Moishezon Spaces and Projectivity Criteria

Abstract

We show that a smooth Moishezon space Y is non-projective if and only if it contains a rational curve such that −[C] ∈ NE(Y ). More generally, this holds if Y has Q-factorial, log terminal singularities. We derive this as a consequence of our main technical result: that we can run the relative minimal model program when the base is a normal algebraic space Y of finite type over a field of characteristic 0. As a second application, we show that every log canonical pair (Y, ∆), where Y is an algebraic space of finite type over a field of characteristic 0 admits a dlt modification that is projective over Y .

The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero

“…See [FS11,Lemma 4.12] for the case when Z = Spec(k), where k is a field. See also [Kol21a,Lemma 21] and [VP,Corollary 1.4] for other versions of Kleiman's criterion for algebraic spaces.…”

“…To discuss the outputs of the relative D-MMP, we index these steps continuously, following [VP,§2]. For each r ∈ R >0 , the r-th output of the π-relative D-MMP with scaling of H (if it exists), denoted by f r : X X r , will mean the composite…”