Related papers: Some QCH Kahler surfaces with zero scalar curvatur…
In this paper we study QCH K\"ahler surfaces, i.e. 4-dimensional Riemannian manifolds (of signature (++++)) admitting a K\"ahler complex structure with quasi-constant holomorphic sectional curvature. We give a detailed description of QCH…
In this paper we describe QCH K\"ahler surfaces $(M,g,J)$ of generalized orthotoric type. We introduce a distinguished orthonormal frame on $(M,g)$ and give the structure equations for $(M,g,J)$. In the case when $I$ is conformally K\"ahler…
In this paper we give classification of K\"ahler surfaces of generalized Calabi type.
We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature K\"ahler metric for every K\"ahler class. We show that there are only two such rational surfaces,…
We show that on Kahler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kahler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated…
This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/
We show that if $(M,\omega)$ is any compact K\"ahler manifold, then the blowup of $M$ at any point furnishes a K\"ahler metric with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $\omega$. It follows that…
In this paper we characterize logarithmic surfaces which admit K\"ahler-Einstein metrics with negative scalar curvature and small edge singularities along a normal crossing divisor.
Motivated by a recent work of Chen-Zheng [8] on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection $\nabla^t $ is either K\"ahler,…
Polyhedral K\"ahler surfaces are a class of complex surfaces, which are flat everywhere except on a two-dimensional skeleton. They are defined as a generalisation of the "gluing a polygon side by side" construction of flat Riemann surfaces.…
We construct two types of Eguchi-Hanson metrics with the negative constant scalar curvature. The type I metrics are K\"{a}hler. The type II metrics are ALH whose total energy can be negative.
Let M be a compact complex surface which admits a Kaehler metric whose scalar curvature has integral zero; and suppose the fundamental group of M does not contain an Abelian subgroup of finite index. Then if M is blown up at sufficiently…
Motivated by the torus stability problem, in this work we study K\"ahler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of K\"ahler metrics with…
It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature K\"ahler metric \cite{ACGT,G,M17}. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature…
We present new constructions of Kaehler metrics with constant scalar curvature on complex surfaces, in particular on certain del Pezzo surfaces. Some higher dimensional examples are provided as well.
K\"ahler-Einstein metrics for polarized families of Calabi-Yau manifolds define a natural hermitian metric on the relative canonical bundle. The fact that the curvature form is equal to the pull-back of the Weil-Petersson form up to a…
In this short note, we study compact K\"ahler surfaces whose universal cover can be realized as a quasi-projective (or quasi-K\"ahler) surface. In particular, we show that such a surface is a quotient of a torus if the universal cover is…
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional curvature is…
We point out how some recent developments in the theory of constant scalar curvature K\"ahler metrics can be used to clarify the existence issue for such metrics in the special case of geometrically ruled complex surfaces.
An almost K\"ahler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal K\"ahler metric in the sense of Calabi [8]. We observe…