Criteria for a fiberwise Fujiki/Kahler family to be locally Moishezon/projective

Abstract

We utilize the theory of non-Kahler loci by S. Boucksom to construct an integral 2-cohomology class whose restriction to a general fiber is big, and then construct a relatively big line bundle via the exponential sequence. This leads to a local Moishezonness criterion for fibrations whose total spaces are in Fujiki class C, generalizing the bimeromorphic version of F. Campana's local projectivity theorem. We further combine a similar idea with the singular Demailly-Paun theorem by T. Collins-V. Tosatti to obtain a local projectivity criterion for fibrations from compact Kahler manifolds, yielding a new proof and a generalization of F. Campana's local projectivity theorem.