Hodge Conjecture Research Articles

Hyperkähler manifolds of Jacobian type

Abstract In this paper we define the notion of a hyperkähler manifold (potentially) of Jacobian type. If we view hyperkähler manifolds as “abelian varieties”, then those of Jacobian type should be viewed as “Jacobian varieties”. Under a minor assumption on the polarization, we show that a very general polarized hyperkähler fourfold F of K 3 [ 2 ] ${K3^{[2]}}$ -type is not of Jacobian type. As a potential application, we conjecture that if a cubic fourfold is rational then its variety of lines is of Jacobian type. Under some technical assumption, it is proved that the variety of lines on a rational cubic fourfold is potentially of Jacobian type. We also prove the Hodge conjecture in degree 4 for a generic F of K 3 [ 2 ] ${K3^{[2]}}$ -type.

Beilinson's Hodge Conjecture for Smooth Varieties

AbstractLet U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, andclr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

On algebraic cycles on a fibre product of families of K3-surfaces

We prove the Hodge conjecture and the standard conjecture of Lefschetz type for fibre squares of smooth projective non-isotrivial families of -surfaces over a smooth projective curve under the assumption that the rank of the lattice of transcendental cycles on a generic geometric fibre of the family is an odd prime. We prove the Hodge conjecture for a fibre product of two non-isotrivial families of -surfaces (possibly with degenerations) under the condition that, for every point of the curve, at least one family has non-singular fibre over this point, and the rank of the lattice of transcendental cycles on a generic geometric fibre of one family is odd and not equal to the corresponding rank for the other.

ON THE INTEGRAL HODGE AND TATE CONJECTURES OVER A NUMBER FIELD

AbstractHassett and Tschinkel gave counterexamples to the integral Hodge conjecture among 3-folds over a number field. We work out their method in detail, showing that essentially all known counterexamples to the integral Hodge conjecture over the complex numbers can be made to work over a number field.

Néron–Severi groups under specialization

Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to Andr\'e's theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is nowhere p-adically dense. Our proof uses the ``p-adic Lefschetz (1,1) theorem'' of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties.

Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

Given a smooth projectivenn-foldYY, withH3,0(Y)=0H^{3,0}(Y)=0, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing codimension22-cycles inYYto the intermediate JacobianJ(Y)J(Y), which is an abelian variety. Assumingn=3n=3, we study in this paper the existence of families of11-cycles inYYfor which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. WhenYYitself is rationally connected with trivial Brauer group, we relate this property to the existence of an integral cohomological decomposition of the diagonal ofYY. We also study this property for cubic threefolds, completing the work of Iliev-Markushevich-Tikhomirov. We then conclude that the Hodge conjecture holds for degree44integral Hodge classes on fibrations into cubic threefolds over curves, with some restriction on singular fibers.

Cohomologie non ramifiée et conjecture de Hodge entière

Building upon the Bloch–Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of Voisin implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Colliot-Thélène and Ojanguren implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain classes of threefolds fibered over a curve, we establish a relation between the integral Hodge conjecture and the computation of the index of the generic fiber.

On the coniveau of certain sub-Hodge structures

We study the generalized Hodge conjecture for certain sub-Hodge structure defined as the kernel of the cup product map with a big cohomology class, which is of Hodge coniveau at least 1. As predicted by the generalized Hodge conjecture, we prove that the kernel is supported on a divisor, assuming the Lefschetz standard conjecture.

Tate twists of Hodge structures arising from abelian varieties of type IV

We show that certain abelian varieties A have the property that for every Hodge structure V in the cohomology of A , every effective Tate twist of V occurs in the cohomology of some abelian variety. We deduce the general Hodge conjecture for certain non-simple abelian varieties of type IV.

The Hodge conjecture for self-products of certain K3 surfaces

We use a result of van Geemen (2008) [vG4] to determine the endomorphism algebra of the Kuga–Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of P 2 which are ramified along six lines.

Algebraic Geometry versus Kähler geometry

These notes discuss Hodge theory in the algebraic and Kahler context. We introduce the notion of (polarized) Hodge structure on a cohomology algebra and show how to extract from it topological restrictions on compact Kahler manifolds, and stronger topological restrictions on projective complex manifolds. The second part of the notes is devoted to the discussion of the Hodge conjecture, showing in particular that there is no way to extend it to the Kahler context. We will also discuss algebraic de Rham cohomology which is specific to projective complex manifolds and allows to formulate a number of arithmetic questions related to the Hodge conjecture.

Some remarks on the Hodge conjecture for abelian varieties

We show how the classical Hodge conjecture for the middle cohomology of an abelian variety is equivalent to the general Hodge conjecture for the middle cohomology of a smooth ample divisor in the abelian variety. This is best suited to abelian varieties with actions of imaginary quadratic fields.

Duality of positive currents and plurisubharmonic functions in calibrated geometry

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold $(X,\phi)$. In particular, on $X$ there exist $\phi$-plurisubharmonic functions, $\phi$-convex domains, $\phi$-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the $\phi$-submanifolds, or more generally, the $\phi$-currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the $\phi$-convex hull of a compact set $K\subset X$ in terms of $\phi$-positive Green's currents on $X$ and Jensen measures on $K$. We also characterize boundaries of $\phi$-currents entirely in terms of $\phi$-plurisubharmonic functions. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.

Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results

This paper is a survey of the subject of variations of Hodge structure (VHS) considered as exterior differential systems (EDS). We review developments over the last twenty-six years, with an emphasis on some key examples. In the penultimate section we present some new results on the characteristic cohomology of a homogeneous Pfaffian system. In the last section we discuss how the integrability conditions of an EDS affect the expected dimension of an integral submanifold. The paper ends with some speculation on EDS and Hodge conjecture for Calabi-Yau manifolds.

Motivic integration and projective bundle theorem in morphic cohomology

AbstractWe reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two K-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are K-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties.

Singularities of admissible normal functions

In a recent paper, M. Green and P. Griffiths used R. Thomas’ work on nodal hypersurfaces to sketch a proof of the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using Morihiko Saito’s mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest.

Rational Tate Classes

In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of “rational Tate classes” on varieties over finite fields having the properties that the algebraic classes would have if the Hodge and Tate conjectures were known. In particular, I prove that there exists at most one “good” such theory. v1 July 20, 2007. First version on the web. v2 November 7, 2007. Completely rewritten; shortened the title. v3 April 29, 2008. Submitted version.

Hodge-Type Conjecture for Higher Chow Groups

Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injective and the Hodge conjecture is true for certain smooth projective varieties over the algebraic closure of the rational number field. We also verify the conjecture on the surjectivity in some cases of the complement of a union of general hypersurfaces in a smooth projective variety.

Singularities of the secant variety

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties.

Slice filtration on motives and the Hodge conjecture (with an appendix by J. Ayoub)

We clarify the expected properties of the slice filtration on triangulated motives from the point of view of the generalized Hodge conjecture. In the appendix, J. Ayoub proves unconditionally that the slice filtration does not respect geometric motives. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)