Related papers: Asymptotics of K\"ahler-Einstein metrics on comple…

In this paper, we consider {\em mixed curvature} $\mathcal{C}_{\alpha,\beta}$ for Hermitian manifolds, which is a convex combination of the first Chern Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam…

We study singular K\"ahler-Einstein metrics that are obtained as non-collapsed limits of polarized K\"ahler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the…

Let $X$ be an arbitrary non-compact hyperbolic Riemann surface, that is, not $\mathbb C$ or $\mathbb C^*$. Given a tuple of holomorphic differentials $\boldsymbol q=(q_2,\cdots,q_n)$ on $X$, one can define a Higgs bundle…

Let $(E,\Phi)\rightarrow (X,\omega_X)$ be a Higgs bundle over a compact K\"ahler manifold. We suppose that the holomorphic vector bundle $E$ decomposes into a direct sum of holomorphic line bundles. In this paper, we give the necessary and…

Given any compact homogeneous space $H/K$ with $H$ simple, we consider the new space $M=H\times H/\Delta K$, where $\Delta K$ denotes diagonal embedding, and study the existence, classification and stability of $H\times H$-invariant…

The main result of this paper gives a new construction of extremal K\"ahler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new…

We consider Hitchin's hyperk\"ahler metric $g$ on the moduli space $\mathcal{M}$ of degree zero $\mathrm{SL}(2)$-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes to infinity along a generic ray in…

In this paper, we consider general $k$th-mixed curvature $\mathcal{C}^{(k)}_{\alpha,\beta}$ ($\beta\neq0$) for Hermitian manifolds, which is a convex combination of the $k$th Chern Ricci curvature and holomorphic sectional curvature. We…

We classify Einstein metrics on $\mathbb{R}^4$ invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. The metrics are either Ricci-flat or of negative Ricci curvature. We show that all…

By revisiting Tian's peak section method, we obtain a localization principle of the Bergman kernels on K\"ahler manifolds with complex hyperbolic cusps, which is a generalization of Auvray-Ma-Marinescu's localization result Bergman kernels…

Let $(X,\omega)$ be a compact normal K\"ahler space, with Hodge metric $\omega$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical K\"ahler metrics, we introduce a geodesic…

We study the Wu metric for the non-convex domains of the form \[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big \}, \] where $ 0 < m < 1/2$.…

We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical K\"ahler manifolds: If $\mathscr{E}$ is a reflexive sheaf over an ACyl K\"ahler manifold, which is asymptotic to a $\mu$-stable holomorphic vector…

We consider the homogeneous space $M=H\times H/\Delta K$, where $H/K$ is an irreducible symmetric space and $\Delta K$ denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of $H\times H$-invariant…

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…

We introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler manifold $X$, depending on a fixed real torus $\mathbb{T}$ in the reduced group of automorphisms of $X$, and two smooth (weight)…

We show that given a quasi-circle $C$ in $\partial_{\infty}\mathbb{H}^3$ (respectively in $\partial_{\infty} \mathbb{ADS}^3$) and a complete conformal metric $h$ on $\mathbb{D}$ whose curvature $K_h$ takes values in a compact subset of…

Let $X$ be a Riemann surface, $K_X \rightarrow X$ the canonical bundle, and $T_X\rightarrow X$ the dual bundle of the canonical bundle. For each integer $r \geq 2$, each $q \in H^0(K_X^r)$, and each choice of the square root $K_X^{1/2}$ of…

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler when the constant is non-zero and must be Chern flat when the constant is zero. The…

Let $\pi: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact K\"ahler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique K\"ahler-Einstein…