Let $X, Y$ be smooth projective varieties and $f:X \times Y \to Y$ be the natural projetion map. Let $\mathcal{F}$ be a locally free sheaf on $X \times Y$. Is it true that $f_*\mathcal{F}$ is locally free on $Y$? If not true in general is there any additional condition on $X, Y$ under which this will hold true?
1 Answer
Let $X = P^1$, $Y = P^3$. We will take $F$ to be an extension $$ 0 \to O(2,1) \to F \to O \oplus O \oplus O \to 0. $$ Of course $F$ is locally free. Note that $$ Ext^1(O,O(2,1)) = H^1(P^1\times P^3,O(2,1)) = H^1(P^1,O(2))\otimes H^0(P^3,O(1)) $$ is a 4dimensional vector space, so we can take $F$ to be the extension corresponding to a 3dimensional subspace in it. Pushing forward to $P^3$ then gives an exact sequence $$ 0 \to f_*F \to O \oplus O \oplus O \to O(1) \to R^1f_*F \to 0 $$ and the middle map corresponds to out choice of 3 linear functions on $P^3$, so $R^1f_*F$ is the structure sheaf of a point and $f_*F$ is the simplest example of a reflexive sheaf which is not locally free.

3$\begingroup$ See Hartshorne, Chapter III, Theorem 12.11 or EGA III, 7.7 for a set of criteria guaranteeing that $f_*F$ is locally free. $\endgroup$ Nov 25, 2014 at 22:53