A differential geometric criterion for Moishezon spaces

Abstract

Proof. Let r r :Y~X be a projective desingularization of X, and let E be the exceptional set in Y. If M is any negative line bundle over Y, then by Proposition 3, we know that lr.(M ®d~(-E)) is a pseudonegative sheaf on X. But let's assume that, in fact, M is so negative that Hi(Y, M ® ¢ ( E)) =0. [For example, this will be true if M ®tg(E) is negative, by Kodaira's Vanishing Theorem.] Let ~ = n . (M ®~(-E)) . We will show that Hi(X, ~ ) = 0. Let { Us} be a locally finite open cover of X, and let {~ij} be a 1-cocycle with respect to this cover. We want to show that there exists a 0-cocycle {fit} with respect to our open cover such that at u = fl~-fli. But {~r1 Us} is a locally finite open cover of Y, and by definition of n, , the cocycle {~o} gives rise in canonical fashion to a cocycle (which we denote {~j}) on {rr-lU~}. Furthermore, since (as is well-known) the natural map from Ht({r~-lU~}, M®t~(-E)) to H I ( Y , M ~ ( E ) ) is one-to-one, there exists a