Degeneration of Kahler-Einstein hypersurfaces in complex torus to generalized pair of pants decomposition

Kähler-Ricci 孤立子退化的存在性

The existence of the Kähler-Ricci soliton degeneration

本研究证明了所有对数法诺对的Hamilton-Tian猜想代数版本，证实其存在典范两步退化至约化一致Ding稳定三元组，且基域为复数域时该三元组有Kähler-Ricci孤立子。

Degeneration of Kahler-Ricci solitons on Fano manifolds

We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent subsequence in the Gromov-Hausdorff topology to a Kahler- Ricci soliton on a Q-Fano variety with log terminal singularities.

Infinite geodesic rays in the space of Kahler potentials

In this paper we explore the connection between special degenerations of algebraic manifolds and geodesics in the space of Kahler metrics. We provide a new and general geometric construction of nontrivial solutions for the geodesic equation. We show how to associate to any special nontrivial degeneration a geodesic of inifite length.

Degeneration of Kahler-Einstein manifolds of negative scalar curvature

Let $π: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact Kähler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique Kähler-Einstein metric on $\mathcal{X}_t= π^{-1}(t)$. We show that as $t\rightarrow 0$, $(\mathcal{X}_t, g_t)$ converges in pointed Gromov-Hausdorff topology to a unique finite disjoint union of complete metric length spaces $\coprod_{α=1}^\mathcal{A} (Y_α, d_α)$ without loss of volume. Each $(Y_α, d_α)$ is a smooth open Kähler-Einstein manifold of complex dimension n outside its clo…

Degeneration of Kahler polarizations to mixed polarizations on toric varieties

Let $(X, ω, J)$ be a toric variety of dimension $2n$ determined by a Delzant polytope. In this paper, we first construct the polarizations $\shP_{k}$ by the Hamiltonian $T^{k}$-action on $X$ (see Theorem 3.11). We will show that $\shP_{k}$ is a singular mixed polarization for $1\le k < n$, and $\shP_{n}$ is a singular real polarization which coincides with the real polarization studied in \cite{BFMN} on the open dense subset of $X$. Then for each $1\le k \le n$, we will find a one-parameter family of Kähler polarizations $\shP_{k,t}$ on $X$ that converges to $\shP_{k}$ (see Theorem 3.12). Fina…

Collapsing of negative Kähler-Einstein metrics

In this paper, we study the collapsing behaviour of negative Kähler-Einstein metrics along degenerations of canonical polarized manifolds. We prove that for a toroidal degeneration of canonical polarized manifolds with the total space $\mathbb{Q}$-factorial, the Kähler-Einstein metrics on fibers collapse to a lower dimensional complete Riemannian manifold in the pointed Gromov-Hausdorff sense by suitably choosing the base points. Furthermore, the most collapsed limit is a real affine Kähler manifold.

Note on equivalences for degenerations of Calabi-Yau manifolds

This note studies the equivalencies among convergences of Ricci-flat Kähler-Einstein metrics on Calabi-Yau manifolds, cohomology classes and potential functions.

奇异Kähler-Einstein度量向切锥的多项式收敛性研究

On polynomial convergence to tangent cones for singular Kähler-Einstein metrics

该研究针对奇异Kähler-Einstein度量的锥形性质，结合Donaldson-Sun两步退化理论与Hein-Sun方法，分别在度量极限和代数奇点芽场景下，给出了度量在奇点处为锥形的充要或充分条件。

Some remarks on conic degeneration and bending of Poincaré-Einstein metrics

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--Einstein metrics on $X$. One 1-parameter subfamily $\{g_t\}_{t>0}$ has the property that as $t\searrow 0$, $g_t$ converges to a PE metric $g_0$ on $X_0$ with conic singularity, while $t^{-1}g_t$ converges to a complete Ricci-flat Kähler metric $\hat{g}_0$ on $X$. Another 1-param…

Convergence of Calabi-Yau manifolds

In this paper, we study the convergence of Calabi-Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi-Yau manifolds.

