2 problems

Let $f\colon X\to Y$ be a finite morphism between smooth irreducible projective curves, and let $V$ be a general vector bundle on $X$. Beauville's conjecture. The pushforward…

Pseudo-effective pushforward conjecture. Any pseudo-effective class $\alpha\in H_{2k}(X,\mathbf{R})$ such that $\varphi_*\alpha=0$ in $H_{2k}(Y,\mathbf{R})$ belongs to the real vec…