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 ?** 



Push-forward of a nef bundle
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.


