Let $f:X\rightarrow Y$ be a finite morphism between normal varieties. Let $E$ be a vector bundle on $X$ and let us consider its pushforwad $f_{*}E$.
Does anyone know an example where $E$ is nef but $f_{*}E$ is not nef ?
Take $f:X\rightarrow Y$ a double covering of smooth varieties, branched along an ample divisor $D$. Then there is a line bundle $L$ such that $D$ is the zero divisor of a section of $L^2$, and $f_*\mathcal{O}_X=\mathcal{O}_X\oplus L^{-1}$, which is certainly not nef.
Let $f:X\rightarrow Y$ be a finite separable morphism of degree two between two smooth curves. Let $R\subset X$ be the ramification divisor and $B = f_{*}R\subset Y$ the branch divisor. Then $$(det f_{*}\mathcal{O}_{X})^{2}\cong\mathcal{O}_{Y}(-B)$$ and $\mathcal{O}_{Y}(-B)$ is not nef.