arXiv research

New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.

Establishes Hermite-Einstein metrics on complex spaces with singularities.

New Einstein RCD spaces found with cone singularities.

New Einstein manifolds split into symmetric and compact parts.

Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.

Newly discovered Eguchi-Hanson metric arises from edge metrics.

Study Einstein-Weyl spaces from Segre quartic surfaces, finding unique geodesics and deformations.

Study on Einstein metrics on complex projective spaces with specific group actions.

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Ampère equations. This type of equations is precisely what is needed in order to construct Kähler-Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähle…

Let $(M,g)$ be a compact Kähler-Einstein manifold with $c_1 > 0$. Denote by $K\to M$ the canonical line-bundle, with total space $X$, and $X_0$ the singular space obtained by blowing down $X$ along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincaré-…

Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.

The paper shows that certain Einstein orbifolds cannot be limits of smooth Einstein metrics.

Study two types of singular Kähler-Einstein metrics on complex varieties.

We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein (n+1)-manifold associated to a maximal domain of a static n-manifold where the static potential is po…

Compact Kahler-Einstein manifolds converge to semi-log canonical models.

Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.

Study of singular metrics with negative scalar curvature on compact manifolds.

Continuity of Kähler-Einstein potentials at singularities proven.

Existence of metrics on non-Kähler varieties, generalizing previous work.

Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.

Let $G/H$ be a connected, simply connected homogeneous space of a compact Lie group $G$. We study $G$-invariant quasi-Einstein metrics on the cohomogeneity one manifold $G/H\times (0,1)$ imposing the so-called monotypic condition on $G/H$. We obtain estimates on the rate of blow-up for these metrics near a singularity …

The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.

Continuous metrics on manifolds with singularities are shown to be Einstein.

Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.

We study Riemannian geometry of canonical Kahler-Einstein currents on projective Calabi-Yau varieties and canonical models of general type with crepant singularities. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original project…

We obtain a compactness result for various classes of Riemannian metrics in dimension four; in particular our method applies to anti-self-dual metrics, Kahler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding m…

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational instantons, but we focus on a different class of singularities. We show that any re…

We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scal…

Study Fano fibrations and Kähler-Ricci flow singularities, proving diameter bounds and curvature estimates.

The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.

The study shows how certain singular Kähler metrics can define Kähler currents and RCD spaces.

We deal with compact Kaehler manifolds M which are acted on by a semisimple compact Lie group G of isometries with codimension one regular orbits. We provide an explicit description of the standard blow-ups of such manifolds along complex singular orbits, in case b_1(M) = 0 and the regular orbits are Levi nondegenerate…

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

Kaehler-Einstein metrics on orbifolds derived from Einstein sequences.

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As…

Constructs Kahler-Einstein metrics near isolated log canonical singularities.

The paper characterizes Einstein metrics using CPE metrics and vacuum static spaces.

The note proves positive currents induced by VKE with mixed singularities.

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of th…

Study on continuity of solutions for complex Monge-Ampère equations with movable singularities.

The Kähler-Ricci flow converges to a negative Kähler-Einstein metric under certain conditions.

Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a Kähler-Einstein metric iff $X$ has semi-log canonical singularities i.e. iff $X$ is…

Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the …

New Einstein metrics constructed on complex line bundle over CP1.

We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytop…

Study on Kähler-Einstein metrics on quasi-projective manifolds.