On the coniveau of certain sub-Hodge structures

TL;DR

This paper investigates the generalized Hodge conjecture for specific sub-Hodge structures, demonstrating that under the Lefschetz standard conjecture, the kernel is supported on a divisor, advancing understanding of Hodge coniveau.

Contribution

It proves that the kernel of certain cup product maps with big classes is supported on a divisor, assuming the Lefschetz standard conjecture, linking Hodge coniveau to geometric support.

Findings

Kernel is supported on a divisor under the conjecture

Supports the generalized Hodge conjecture in specific cases

Links Hodge coniveau to geometric support

Abstract

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.