Space of Kähler metrics (V)-- Kähler quantization

An Obata-type characterization of Calabi metrics on line bundles

We characterize those complete K{ä}hler manifolds supporting a nonconstant real-valued function with critical points whose Hessian is complex linear, has pointwise two eigenvalues and whose gradient is a Hessian-eigenvector.

On pseudo-Hermitian Einstein spaces

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 from K{ä}hler-Einstein metrics. Moreover, we discuss the corresponding Fefferman metrics of the pseudo-Hermitian Einstein structures. These Fefferman metrics are never Einstein, but they are locally always conformally Einstein.

Hyper-K{ä}hler Hierarchies and their twistor theory

A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-Kähler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended space-time ${\cal N}$ is constructed whose extra dimensions corresp…

Kähler-Einstein metrics on group compactifications

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 continuity method and its translation into a real Monge-Amp{è}re equation, using the invariance under the action of a maximal compact subgroup $K\times K$.

全纯向量丛全空间上的全纯双截曲率

Holomorphic Bisectional Curvature on Total Spaces of Holomorphic Vector Bundles

该研究针对紧复流形上全纯向量丛全空间，建立了具指定全纯双截曲率符号的完备Kähler度量存在的充要条件，还得到非负情形的度量分裂、排除正维基的严格正曲率并构造非正情形的反例。

Balanced Metric and Berezin Quantization on the Siegel-Jacobi Ball

We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.

The Simanca metric admits a regular quantization

Let $g_S$ be the Simanca metric on the blow-up $\tilde{\mathbb{C}}^2$ of $\mathbb{C}^2$ at the origin. We show that $(\tilde{\mathbb{C}}^2,g_S)$ admits a regular quantization. We use this fact to prove that all coefficients in the Tian-Yau-Zelditch expansion for the Simanca metric vanish and that a dense subset of $(\tilde{\mathbb{C}}^2, g_S)$ admits a Berezin quantization

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 …

Kähler 度量 Mabuchi 空间中的实解析测地线与量子化

Real-analytic geodesics in the Mabuchi space of Kähler metrics and quantization

本文研究 Kähler 度量 Mabuchi 空间中测地线的量子化逼近，证明实解析初值在短时间内量子化 Bergman 测地线收敛于 Mabuchi 测地线，并指出边值问题通常无解。

Quantization of Hitchin's equations for Higgs Bundles I

We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit towards the solution of the Hitchin equation. We relate the existence of balanced metrics to the Gieseker stability of the Higgs bundle.

辛纤维化的量子化与典范度量

Quantization of symplectic fibrations and canonical metrics

本文通过Berezin-Toeplitz量子化与辛纤维化分步量子化的联系，研究向量丛的谱间隙及Donaldson迭代收敛到平衡度量的速率，并在标量情形给出精细估计。

A quantization proof of the uniform Yau-Tian-Donaldson conjecture

Using quantization techniques, we show that the $δ$-invariant of Fujita-Odaka coincides with the optimal exponent in certain Moser-Trudinger type inequality. Consequently we obtain a uniform Yau-Tian-Donaldson theorem for the existence of twisted Kähler-Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger-Colding-Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature Kähler metrics is also given.

Some special Kahler metrics on SL(2,C) and their holomorphic quantization

The group SU(2)*SU(2) acts naturally on SL(2,C) by simultaneous right and left multiplication. We study the Kahler metrics invariant under this action using global Kahler potentials. The volume growth and various curvature quantities are then explicitly computable. Examples include metrics of positive, negative and zero Ricci curvature, and the 1-lump metric of the CP^1-model on a sphere. We then look at the holomorphic quantization of these metrics, where some physically satisfactory results on the dimension of the Hilbert space can be obtained. These give rise to an interesting geometrical c…

几何多重位势理论中的量子化

Quantization in geometric pluripotential theory

本文旨在建立 Kähler 量子化的 Fubini--Study 度量空间与多重位势理论有限能量空间之间的联系，证明 Kähler 位势空间上的 Finsler 结构可被量子化，并应用于屋顶包络、Lidskii 型不等式及有限能量位势的逼近。

Griffiths 极值性、范数的插值与 Kähler 量子化

Griffiths extremality, interpolation of norms, and Kähler quantization

本文引入了 Griffiths 极值 Finsler 度量，证明其由复 Monge--Ampère 型方程刻画并求解了 Dirichlet 问题，同时展示了其在 Kähler 几何 PDE 解量子化中的应用。

