A priori estimates for Donaldson’s equation over compact Hermitian manifolds

Abstract

In this paper we prove a priori estimates for Donaldson equation’s $$\begin{aligned} \omega \wedge (\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n-1} =e^{F}(\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n}, \end{aligned}$$ over a compact complex manifold $$X$$ of complex dimension $$n$$ , where $$\omega $$ and $$\chi $$ are arbitrary Hermitian metrics. Our estimates answer a question of Tosatti-Weinkove (Asian J. Math. 14:19–40, 2010).