Let $X$ be a compact Kähler manifold, and $F$ a coherent sheaf. Does $F$ admit a locally free resolution
$$E^* \to F \to 0?$$
Of course it will be enough to construct a surjection
$$E^0 \to F \to 0$$
for a locally free sheaf $E^0$.
I know that if $X$ is projective, this can be obtained by twisting with an ample line bundle to obtain a surjection
$$H^0(F \otimes \mathcal O(n)) \otimes \mathcal O_X \to F \otimes \mathcal O(n) \to 0.$$
According to Appendix B of Fulton's Intersection Theory, this statement still holds more generally for smooth schemes $X$. But what if $X$ is not algebraic?
Best Answer
For a finite length resolution $ E^{\bullet} $, this is false by a theorem of Voisin, that any holomorphic vector bundle on a general torus of dimension > 2 has vanishing Chern classes $ c_i $ for every $ i > 0 $. Then a finite length resolution of the ideal sheaf of a point on such a torus does not exist, due to the Whitney formula.
The reference is Voisin’s Survey specifically, Theorem 13 and the ensuing discussion.