Push-forward of locally free sheaves

Question

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?

Answer 1

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 4-dimensional vector space, so we can take $F$ to be the extension corresponding to a 3-dimensional 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.