Related papers: Complex Hessian Equation on K\"ahler Manifold

We give an elementary introduction to hyperk\"ahler manifolds, survey some of their interesting properties and some open problems.

We prove the symmetry of solutions to overdetermined problems for a class of fully nonlinear equations, namely Hessian quotient equations and Hessian quotient curvature equations. Our approach is based on establishing a Rellich-Pohozaev…

In this paper we develop an analogue of the Berkovich analytification for non-necessarily algebraic complex spaces. We apply this theory to generalize to arbitrary compact K\"ahler manifolds a result of Chi Li, proving that a stronger…

In this paper, we derive a new monotonicity formula for the plurisuhbarmonic functions on complete K\"ahler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of…

We study various constraints on the Beauville quadratic form and the Huybrechts-Riemann-Roch polynomial for hyper-K\"ahler manifolds, mostly in dimension 6 and in the presence of an isotropic class. In an appendix, Chen Jiang proves that in…

We study the Hermitian curvature flow of locally homogeneous non-K\"ahler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a…

This paper is the first step in a systematic project to study examples of K\"ahler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact K\"ahler surface with $H>0$ must be rational…

In this paper, we consider the Dirichlet problem for a class of prescribed Hessian quotient type curvature equations with homogeneous boundary data in Minkowski space. By establishing the a priori C2 estimates, we obtain the existence…

In this paper, we establish new Laplacian comparison theorems and rigidity theorems for complete K\"ahler manifolds under new curvature notions that interpolate between Ricci curvature and holomorphic bisectional curvature.

In this paper we introduce a set of equations on a principal bundle over a compact complex manifold coupling a connection on the principal bundle, a section of an associated bundle with K\"ahler fibre, and a K\"ahler structure on the base.…

Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact…

We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…

We prove the subsolution theorem for the complex Hessian equations in a smoothly bounded strongly $m$-pseudoconvex domain, $1 < m < n$, in $\bC^n$.

In this paper we address the following question regarding the regularity of geodesics in the space of K\"ahler potentials. Given a geodesic which is highly regular, and has smooth boundary value, can we expect that it is actually smooth? We…

In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive…

In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the K\"ahler and non-K\"ahler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight…

We introduce a cohomological obstruction to solving the constant scalar curvature K\"ahler (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. Geometrically this gives an obstruction for a manifold to…

Our recent work about fully non-linear elliptic equations on compact manifolds with a flat hyperk\"ahler metric is hereby extended to the parabolic setting. This approach will help us to study some problems arising from hyperhermitian…

We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the…

In this note we show that on any compact subdomain of a K\"ahler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized…