Related papers: Variation of singular K\"ahler-Einstein metrics: K…

Let G be a nontrivial finite subgroup of U(m) acting freely on C^m - 0. Then C^m/G has an isolated quotient singularity at 0. Let X be a resolution of C^m/G, and g a Kahler metric on X. We say that g is Asymptotically Locally Euclidean…

We show that the Calabi-Yau metrics with isolated conical singularities of Hein-Sun admit polyhomogeneous expansions near their singularities. Moreover, we show that, under certain generic assumptions, natural families of smooth Calabi-Yau…

Let $f:\,X \to \mathbb{P}^1$ be a non-isotrivial semi-stable family of varieties of dimension $m$ over $\mathbb{P}^1$ with $s$ singular fibers. Assume that the smooth fibers $F$ are minimal, i.e., their canonical line bundles are semiample.…

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-K\"ahler manifolds, and arise independently in mathematical physics. We reinterpret this condition…

In this paper, we study MRC fibrations of compact K\"ahler manifolds with partially semi-positive curvature. We first prove that a compact K\"ahler manifold is rationally connected if its tangent bundle is BC-$p$ positive for all $1\leq…

We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.

Every compact K\"ahler manifold with negative first Chern class admits a unique metric $g$ such that $\text{Ric}(g) = -g$. Understanding how families of these metrics degenerate gives insight into their geometry and is important for…

We prove that a Kahler supermetric on a supermanifold with one complex fermionic dimension admits a super Ricci-flat supermetric if and only if the bosonic metric has vanishing scalar curvature. As a corollary, it follows that Yau's theorem…

We present a method to construct approximate analytic expressions for Ricci-flat K\"ahler metrics on Calabi-Yau threefolds with explicit dependence on the K\"ahler moduli. Our strategy combines numerical data obtained from machine learning…

We study two different natural notions of singular K\"ahler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat K\"ahler cone metrics that arise as non-collapsed limits of sequences of K\"ahler-Einstein…

The author has proved that a crepant resolution Y of a Ricci-flat K\"{a}hler cone X admits a complete Ricci-flat K\"{a}hler metric asymptotic to the cone metric in every K\"{a}hler class in H^2_c(Y,\R). These manifolds are generalizations…

This note gives a simple formula for the unique asymptotically conical Calabi-Yau metrics on the canonical bundle of a flag variety known to exist by the work of R. Goto and others. This is done by generalizing the well known Calabi Ansatz…

Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize…

Let $(M^n, g)$ be a compact K\"ahler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact K\"ahler manifold $N^k$ with $c_1 < 0$. This confirms a…

We show that a polarized affine variety admits a Ricci flat K\"ahler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to K\"ahler cones, or equivalently,…

In this paper, we show that along $\mathbb Q$-Fano fibration, when general fibres, base and central fiber (with at worst Kawamata log terminal singularities)are K-poly stable then there exists a relative K\"ahler-Einstein metric. We…

We present a simple derivation of the Ricci-flat Kahler metric and its Kahler potential on the canonical line bundle over arbitrary Kahler coset space equipped with the Kahler-Einstein metric.

The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost K\"ahler manifolds, providing an explicit computation for a family of almost K\"ahler…

We construct a family of $6$-dimensional compact manifolds $M(A)$, which are simultaneously diffeomorphic to complex Calabi-Yau manifolds and symplectic Calabi-Yau manifolds. They have fundamental groups $\mathbb{Z} \oplus \mathbb{Z}$,…

Let $(X,L_{X})$ be an $n$-dimensional polarized manifold. Let $D$ be a smooth hypersurface defined by a holomorphic section of $L_{X}$. In this paper, we show the existence of a complete K\"{a}hler metric on $X \setminus D$ whose scalar…