Related papers: A Holomorphic Splitting Theorem

Let $M$ be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ for some compact connected Ricci-flat manifold $X$. We begin by proving general structure…

A notion of asymptotically conical K\"ahler orbifold is introduced, and, following previous existence results in the setting of asymptotically conical manifolds, it is shown that a certain complex Monge-Amp\'ere equation admits a rapidly…

We study a class of asymptotically cylindrical Ricci-flat K\"ahler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso…

We prove that the deformation theory of compactifiable asymptotically cylindrical Calabi-Yau manifolds is unobstructed. This relies on a detailed study of the Dolbeault-Hodge theory and its description in terms of the cohomology of the…

In 1978, Yau confirmed a conjecture due to Calabi stating the existence of K\"ahler metrics with prescribed Ricci forms on compact K\"ahler manifolds. A version of this statement for effective orbifolds can be found in the literature. In…

A new class of Riemannian metrics, called octonionic K\"ahler, is introduced and studied on a certain class of 16-dimensional manifolds. It is an octonionic analogue of K\"ahler metrics on complex manifolds and of HKT-metrics of…

In this paper, we study the local properties of the moduli space of a polarized Calabi-Yau manifold. Let $U$ be a neighborhood of the moduli space. Then we know the universal covering space $V$ of $U$ is a smooth manifold. Suppose $D$ is…

Tian and Yau constructed a complete Ricci-flat K\"ahler metric on the complement of an ample and smooth anticanonical divisor. We inquire into the behaviour of this metric towards the boundary divisor and prove a slow decay rate of the…

Let $B$ be a K\"ahler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding K\"ahler-Ricci solitons on the total…

Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau…

We outline a method to determine analytic K\"ahler potentials with associated approximately Ricci-flat K\"ahler metrics on Calabi-Yau manifolds. Key ingredients are numerically calculating Ricci-flat K\"ahler potentials via machine learning…

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 prove that a complete noncompact K\"ahler surface with positive and bounded sectional curvature is biholomorphic to $\mathbb{C}^2$. This result confirms a special case of Yau's conjecture that a complete noncompact K\"ahler $n$-manifold…

We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact…

We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon…

We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant…

Given a compact complex $n$-fold $X$ satisfying the $\partial\bar\partial$-lemma and supposed to have a trivial canonical bundle $K_X$ and to admit a balanced (=semi-K\"ahler) Hermitian metric $\omega$, we introduce the concept of…

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 $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and ${\cal M}_{\xi}$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\geq 2$, over $X$. Take a smooth anticanonical…

An old theorem of Weil and Kodaira says that for a compact K\"ahler manifold $X$ there is a closed logarithmic $1$-form with residue divisor $D$ if and only if $D$ is homologous to zero in $H_{2n-2}(X,\mathbb C)$. In the first part of this…