Kernels of numerical pushforwards

Let $π: X \to Y$ be a morphism of projective varieties and consider the pushforward map $π_*: N_k(X) \to N_k(Y)$ 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 $π_*$ is spanned by k-cycles contracted by $π$.

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