Hodge Theory and Geometry
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Algebraic Cycles from a Computational Point of View
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 392 (2008) 128–140 www.elsevier.com/locate/tcs Algebraic cycles from a computational point of view Carlos Simpson CNRS, Laboratoire J. A. Dieudonne´ UMR 6621, Universite´ de Nice-Sophia Antipolis, 06108 Nice, France Abstract The Hodge conjecture implies decidability of the question whether a given topological cycle on a smooth projective variety over the field of algebraic complex numbers can be represented by an algebraic cycle. We discuss some details concerning this observation, and then propose that it suggests going on to actually implement an algorithmic search for algebraic representatives of classes which are known to be Hodge classes. c 2007 Elsevier B.V. All rights reserved. Keywords: Algebraic cycle; Hodge cycle; Decidability; Computation 1. Introduction Consider the question of representability of a topological cycle by an algebraic cycle on a smooth complex projective variety. In this paper we point out that the Hodge conjecture would imply that this question is decidable for varieties defined over Q. This is intuitively well-known to specialists in algebraic cycles. It is interesting in the context of the present discussion, because on the one hand it shows a relationship between an important question in algebraic geometry and the logic of computation, and on the other hand it leads to the idea of implementing a computer search for algebraic cycles as we discuss in the last section. This kind of consideration is classical in many areas of geometry. For example, the question of deciding whether a curve defined over Z has any Z-valued points was Hilbert’s tenth problem, shown to be undecidable in [10,51, 39]. -
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. -
The Integral Hodge Conjecture for Two-Dimensional Calabi-Yau
THE INTEGRAL HODGE CONJECTURE FOR TWO-DIMENSIONAL CALABI–YAU CATEGORIES ALEXANDER PERRY Abstract. We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves. 1. Introduction Let X be a smooth projective complex variety. The Hodge conjecture in degree n for X states that the subspace of Hodge classes in H2n(X, Q) is generated over Q by the classes of algebraic cycles of codimension n on X. This conjecture holds for n = 0 and n = dim(X) for trivial reasons, for n = 1 by the Lefschetz (1, 1) theorem, and for n = dim(X) − 1 by the case n = 1 and the hard Lefschetz theorem. In all other degrees, the conjecture is far from being known in general, and is one of the deepest open problems in algebraic geometry. There is an integral refinement of the conjecture, which is in fact the version originally proposed by Hodge [Hod52]. Let Hdgn(X, Z) ⊂ H2n(X, Z) denote the subgroup of inte- gral Hodge classes, consisting of cohomology classes whose image in H2n(X, C) is of type (n,n) for the Hodge decomposition. -
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. -
Introduction to Hodge Theory
INTRODUCTION TO HODGE THEORY DANIEL MATEI SNSB 2008 Abstract. This course will present the basics of Hodge theory aiming to familiarize students with an important technique in complex and algebraic geometry. We start by reviewing complex manifolds, Kahler manifolds and the de Rham theorems. We then introduce Laplacians and establish the connection between harmonic forms and cohomology. The main theorems are then detailed: the Hodge decomposition and the Lefschetz decomposition. The Hodge index theorem, Hodge structures and polariza- tions are discussed. The non-compact case is also considered. Finally, time permitted, rudiments of the theory of variations of Hodge structures are given. Date: February 20, 2008. Key words and phrases. Riemann manifold, complex manifold, deRham cohomology, harmonic form, Kahler manifold, Hodge decomposition. 1 2 DANIEL MATEI SNSB 2008 1. Introduction The goal of these lectures is to explain the existence of special structures on the coho- mology of Kahler manifolds, namely, the Hodge decomposition and the Lefschetz decom- position, and to discuss their basic properties and consequences. A Kahler manifold is a complex manifold equipped with a Hermitian metric whose imaginary part, which is a 2-form of type (1,1) relative to the complex structure, is closed. This 2-form is called the Kahler form of the Kahler metric. Smooth projective complex manifolds are special cases of compact Kahler manifolds. As complex projective space (equipped, for example, with the Fubini-Study metric) is a Kahler manifold, the complex submanifolds of projective space equipped with the induced metric are also Kahler. We can indicate precisely which members of the set of Kahler manifolds are complex projective, thanks to Kodaira’s theorem: Theorem 1.1. -
