2 problems

Smoothness criterion. The morphism $f$ is smooth if and only if there is a holomorphic $1$-form $\omega\in H^0(A,\Omega_A^1)$ such that $f^*\omega$ has no zero.

Direct-image conjecture for strict nefness. If $K_X^{-1}$ is strictly nef, then $K_Y^{-1}$ is strictly nef.