Riemannian geometry of Kahler-Einstein currents

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 projective variety, with well-defined tangent cones. We also prove a special degeneration for Kahler-Einstein manifolds of general type as an approach to establish the compactification of the moduli space of Kahler-Einstein manifolds of general type. A number of applications are given f…

（对数）Kähler-Einstein 度量气泡的若干模型

Some models for bubbling of (log) Kähler-Einstein metrics

本文研究了复一维与二维（对数）Kähler-Einstein 度量非塌缩退化的度量气泡树，并提出高维猜想图景。

Generalized Kähler-Einstein metric along $\mathbb Q$-Fano fibration

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 Kähler-Einstein metric. We introduce the fiberwise Kähler-Einstein foliation and we mention that the main difficulty to obtain higher estimates is to solve relative CMA equation along such foliation. We propose a program such that for finding a pair of canonical metric $(ω_X,ω_B)$, which satisfies in $$Ric(ω_X)=π^*ω_B+π^*(ω_{WP})+[\mathcal N]$$ on K-poly stable degeneration $π:X\to B$, where $…

Hybrid convergence of Kähler-Einstein measures

We compute the hybrid limit (in the sense of Boucksom-Jonsson) of the family of Kähler-Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial valuations, determined by the natural algebro-geometric limit of the family. We also make some remarks on the non-archimedean Monge-Ampère operator and hybrid continuity of Kähler-Einstein potentials in this context.

Compact Moduli Spaces of Del Pezzo Surfaces and Kähler-Einstein metrics

We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

K Energy and K stability on Hypersurfaces

In this paper, we study the limiting properties of the $K$ energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. Kähler-Einstein metrics and the generalized Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of hypersurfaces. In particular, we allow the center fiber of a special degeneration (a degeneration by a one-parameter group) to have multiplicity great than 1.

Riemannian geometry of Kahler-Einstein currents II: an analytic proof of Kawamata's base point free theorem

It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and $L^2$-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Riemannian structure on canonical models. Our approach can be viewed as the Kodaira embedding theorem on singular metric spaces with canonical Kahler metrics.

K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functional along weak geodesic rays in the space of all bounded positively curved metrics on the anti-canonical line bundle of X and also gives a new proof in the non-singular case. One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerat…

Conical Kahler-Einstein metric revisited

In this paper we introduce the "interpolation-degneration" strategy to study Kahler-Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By "interpolation" we show the angles in $(0, 2π]$ that admit a conical Kahler-Einstein metric form an interval; and by "degeneration" we figure out the boundary of the interval. As a first application, we show that there exists a Kahler-Einstein metric on $P^2$ with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in $(π/2, 2π]$. When th…

复结构退化与Calabi-Yau度量的坍缩

Complex structure degenerations and collapsing of Calabi-Yau metrics

研究极化Calabi-Yau流形退化族的度量坍缩问题，针对射影空间中Calabi-Yau超曲面退化为两Fano超曲面横截并的情形，用粘合与奇异摄动法构造环面对称Kähler度量，建立度量坍缩与复结构退化的精确关系，全维结果含复二维情形均为新发现，并探讨推广可能。

中间复结构极限的通用正则性

Generic regularity of intermediate complex structure limits

研究近中间复结构极限处Calabi-Yau流形的极化退化，并将潜在的C^0收敛提升为通用区域的度量收敛结果。

Weighted Bott-Chern and Dolbeault cohomology for LCK-manifolds with potential

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the $dd^c$-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds.…

锥情形下的Donaldson-Sun理论

Donaldson-Sun Theory in the Conic Case

将Donaldson-Sun度量切锥理论推广到边界系数为有理数有限子集的锥Kähler-Einstein对的非塌缩Gromov-Hausdorff极限，证明对数度量切锥的唯一性，关联稳定退化机制，构造反例表明Kähler-Einstein假设对切锥刚性的必要性。

Topological and Geometric Obstructions on Einstein-Hilbert-Palatini Theories

In this article we introduce $A$-valued Einstein-Hilbert-Palatini functional ($A$-EHP) over a n-manifold $M$, where $A$ is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if $A$ is weak $(k,s)$-solvable, then $A$-EHP is non-null only if $n<k+s+3$. We prove that essentially all algebras modeling classical geometries (except semi-Riemannian geometries with specific signatures) satisfy this condition for $k=1$ and $s=2$, including Hitchin's generalized complex geometry, Pantilie's generalized quaternio…