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, . -
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. -
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. -
Hodge Theory in Combinatorics
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 55, Number 1, January 2018, Pages 57–80 http://dx.doi.org/10.1090/bull/1599 Article electronically published on September 11, 2017 HODGE THEORY IN COMBINATORICS MATTHEW BAKER Abstract. If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a polynomial in n, called the chromatic polynomial of G.Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read’s conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh’s result was subsequently refined and generalized by Huh and Katz (also in 2012), again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of the Huh and Huh–Katz papers left open the more general Rota–Welsh conjecture, where graphs are generalized to (not neces- sarily representable) matroids, and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh– Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. -
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. -
Completions of Period Mappings: Progress Report
COMPLETIONS OF PERIOD MAPPINGS: PROGRESS REPORT MARK GREEN, PHILLIP GRIFFITHS, AND COLLEEN ROBLES Abstract. We give an informal, expository account of a project to construct completions of period maps. 1. Introduction The purpose of this this paper is to give an expository overview, with examples to illus- trate some of the main points, of recent work [GGR21b] to construct \maximal" completions of period mappings. This work is part of an ongoing project, including [GGLR20, GGR21a], to study the global properties of period mappings at infinity. 1.1. Completions of period mappings. We consider triples (B; Z; Φ) consisting of a smooth projective variety B and a reduced normal crossing divisor Z whose complement B = BnZ has a variation of (pure) polarized Hodge structure p ~ F ⊂ V B ×π (B) V (1.1a) 1 B inducing a period map (1.1b) Φ : B ! ΓnD: arXiv:2106.04691v1 [math.AG] 8 Jun 2021 Here D is a period domain parameterizing pure, weight n, Q{polarized Hodge structures on the vector space V , and π1(B) Γ ⊂ Aut(V; Q) is the monodromy representation. Without loss of generality, Φ : B ! ΓnD is proper [Gri70a]. Let } = Φ(B) Date: June 10, 2021. 2010 Mathematics Subject Classification. 14D07, 32G20, 32S35, 58A14. Key words and phrases. period map, variation of (mixed) Hodge structure. Robles is partially supported by NSF DMS 1611939, 1906352. 1 2 GREEN, GRIFFITHS, AND ROBLES denote the image. The goal is to construct both a projective completion } of } and a surjective extension Φe : B ! } of the period map. We propose two such completions T B Φ }T (1.2) ΦS }S : The completion ΦT : B ! }T is maximal, in the sense that it encodes all the Hodge- theoretic information associated with the triple (B; Z; Φ). -
Hodge Theory
HODGE THEORY PETER S. PARK Abstract. This exposition of Hodge theory is a slightly retooled version of the author's Harvard minor thesis, advised by Professor Joe Harris. Contents 1. Introduction 1 2. Hodge Theory of Compact Oriented Riemannian Manifolds 2 2.1. Hodge star operator 2 2.2. The main theorem 3 2.3. Sobolev spaces 5 2.4. Elliptic theory 11 2.5. Proof of the main theorem 14 3. Hodge Theory of Compact K¨ahlerManifolds 17 3.1. Differential operators on complex manifolds 17 3.2. Differential operators on K¨ahlermanifolds 20 3.3. Bott{Chern cohomology and the @@-Lemma 25 3.4. Lefschetz decomposition and the Hodge index theorem 26 Acknowledgments 30 References 30 1. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory. In Section 2, we first describe a key motivation behind the Hodge theory for compact, closed, oriented Riemannian manifolds: the observation that the differential forms that satisfy certain par- tial differential equations depending on the choice of Riemannian metric (forms in the kernel of the associated Laplacian operator, or harmonic forms) turn out to be precisely the norm-minimizing representatives of the de Rham cohomology classes. This naturally leads to the statement our first main theorem, the Hodge decomposition|for a given compact, closed, oriented Riemannian manifold|of the space of smooth k-forms into the image of the Laplacian and its kernel, the sub- space of harmonic forms. We then develop the analytic machinery|specifically, Sobolev spaces and the theory of elliptic differential operators|that we use to prove the aforementioned decom- position, which immediately yields as a corollary the phenomenon of Poincar´eduality.