Integrability of (ω,m)-subharmonic functions on compact Hermitian manifolds

In this paper, we generalize the Bochner-Kodaira formulas to the case of Hermitian complex (possibly non-holomorphic) vector bundles over compact Hermitian (possibly non-Kahler) manifolds. As applications, we get the complex analyticity of harmonic maps between compact Hermitian manifolds.

Fully non-linear degenerate elliptic equations in complex geometry

We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge–Ampere, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kahler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.

The complex Monge–Ampère type equation on compact Hermitian manifolds and applications

Given a Hermitian manifold (Mn, g), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $${\nabla ^s} = \left( {1 - \frac{s}{2}} \right){\nabla ^c} + \frac{s}{2}{\nabla ^b}$$ the s-Gauduchon connection of M, where ∇c and ∇b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0 (a flat Chern connection) or s = 2 (a flat Bismut connection) are classified respectively by Boothby in the 1950s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either $$s \geqslant 4 + 2\sqrt 3 \approx 7.46$$ or $$s \leqslant 4 - 2\sqrt 3 \approx 0.54$$ and s ≠ 0, then g is Kähler. We also show that, when n = 2, g is always Kähler unless s = 2. Therefore non-Kähler compact Gauduchon flat surfaces are exactly isosceles Hopf surfaces.

Soit M une variete hermitienne compacte. Alors: a) u+v>0⇒P m =0, ∀ m >0; b) u+v≥0⇒P m ∈{0,1}, ∀ m >0, ou u et v sont les deux courbures scalaires de geometrie hermitienne, et P m et Q m sont les miemes plurigenres et le mieme plurigenre dual de M

This paper divides into two parts. Let $(X,\omega )$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $\omega $ satisfies the assumption that $\partial \overline {\partial }\omega ^k=0$ for all $k$, we generalize the volume of the co

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D -invariant subbundle (resp. if it is the D -invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to the existence of a so-called harmonic metric in E . In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X,g) . We propose new notions of g -(poly-)stability of such bundles, and of g -Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if g is a Kähler metric, but are different in general. Our main result is that the g -polystability in our sense is equivalent to the existence of a g -Hermitian-Einstein metric. Our notion of a g -Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of Kähler manifolds [S1], [S2], [S3].

We prove the bounded subsolution theorem for the complex Monge–Ampère type equation, with the right-hand side being a positive Radon measure, on a compact Hermitian manifold with boundary.

Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree

In this paper, we show the existence and uniqueness of bounded solutions of the degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and obtain the asymptotics of these solutions. As applications, we give partial answers to the Tosatti-Weinkove conjecture and Demailly-Păun conjecture.

We show that, up to scaling, the complex Monge-Ampère equation on compact Hermitian manifolds always admits a smooth solution.

In this paper, we describe how pluripotential methods can be applied to study weak solutions of the complex Monge–Ampère equation on compact Hermitian manifolds. We indicate the differences between Kähler and non-Kähler setting. The results include a priori estimates, existence and stability of solutions.

<abstract><p>In this paper, we consider the parabolic Hessian quotient equation on compact Hermitian manifolds. By setting up a priori estimates of the admissible solutions, we prove the long-time existence of the solution to the parabolic Hessian quotient equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.</p></abstract>

In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 {\Gamma }_{k-1} cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.