which appear as the mirrors of Fano varieties in mirror symmetry. Mirror symmetry is a phenomenon that arose in theoretical physics in the late 1980s. It says that to a given Calabi–Yau variety W there should be a dual Calabi–Yau variety W ∨ so that the A-model TQFT on W is equivalent to the B-model TQFT on W ∨ and vice versa. The A- and B-model TQFTs associated to a Calabi–Yau variety are built up from symplectic and algebraic data respectively. Consequently the symplectic geometry of W should be related to the algebraic geometry of W ∨ and vice versa. A number of precise and interrelated mathematical approaches to mirror symmetry have been studied intensely over the last several decades. Notable approaches to studying mirror symmetry include homological mirror symmetry [Kon95], SYZ mirror symmetry [SYZ96], and the more classical enumerative mirror symmetry.
    [Show full text]
  • Introduction to Hodge Theory and K3 Surfaces Contents
    Introduction to Hodge Theory and K3 surfaces Contents I Hodge Theory (Pierre Py) 2 1 Kähler manifold and Hodge decomposition 2 1.1 Introduction . 2 1.2 Hermitian and Kähler metric on Complex Manifolds . 4 1.3 Characterisations of Kähler metrics . 5 1.4 The Hodge decomposition . 5 2 Ricci Curvature and Yau's Theorem 7 3 Hodge Structure 9 II Introduction to Complex Surfaces and K3 Surfaces (Gianluca Pacienza) 10 4 Introduction to Surfaces 10 4.1 Surfaces . 10 4.2 Forms on Surfaces . 10 4.3 Divisors . 11 4.4 The canonical class . 12 4.5 Numerical Invariants . 13 4.6 Intersection Number . 13 4.7 Classical (and useful) results . 13 5 Introduction to K3 surfaces 14 1 Part I Hodge Theory (Pierre Py) Reference: Claire Voisin: Hodge Theory and Complex Algebraic Geometry 1 Kähler manifold and Hodge decomposition 1.1 Introduction Denition 1.1. Let V be a complex vector space of nite dimension, h is a hermitian form on V . If h : V × V ! C such that 1. It is bilinear over R 2. C-linear with respect to the rst argument 3. Anti-C-linear with respect to the second argument i.e., h(λu, v) = λh(u; v) and h(u; λv) = λh(u; v) 4. h(u; v) = h(v; u) 5. h(u; u) > 0 if u 6= 0 Decompose h into real and imaginary parts, h(u; v) = hu; vi − i!(u; v) (where hu; vi is the real part and ! is the imaginary part) Lemma 1.2. h ; i is a scalar product on V , and ! is a simpletic form, i.e., skew-symmetric.
    [Show full text]
  • Hodge Theory and Geometry
    HODGE THEORY AND GEOMETRY PHILLIP GRIFFITHS This expository paper is an expanded version of a talk given at the joint meeting of the Edinburgh and London Mathematical Societies in Edinburgh to celebrate the centenary of the birth of Sir William Hodge. In the talk the emphasis was on the relationship between Hodge theory and geometry, especially the study of algebraic cycles that was of such interest to Hodge. Special attention will be placed on the construction of geometric objects with Hodge-theoretic assumptions. An objective in the talk was to make the following points: • Formal Hodge theory (to be described below) has seen signifi- cant progress and may be said to be harmonious and, with one exception, may be argued to be essentially complete; • The use of Hodge theory in algebro-geometric questions has become pronounced in recent years. Variational methods have proved particularly effective; • The construction of geometric objects, such as algebraic cycles or rational equivalences between cycles, has (to the best of my knowledge) not seen significant progress beyond the Lefschetz (1; 1) theorem first proved over eighty years ago; • Aside from the (generalized) Hodge conjecture, the deepest is- sues in Hodge theory seem to be of an arithmetic-geometric character; here, especially noteworthy are the conjectures of Grothendieck and of Bloch-Beilinson. Moreover, even if one is interested only in the complex geometry of algebraic cycles, in higher codimensions arithmetic aspects necessarily enter. These are reflected geometrically in the infinitesimal structure of the spaces of algebraic cycles, and in the fact that the convergence of formal, iterative constructions seems to involve arithmetic 1 2 PHILLIP GRIFFITHS as well as Hodge-theoretic considerations.
    [Show full text]
  • 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”.
    [Show full text]
  • Mixed Hodge Structures with Modulus Is Abelian
    MIXED HODGE STRUCTURES WITH MODULUS FLORIAN IVORRA AND TAKAO YAMAZAKI Abstract. We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato-Russell’s Albanese varieties with modulus to 1-motives. Contents 1. Introduction 1 2. Mixed Hodge structures with modulus 3 3. Laumon 1-motives 8 4. Cohomology of a variety with modulus 12 5. Duality 21 6. Picard and Albanese 1-motives 24 7. Relation with enriched and formal Hodge structures 26 References 29 1. Introduction 1.1. Background. Unlike K-theory, classical cohomology theories, such as Betti cohomology, étale cohomology or motivic cohomology (in particular Chow groups) are not able to distinguish a smooth variety from its nilpotent thickenings. This inability to detect nilpotence makes those cohomologies not the right tool to study arXiv:1712.07423v2 [math.AG] 12 Jun 2018 non-homotopy invariant phenomena. One very important situation where these kind of phenomena occur, is at the boundary of a smooth variety. More precisely if X is a smooth proper variety and D is an effective Cartier Divisor on X, then D can be seen as the non-reduced boundary at infinity of the smooth variety X := X \ D.
    [Show full text]
  • On the Mixed Hodge Structure Associated to Hypersurface Singularities
    ON THE MIXED HODGE STRUCTURE ASSOCIATED TO HYPERSURFACE SINGULARITIES MOHAMMAD REZA RAHMATI Abstract. Let f : Cn+1 ! C be a germ of hyper-surface with isolated singularity. By the work of Saito M. and P. Deligne the extension of the VMHS associated to cohomologies of the fibers of an isolated singularity in the affine space (Also called vanishing cohomology in this case) after the extension, on the puncture defines another MHS whose Hodge filtration is automatically opposite to the former Hodge filtration (Limit Hodge filtration of Schmid in pure case and Steenbrink in mixed case). Now the polarization deforms to Grothendieck residue pairing modified by a varying sign on the Hodge graded pieces. This also proves the existence of a Riemann-Hodge bilinear relation for Grothendieck pairing and allow to calculate the signature of Grothendieck pairing. The above form polarizes the complex VHS of G. Pearlstein et al. Introduction One of the important subject of study in Hodge theory and D-modules is the behaviour of the underlying variation of (mixed) Hodge structures in the extensions. We will consider the VMHS associated to isolated hypersurface singularities in the affine space Cn+1. The mixed Hodge structure would be the Steenbrink limit mixed Hodge structure. Classically there are two equivalent ways to define this MHS. One method which is actually due to J. Steenbrink himself is by applying a spectral sequence argument to the resolution of the singularity in projective fibration followed with Invariant cycle theorem. Another method which is equivalent to the first is to define it by the structure of lattices in the Gauss-Manin system associated to VMHS on the punctured disc.
    [Show full text]