Generalized Hodge Conjecture Research Articles

On the Motive of Codimension 2 Linear Sections of Gr(3, 6)

We consider Fano sevenfolds Y obtained by intersecting the Grassmannian Gr(3, 6) with a codimension 2 linear subspace (with respect to the Plücker embedding). We prove that the motive of Y is Kimura finite-dimensional. We also prove the generalized Hodge conjecture for all powers of Y.

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup constructed here is in fact the whole monodromy group. As an application we prove the Hodge conjecture for the generic abelian fourfold of Weil type with complex multiplication by an arbitrary imaginary quadratic number field K, but only for polarizations with discriminant 1. The latter result is inspired by a recent observation of O'Grady that the third intermediate Jacobians of smooth projective varieties of generalized Kummer deformation type form complete families of abelian fourfolds of Weil type. Finally, we prove the surjectivity of the Abel-Jacobi map from the Chow group of co-dimension two algebraic cycles homologous to zero on every projective irreducible holomorphic symplectic manifold Y of Kummer type onto the third intermediate Jacobian of Y, as predicted by the generalized Hodge Conjecture.

Generic cycles, Lefschetz representations, and the generalized Hodge and Bloch conjectures for Abelian varieties

We prove Bloch's conjecture for correspondences on powers of complex abelian varieties, that are "generically defined". As an application we establish vanishing results for (skew-)symmetric cycles on powers of abelian varieties and we address a question of Voisin concerning (skew-)symmetric cycles on powers of K3 surfaces in the case of Kummer surfaces. We also prove Bloch's conjecture in the following situation. Let $\gamma$ be a correspondence between two abelian varieties $A$ and $B$ that can be written as a linear combination of products of symmetric divisors. Assume that $A$ is isogenous to the product of an abelian variety of totally real type with the power of an abelian surface. We show that $\gamma$ satisfies the conclusion of Bloch's conjecture. A key ingredient consists in establishing a strong form of the generalized Hodge conjecture for Hodge sub-structures of the cohomology of $A$ that arise as sub-representations of the Lefschetz group of $A$. As a by-product of our method, we use a strong form of the generalized Hodge conjecture established for powers of abelian surfaces to show that every finite-order symplectic automorphism of a generalized Kummer variety acts as the identity on the zero-cycles.

Convexity of complements of tropical varieties, and approximations of currents

The goal of this note is to affirm a local version of the conjecture of Nisse--Sottile [NS16] on higher convexity of complements of tropical varieties, while providing a family of counter-examples for the global Nisse--Sottle conjecture in any codimension and dimension higher than 1. Moreover, it will be shown that, surprisingly, this family also provides a family of counter-examples for the generalized Hodge conjecture for positive currents in these dimensions, and gives rise to further approximability obstruction.

A tropical approach to a generalized Hodge conjecture for positive currents

Demailly showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents associated to subvarieties, and asked whether any strongly positive (p,p)-dimensional closed current with rational cohomology class can be approximated by positive linear combinations of integration currents associated to subvarieties. Using tropical geometry, we construct a (p,p)-dimensional current on a smooth projective variety that does not satisfy the latter statement.

Torsion 1-cycles and the coniveau spectral sequence

We relate the torsion part of the Abel-Jacobi kernel in the Griffiths group of 1-cycles to a birational invariant analogous to the degree 4 unramified cohomology and an invariant associated to the generalized Hodge conjecture in degree 2{\dim}(X)-3 . We also describe in terms of \mathcal H -cohomology the Griffiths group of 1-cycles and the group of torsion cycles algebraically equivalent to zero of arbitrary dimension.

Refined Motivic Dimension

AbstractWe define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.

Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes

AbstractSuppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D and that U is the complement of the ramification locus in Y. The first theorem in this paper implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance, this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to more general nonabelian covers.

A remark on the generalized Hodge conjecture

It is well known that for a smooth, projective variety \(X\) over \({\mathbb {C}}\) we have \(N^{p}H^{i}(X,{\mathbb {Q}})\subset F^{p} H^{i}(X,{\mathbb {C}})\cap H^{i}(X,{\mathbb {Q}})\), where \(N^{\bullet }\) and \(F^{\bullet }\) are respectively the coniveau and Hodge filtrations. In general this inclusion is strict. We introduce a natural subspace \(S^{p,i}\subset F^{p}H^{i}(X,{\mathbb {C}})\) such that \(N^{p}H^{i}(X,{\mathbb {Q}})= S^{p,i}\cap H^{i}(X,{\mathbb {Q}})\) holds true for any \(i,p\). The main technical tool is the use of semi-algebraic sets, which are available by the triangulation of complex projective varieties.

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.

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.

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)

On 𝑝-adic intermediate Jacobians

For an algebraic variety X X of dimension d d with totally degenerate reduction over a p p -adic field (definition recalled below) and an integer i i with 1 ≤ i ≤ d 1\leq i\leq d , we define a rigid analytic torus J i ( X ) J^i(X) together with an Abel-Jacobi mapping to it from the Chow group C H i ( X ) h o m CH^i(X)_{hom} of codimension i i algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over C \bf {C} . We compare and contrast the complex and p p -adic theories. Finally, we examine a special case of a p p -adic analogue of the Generalized Hodge Conjecture.

Coniveau and the Grothendieck group of varieties

It is shown that if the generalized Hodge conjecture, or some weaker form of it, holds for a Calabi-Yau variety then it holds for any Calabi-Yau variety birationally equivalent to it. The key idea is to construct suitable homomorphisms between the Grothendieck group of varieties and the Grothendieck group of the exact category of filtered Hodge structures. This along with motivic integration yields the first statement.

Filtrations on Chow Groups and Transcendence Degree

For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending on the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of transcendence degree one. This may be viewed as a refinement of results of Nori, Schoen, and Green–Griffiths–Paranjape. For higher graded pieces we have a similar assertion assuming a conjecture of Beilinson and Grothendieck’s generalized Hodge conjecture.

On the General Hodge Conjecture for Abelian Varieties of CM-type

The General Hodge Conjecture for abelian varieties of CM-type is shown to be implied by the usual Hodge Conjecture for those up to codimension two.

Hodge-theoretic invariants for algebraic cycles

We use Hodge theory to define a filtration on the Chow groups of a smooth, projective algebraic variety. Assuming the generalized Hodge conjecture and a conjecture of Bloch-Beilinson, we show that this filtration terminates at the codimension of the algebraic cycle class, thus providing a complete set of period-type invariants for a rational equivalence class of algebraic cycles.

General Hodge conjecture for abelian varieties of CM-type

The general Hodge conjecture for abelian varieties of CM-type is shown to be implied by the usual Hodge conjecture for those up to codimension two.

Hodge Structures on Abelian Varieties of Type III

We show that the usual Hodge conjecture implies the general Hodge conjecture for certain abelian varieties of type III, and use this to deduce the general Hodge conjecture for all powers of certain 4-dimensional abelian varieties of type III. We also show the existence of a Hodge structure M such that M occurs in the cohomology of an abelian variety, but the Tate twist M(1) does not occur in the cohomology of any abelian variety, even though it is effective.