An obstruction to asymptotic semistability and approximate critical metrics

Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures

We study the basic properties of Higgs sheaves over compact Kähler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and we show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we proved some prop…

An obstruction to the existence of constant scalar curvature Kähler metrics

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope $μ$ for a projective manifold and for each of its subschemes, and show that if $X$ is cscK then $μ(Z)\leμ(X)$ for all subschemes $Z$. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as del Pezzo surfaces and projective bundles. If $\PP(E)\to B$ is a projective bundle which admits a cscK metric in a rational Kähler class with sufficiently small fibres, then $E$ is a slope semist…

Scalar curvature and asymptotic Chow stability of projective bundles and blowups

The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme. These invariants are calculated in two special cases. The first is a projective bundle over a curve of genus at least two, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the underlying vector bundle is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarizati…

一致K-稳定环面簇的渐近Chow稳定性

Asymptotic Chow stability of uniformly K-stable toric varieties

本文研究极化环面簇，在Futaki-Ono不变量消失的假设下，给出了一致K-稳定极化环面簇(X,L)渐近Chow多稳定的充分判据，并证明所有Futaki-Ono不变量消失的一致K-稳定光滑极化环面簇都是渐近Chow多稳定的。

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

We use the correspondence between extremal Sasaki structures and weighted extremal Kahler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors, and Lahdili's theory of weighted K-stability in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the correspondin…

A conical approximation of constant scalar curvature Kähler metrics of Poincaré type

Let $(X,L_X)$ be a polarized manifold and $D$ be a smooth hypersurface such that $D \in | L_X |$. In this paper, we show that if there is no nontrivial holomorphic vector field on $D$ and ${\rm Aut}_0 ((X,L_X); D)$ is trivial, then constant scalar curvature Kähler metrics of Poincaré type on $X \setminus D$ can be approximated by constant scalar curvature Kähler metrics with cone singularities of sufficiently small angle along $D$. This result implies log K-semistability of $((X,L_X);D)$ with angle 0.

Geometric quantization of coupled Kähler-Einstein metrics

We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the ``balanced metrics''. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce some algebro-geometric obstruction which interpolates between the Donaldson-Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the …

高维紧致爱因斯坦orbifold的解析

Resolution of compact Einstein orbifolds in general dimensions

研究高维紧致爱因斯坦流形非塌缩序列的Gromov-Hausdorff极限为紧致爱因斯坦orbifold，证明其存在有限奇点，并给出超几何orbifold的负障碍物，简化了Ozuch四维工作中的技术难点。

CR Yamabe 不变量与常数量曲率 Sasaki 度量

The CR Yamabe invariant and constant scalar curvature Sasaki metrics

本文利用 CR Yamabe 问题的工具，将 CR Yamabe 不变量与常横截数量曲率 Sasaki 结构及 K-稳定性联系起来，并给出极化紧复流形 K-半稳定的新数值判据。

关于Wess-Zumino-Witten方程和Harder-Narasimhan势

About Wess-Zumino-Witten equation and Harder-Narasimhan potentials

研究极化复射影流形中Wess-Zumino-Witten方程近似解的代数障碍，并推广为Monge-Ampère型方程，应用时证明了Andréotti-Grauert定理的反向结论。

Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice of contact form $θ$), i.e., is an invariant of the contact structure …

Balanced metrics for Kähler-Ricci solitons and quantized Futaki invariants

We show that a Kähler-Ricci soliton on a Fano manifold can always be smoothly approximated by a sequence of relative anticanonically balanced metrics, also called quantized Kähler-Ricci solitons. The proof uses a semiclassical estimate on the spectral gap of an equivariant Berezin transform to extend a strategy due to Donaldson, and can be seen as the quantization of a method due to Tian and Zhu, using quantized Futaki invariants as obstructions for quantized Kähler-Ricci solitons. As corollaries, we recover the uniqueness of Kähler-Ricci solitons up to automorphisms, and show how our result a…

关于常负曲率流形的近似反谱定理

An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature

本文研究在常负曲率约束下，通过构造具有不同拓扑复杂度的闭合流形，近似任意有限递增的目标谱列表，同时讨论曲率归一化下的普遍障碍。