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EDIT. Karl Schwede has given a different approach to solve this question, which is very clear. However, I still want to figure out how to deal with this problem along Ein's line, and waiting for the ...

Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers $p$ and $q$ such ...