Kernels of numerical pushforwards

TL;DR

This paper investigates the kernel of the pushforward map on numerical cycle classes induced by a morphism of projective varieties, showing it is spanned by contracted cycles under certain fiber conditions.

Contribution

It establishes a criterion for the kernel of the pushforward to be generated by contracted cycles when fiber Chow groups are minimal.

Findings

Kernel of pushforward spanned by contracted cycles

Condition on fiber Chow groups for kernel description

Provides a geometric criterion for kernel structure

Abstract

Let $\pi: X \to Y$ be a morphism of projective varieties and consider the pushforward map $\pi_*: N_k(X) \to N_k(Y)$ of numerical cycle classes. We show that when the Chow groups of points of the fibers are as simple as they can be, then the kernel of $\pi_*$ is spanned by k-cycles contracted by $\pi$.