Pseudo-effective classes and pushforwards

TL;DR

This paper explores conjectures relating pseudo-effective classes and their pushforwards under morphisms of complex projective varieties, proving some cases and linking to Grothendieck's Hodge conjecture.

Contribution

It formulates conjectures on pseudo-effective classes and proves them for curves and divisors, also connecting to a major conjecture in Hodge theory.

Findings

Conjectures are proven for classes of curves and divisors.

One conjecture implies Grothendieck's generalized Hodge conjecture for certain varieties.

Establishes relations between pseudo-effective classes and contracted varieties.

Abstract

Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted by the morphism. We prove these conjectures for classes of curves or divisors. We also prove that one of these conjectures implies Grothendieck's generalized Hodge conjecture for varieties with Hodge coniveau at least 1.