Hermitian null loci

We establish a transcendental generalization of Nakamaye’s theorem to compact complex manifolds when the form is not assumed to be closed. We apply the recent analytic technique developed by Collins–Tosatti to show that the non-Hermitian locus of a nef and big $(1,1)$-form, which is not necessarily closed, on a compact complex manifold equals the union of all positive-dimensional analytic subvarieties where the restriction of the form is not big (null locus). As an application, we can give an alternative proof of the Nakai–Moishezon criterion of Buchdahl and Lamari for complex surfaces and generalize this result in higher dimensions. This is also used for studying degenerate complex Monge–Ampère equations on compact Hermitian manifolds. Finally, we investigate finite time non-collapsing singularities of the Chern–Ricci flow, partially answering a question raised by Tosatti and Weinkove.