Cycle Map for Strictly Decomposable Cycles

TL;DR

This paper introduces a new class of cycles called nondegenerate, strictly decomposable cycles, and demonstrates their non-vanishing under a refined cycle map, revealing new insights into higher cycles and their indecomposability.

Contribution

It defines nondegenerate, strictly decomposable cycles and proves their non-vanishing under a refined cycle map, extending Nori's argument and exploring higher cycle indecomposability.

Findings

Nondegenerate, strictly decomposable cycles have non-vanishing images under the refined cycle map.

Existence of uncountably many indecomposable higher cycles on products with varieties having a nonzero global 1-form.

Higher cycles not annihilated by the reduced higher Abel-Jacobi map are linked to indecomposable cycles.

Abstract

We introduce a class of cycles, called nondegenerate, strictly decomposable cycles, and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed Hodge structures does not vanish. This class contains certain cycles in the kernel of the Abel-Jacobi map. The construction gives a refinement of Nori's argument in the case of a self-product of a curve. As an application, we show that a higher cycle which is not annihilated by the reduced higher Abel-Jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1-form.