Coniveau 2 complete intersections and effective cones

TL;DR

This paper proposes a strategy to approach the generalized Hodge conjecture for coniveau 2 complete intersections and introduces a conjecture on effective cones of cycle classes, linking the two problems.

Contribution

It introduces a new strategy for the generalized Hodge conjecture and formulates a conjecture on effective cones, connecting these two areas of algebraic geometry.

Findings

Generalized Hodge conjecture for coniveau 2 would follow from a specific case of the effectiveness conjecture

Proposes a new approach to attacking the generalized Hodge conjecture

States a conjecture on the cones of effective cycle classes in intermediate dimensions

Abstract

The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness conjecture.