I. Hodge theoretic approach to algebraic cycles

The aim is to generalize Abel's theorem to higher dimension. More precisely we search for Hodge theoretic invariants which capture the Chow group (namely the group of algebraic cycles modulo rational equivalence) of a complex algebraic variety. The first important step has been made by Griffiths. He defined so-called Abel-Jacobi map from Chow groups to intermediate Jacobians which are complex tori determined by the Hodge structure of the variety. As Mumford discovered, Griffiths Abel-Jacobi map have large kernel in general.

In [09b] Hodge theoretic invariants for algebraic cycles are introduced following the philosophy of mixed motives by Bloch and Beilinson. It is shown that these invariants are able to capture algebraic cycles lying in the kernel of Griffiths Abel-Jacobi map. Working with a candidate Bloch-Beilinson filtration on Chow groups studied in [96a], we construct a space of arithmetic Hodge theoretic invariants and corresponding map from the graded quotients of the filtrations on Chow groups to the invariant spaces, and determine conditions on $X$ for which the kernel and image of the map are ``uncountably large''.

II. Beilinson-Hodge conjecture

The infinitesimal method in Hodge theory is fruitful in various aspects of algebraic geometry. The idea originates from Griffiths work where the Poincar'e residue representation of cohomology of a hypersurface played a crucial role in proving the infinitesimal Torelli theorem for hypersurfaces. Since then many important applications of the idea have been made to geometric problems such as the Noether-Lefschetz theorem for Hodge cycles and study of algebraic cycles.

In this part of research we apply the method to study an analog of the Noether-Lefschetz theorem in the context of Beilinson's Hodge conjecture. The conjecture is an analog of the Hodge conjecture and predict that certain cohomology class (called Beilinson-Hodge cycles) of non-compact complex algebraic manifolds come from Bloch's higher cycles via regulator maps. It has been known to holds only for the one-dimensional case, which is in fact equivalent to Abel's theorem for Riemann surfaces.

In [06a] we give an estimate of the codimension of the Noether-Lefschetz locus in the moduli space of open complete intersections. It implies that Beilinson's Hodge conjecture holds for general open complete intersections.

In [08a] we take a closer look at the Noether-Lefschetz locus for Beilinson-Hodge cycles and give an explicit description of its irreducible component of maximal dimension in a certain case and discover a surprising phenomina: there are infinitely many components of maximal dimension. (see [L1] for a summary on these works).

In [07b] we study a variant of Beilinson's Hodge conjecture in the following setting: Given a family of open complete intersections over a base S, one can define by aid of theory of mixed Hodge modules, the space of Beilinson-Hodge cycles in the cohomology group of S with coefficient in the local system arising from the family.

III. Motivic cohomology of arithmetic schemes

Motivic cohomology of schemes is an important object to study in arithmetic geometry. For regular schemes it agrees with Bloch's higher Chow groups which generalizes Chow groups. Such objects of arithmetic importance as the ideal class group and the unit group of the ring $O_K$ of integers in a number field $K$, are also motivic cohomology of $Spec(O_K)$. Conjecturally it is closely related to $L$-functions of arithmetic schemes (namely schemes of finite type over a finite field or the ring of rational integers). An important open problem is the conjecture that motivic cohomology of arithmetic schemes should be finitely generated. The conjecture generalizes the known finiteness results on the ideal class group and the unit group of $O_K$ and the Mordel-Weil theorem on the group of rational points on an abelian variety over a number field. There have been only few results on the conjecture except these cases.

In [P2] the problem is related to a conjecture of Kato on the acyclicity of a certain complexes of Bloch-Ogus type associated to an arithmetic scheme $X$. In case $X=Spec(O_K)$ as above, the Kato conjecture rephrases a fundamental fact in number theory concerning the Brauer group of $K$, the Hasse principle for central simple algebras over $K$.

We are now able to prove the Kato conjecture for a smooth projective variety over a finite field under the assumption of a certain strong form of resolution of singularities. Combined with [P4] where the assumption is proved to hold for surfaces, the result gives rise to a new finiteness result for motivic cohomology (see [L2] for a summary of this).

It is not difficult to extend the method of [P2] to study the Kato conjecture and motivic cohomology of arithmetic schemes over the ring of integers in a local field, at least restricted to the prime-to-$p$ part, where $p$ is the residue characteristic of the local field. In order to deal with the $p$-part and the case of arithmetic schemes over the ring of integers in a number field, one need develop a new input from $p$-adic Hodge theory. This is a work in progress.

IV. Cycle class map for arithmetic schemes

An 'etale cycle class map relates Chow groups or higher Chow groups with finite coefficients of a scheme to its 'etale cohomology group and is an important object to study in arithmetic geometry. If one had its bijectivity or injectivity, it would bring about various important information (for example finiteness) on the structure of the Chow groups or higher Chow groups, which is usually very difficult to compute. So far, few results on such injectivity had been known. Far from that, there have been examples of varieties over a number field or a local field whose cycle maps are not injective.

In [P5] a new viewpoint is introduced on the injectivity problem by investigating cycle maps for an arithmetic scheme, more precisely for a regular proper flat scheme $X$ over the ring $O_K$ of integers in a number field $K$. A main result relates the injectivity of torsion cycle maps for cycles of codimension two with a conjecture of Bloch-Kato which characterizes the image of the $p$-adic regulator maps from higher Chow groups to continuous \'etale cohomology of $X_K$, the generic fiber of $X$, by using $p$-adic Hodge theory. We have obtained its injectivity when the geometric genus of $X_K$ is zero.

In [09a] a result is obtained concerning the cycle map for $CH_1(X)$, the Chow group of 1-cycles on a regular proper flat scheme $X$ over the ring $O_K$ of integers in a local field $K$. We prove the bijectivity of the prime-to-$p$ part of the cycle map, where $p$ is the characteristic of the residue field of $O_K$. As an application we get a new finiteness result of $CH_1(X)$ modulo $n$ for an integer $n$ prime-to $p$.

In [07a] we disprove a variant over a local field $K$ of the Bloch-Kato conjecture on the image of the $p$-adic regulator maps, mentioned in [P5]. Combined with [09a], it gives rise to the first example of a surface over $K$ whose Chow group of zero-cycles contains an infinite torsion subgroup. Key techniques in the proof are imported from Hodge theory; the theory of mixed Hodge modules and the infinitesimal method in Hodge theory.

See [L3-1] [L3-2] for a summary on the works [09a] and [07a].