Deformation quantization and Kähler geometry with moment map

In the first part of this paper we outline the constructions and properties of Fedosov star product and Berezin-Toeplitz star product. In the second part we outline the basic ideas and recent developments on Yau-Tian-Donaldson conjecture on the existence of Kähler metrics of constant scalar curvature. In the third part of the paper we outline recent results of both authors, and in particular show that the constant scalar curvature Kähler metric problem and the study of deformation quantization meet at the notion of trace (density) for star product. We formulate a cohomology formula for the inv…

具有T-对称性的Kähler流形上几何量子化的极限

Limit of geometric quantizations on Kähler manifolds with T-symmetry

本文研究具有T-对称性的紧致Kähler流形上几何量子化的极限行为，证明当$t$趋于无穷时Kähler极化对应的量子空间在T-等变意义下收敛到混合极化对应的量子空间，并给出权空间层面的收敛结果。

Balanced metrics and noncommutative Kaehler geometry

In this paper we show how Einstein metrics are naturally described using the quantization of the algebra of functions on a Kahler manifold M. In this setup one interprets M as the phase space itself, equipped with the Poisson brackets inherited from the Kahler 2-form. We compare the geometric quantization framework with several deformation quantization approaches. We find that the balanced metrics appear naturally as a result of setting the vacuum energy to be the constant function on the moduli space of semiclassical vacua. In the classical limit these metrics become Kahler-Einstein (when M a…

The probabilistic vs the quantization approach to Kähler-Einstein geometry

In the probabilistic construction of Kähler-Einstein metrics on a complex projective algebraic manifold X - involving random point processes on X - a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kähler-Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler-Einstein geometry.

Relative Kahler-Ricci flows and their quantization

Let X be a complex manifold fibered over the base S and let L be a relatively ample line bundle over X. We define relative Kahler-Ricci flows on the space of all Hermitian metrics on L with relatively positive curvature. Mainly three different settings are investigated: the case when the fibers are Calabi-Yau manifolds and the case when L is the relative (anti-) canonical line bundle. The main theme studied is whether posivity in families is preserved under the flows and its relation to the variation of the moduli of the complex structures of the fibres. The quantization of this setting is als…

Geometric Quantization of the moduli space of the vortex equations on a Riemann surface

In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.

Geometric Quantization of the Hitchin System

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the…

从无穷远处、量子化与紧致热带阿米巴视角看 Toric Kähler 度量

Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas

本文研究了紧致 toric 流形上 toric Kähler 度量空间在无穷远处的极限，揭示了 Kähler 极化在 Lagrangian 纤维上的退化，实现了全纯与实极化几何量子化的连续插值，并利用度量退化证明了紧致超曲面阿米巴极限可由热带阿米巴描述。

通过覆盖增长估计研究黎曼流形上的渐近量化

Asymptotic quantization on Riemannian manifolds via covering growth estimates

该研究针对黎曼流形上的渐近量化问题，提出了基于球面覆盖数增长的更一般积分条件，并估计了Ricci曲率下界等情形下的覆盖增长，为非光滑空间的相关研究提供了思路。

形变量子化与内蕴非交换微分几何

Deformation quantization and intrinsic noncommutative differential geometry

本文提出非交换微分几何的内蕴表述，给出协变导数、曲率及Bianchi恒等式的几何定义，证明特定条件下Ricci曲率等价，并计算球对称等距嵌入的量子涨落与曲率，表明引力量子化可重整化，最后定义准连接并证明第一Bianchi恒等式。

The Cauchy problem for the homogeneous Monge-Ampere equation, I. Toeplitz quantization

The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow defined by the Cauchy data, analytically continuing the quantization, and then taking a kind of logarithmic classical limit. In this article, we prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) this "quantum analytic continuation potential" coincid…

正拉格朗日量的量子化

Quantization of positive Lagrangians

本研究针对近Calabi-Yau流形中的正拉格朗日子流形，利用量子化映射与Hörmander估计，证明相对Kähler类为有理斜率时其度量诱导路径度量，为拉格朗日子流形的量子化提供关键结果。

几何量子化中的谱收敛——非奇异拉格朗日纤维化情形

Spectral convergence in geometric quantization -- the case of non-singular Langrangian fibrations

该研究利用度量测度空间收敛理论研究几何量子化，证明了预量子化辛流形上Kähler极化收敛于非奇异实极化时$\bar{\bar{∂}}$-拉普拉斯算子的谱及量子希尔伯特空间的收敛性，还得到近Kähler量子化的类似结果。