descent for birational morphisms

Question

Let $X$ and $Y$ be two smooth projective surfaces and $f:X\rightarrow Y$ a birational morphism. Let $E$ be the exceptional curve on $X$ that is contracted. Now EGA III Theorem 4.1.5 states that the functor $\mathcal{F}\mapsto f^*\mathcal{F}$ is an equivalence between categories of locally free $\mathcal{O}_Y$-modules of rank $r$ and locally free $\mathcal{O}_X$-modules $\mathcal{E}$ of the same rank such that the restriction of $\mathcal{E}$ to the formal completition $\mathcal{X}=X_{/E}$ is trival. Is it possible to generalize this Theorem for arbitrary smooth projective schemes over a field of characteristic zero with birational proper morphism between them?

Answer 1

If $f:X \to Y$ is a blowup with smooth center $Z$ and exceptional divisor $E$ then the subcategory $Lf^*(D(Y)) \subset D(X)$ of the derived category of coherent sheaves on $X$ identifies with the subcategory of objects such that their (derived) restriction to $E$ is trivial on fibers of $E$ over $Z$ (in the derived sense). For vector bundles you can forget about derived stuff --- a vector bundle on $X$ is a pullback from $Y$ if and only if it restricts trivially onto any fiber of $E$ over $Z$. Note that you don't need to consider the formal completion.

Further, if the morphism $f:X \to Y$ is a composition of blowups with smooth centers (if the strong factorization holds) then you can iterate the above description. If there is no strong factorization, probably you can use weak factorization instead (which always exists!) and also get some description.

EDIT. Here is a sketch of a proof of the statement about the criterion for a bundle to be pullback. Assume that $F$ is a vector bundle on $X$ and $F_{|E}$ restricts trivially to any fiber of $E = P_Z(N)$ over $Z$. Let $p:P_Z(N) \to Z$ be the projection and $O_E(-1)$ --- relative $O(-1)$. For any object $G \in D(Z)$ one has $$ Ext^\bullet(F,i_*(p^*G\otimes O_E(-k))) = Ext^\bullet(F_{|E},p^*G\otimes O_E(-k)) = H^\bullet(Z,Rp_*(F^\vee_{|E}\otimes p^*G(-k))) = 0 $$ for $k = 1,\dots,r(N)-1$ since $Rp_*(F^\vee_{|E}\otimes p^*G(-k)) = G\otimes^L Rp_*(F_{|E}(-k)) = 0$. Thus $F$ is contained in the first component of a semiorthogonal decomposition $$ D(X) = \langle i_*(O_E(1-r(N))\otimes p^*D(Z)), \dots, i_*(O_E(-1)\otimes p^*D(Z)), Lf^*(D(Y)) \rangle, $$ where $f:X \to Y$ is the blowup and $i:E \to X$ is the embedding. Thus $F \cong Lf^*F'$ for some object $F' \in D(Y)$. One can also check that $F'$ is a vector bundle.