Algebraic Cycles and Mumford-Griffiths Invariants

TL;DR

This paper constructs arithmetic Hodge theoretic invariants related to algebraic cycles on projective manifolds, exploring their properties within a conjectural filtration framework and identifying conditions for large kernel and image.

Contribution

It introduces a new space of invariants and a map linked to the Bloch-Beilinson filtration on Chow groups, advancing the understanding of algebraic cycles and their Hodge-theoretic properties.

Findings

Defined a space of arithmetic Hodge invariants $ abla J^{r, u}(X)$.

Constructed a map $_{X}^{r, u}$ relating Chow groups to these invariants.

Identified conditions under which the kernel and image of the map are uncountably large.

Abstract

Let $X$ be a projective algebraic manifold and let $CH^r(X)$ be the Chow group of algebraic cycles of codimension $r$ on $X$, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration $\{F^{\nu}\}_{\nu\geq 0}$ on $CH^r(X)\otimes {\Bbb Q}$ due to the second author, we construct a space of arithmetic Hodge theoretic invariants $\nabla J^{r,\nu}(X)$ and corresponding map $\phi_{X}^{r,\nu} : Gr_{F}^{\nu}CH^r(X)\otimes {\Bbb Q} \to \nabla J^{r,\nu}(X)$, and determine conditions on $X$ for which the kernel and image of $\phi_{X}^{r,\nu}$ are ``uncountably large''.