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?


1 Answer 1


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.

  • $\begingroup$ Interesting, thanks for the reference! If I'm not mistaken, Serre's criterion for regular local rings (homological dimension is finite) the implies there are no infinite free resolutions. $\endgroup$ Commented Jun 17 at 18:57

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