arXiv research

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …

Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.

The paper studies Ricci curvature on Kähler-Ricci flow.

In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curva…

Study Fano fibrations and Kähler-Einstein metrics on their bases.

Study proves most Fano threefolds are Kähler-Einstein.

Existence of Kähler-Einstein metrics on compactifications of Lie groups.

Survey on Kähler-Einstein and weighted solitons on Fano manifolds.

We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large an…

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit …

Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.

We show that the anti-canonical volume of an $n$-dimensional Kähler-Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\mathrm{lct}^n\cdot\mathrm{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fuji…

The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.

Paper discusses conditions for deforming coupled Kähler-Einstein metrics.

New research confirms Kähler-Einstein metrics for all Fano threefolds of degree 22.

Decomposes Q-Fano Kähler-Einstein varieties into simpler components.

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. …

The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.

The paper examines conditions for Kähler-Einstein metrics on deformations of Fano manifolds.

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being…

We construct a geometrically compactified moduli algebraic space of Kahler-Einstein Fano manifolds.

Study shows polystability of tangent and canonical sheaves on Kähler-Einstein log Fano pairs.

Proves conic version of YTD conjecture on log Fano manifolds.

Surveying Kähler-Einstein metrics and related conjectures.

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…

Proof of flow convergence on Fano manifolds.

Analyzes Kähler-Einstein metrics on families of Fano varieties.

New examples found of complex manifolds with special metrics.

Fano varieties get Kähler-Einstein metrics if they are uniformly K-stable.

Uniform K-stability implies Kähler-Einstein metric for Q-Fano varieties.

The study finds quasi-Einstein metrics on sphere bundles.

We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its canonical compactification.

Kähler-Einstein metrics found on special types of symmetric varieties.

The paper finds Kähler-Einstein metrics on certain degenerations.

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …

Classifies K-stable Fano varieties and finds new examples.

Solves modified conjecture for Fano manifolds using Ding stability.

New proof of Kähler-Einstein Fano manifold $L^\infty$ estimates.

New approach proves K-stability of Fano varieties.

Proves Yau-Tian-Donaldson conjecture for certain singular Fano varieties.

Proof confirms condition for Kähler-Einstein metrics on toric Fano manifolds.

Study finds Kähler-Einstein metrics on two Pasquier varieties.

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

We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…

Mabuchi's metric correlates with a specific stability condition for Fano manifolds.

The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galick…

We survey recent results on the existence of Kähler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties. Moreover, we give some applications and we discuss some natural problems which dese…