Hodge Conjecture For Varieties Research Articles

Variational Hodge conjecture for complete intersections on hypersurfaces in projective space

In this paper, we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.

Periods of complete intersection algebraic cycles

For every even number $n$, and every $n$-dimensional smooth hypersurface of $\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its $\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis and the polarization. As an application, we use this information to address variational Hodge conjecture for a non complete intersection algebraic cycle. We prove that the locus of general hypersurfaces containing two linear cycles whose intersection is of dimension less than $\frac{n}{2}-\frac{d}{d-2}$, corresponds to the Hodge locus of any integral combination of such linear cycles.

A Variational Tate Conjecture in crystalline cohomology

Given a smooth, proper family of varieties in characteristic p > 0 , and a cycle z on a fibre of the family, we consider a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of z , whether z extends cohomologically to the entire family. This is a characteristic p analogue of Grothendieck’s Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension; the former result can be used to reduce the \ell -adic Tate conjecture for divisors over finite fields to the case of surfaces.

$p$-Adic Deformation of Motivic Chow Groups

For a smooth projective scheme Y over W(k) we consider an element in the motivic Chow group of the reduction Y_{m} over the truncated Witt ring W_{m}(k) and give a “Hodge” criterion – using the crystalline cycle class in relative crystalline cohomology – for the element to lift to the continuous Chow group of the associated p -adic formal scheme Y_\bullet . The result extends previous work of Bloch–Esnault–Kerz on the p -adic variational Hodge conjecture to a relative setting. In the course of the proof we derive two new results on the relative de Rham–Witt complex and its Nygaard filtration, and work with a relative version of syntomic complexes to define relative motivic complexes for a smooth lifting of Y_{m} over the ind-scheme \mathrm{Spec}\,W_{\bullet}(W_{m}(k)) .

Semi-regular varieties and variational Hodge conjecture

Following [1,2], we know that semi-regular sub-varieties satisfy the variational Hodge conjecture, i.e., given a family of smooth projective varieties π:X→B, a special fiber Xo and a semi-regular subvariety Z⊂Xo, the cohomology class corresponding to Z remains a Hodge class (as Xo deforms along B) if and only if Z remains an algebraic cycle. In this article, we investigate examples of such sub-varieties. In particular, we prove that any smooth projective variety Z of dimension n is a semi-regular sub-variety of a smooth projective hypersurface in P2n+1 of large enough degree.

Deformation of algebraic cycle classes in characteristic zero

We study a formal deformation problem for rational algebraic cycle classes motivated by Grothendieck’s variational Hodge conjecture. We argue that there is a close connection between the existence of a Chow‐Kunneth decomposition and the existence of expected deformations of cycles. This observation applies in particular to abelian schemes.

A case of the deformational Hodge conjecture via a pro Hochschild–Kostant–Rosenberg theorem

Following ideas of Bloch, Esnault, and Kerz, we establish the deformational part of Grothendieck's variational Hodge conjecture for proper, smooth schemes over K〚t〛, where K is an algebraic extension of Q. The main tool is a pro Hochschild–Kostant–Rosenberg theorem for Hochschild homology.

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.

A Semiregularity Map for Modules and Applications to Deformations

We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel–Jacobi map for submanifolds.

Periods of linear algebraic cycles

In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the homology class of such algebraic cycles. This is known as the variational Hodge conjecture.