We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature Kahler metrics in a given rational cohomology class. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.

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Scalar Curvature and Projective Embeddings, I

On considere l'existence et la regularite des solutions d'equations aux derivees partielles, la construction de solutions fondamentales explicites et les valeurs propres d'operateurs de Schrodinger

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the uncertainty principle

Hilbert's 17th Problem asks whether a real positive semide nite (psd) polynomial in several variables must be a sum of squares of rational functions. This paper gives a survey of the literature on two closely related questions: What can be said about a psd polynomial that is not a sum of squares of polynomials? How can one write a given psd polynomial as a sum of squares of rational functions? These questions go back to Hilbert himself, and his interest in them predated the 1900 Paris Congress.

Some concrete aspects of Hilbert's 17th Problem

This survey is an overview of some of the better known quantization techniques (for systems with finite numbers of degrees-of-freedom) including in particular canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. We have attempted to give an account sufficiently in depth to convey the general picture, as well as to indicate the mutual relationships between various methods, their relative successes and shortcomings, mentioning also open problems in the area. Finally, even for approaches for which lack of space or expertise prevented us from treating them to the extent they would deserve, we have tried to provide ample references to the existing literature on the subject. In all cases, we have made an effort to keep the discussion accessible both to physicists and to mathematicians, including non-specialists in the field.

Quantization methods: a guide for physicists and analysts

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kahler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kahler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kahler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kahler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.

A variational approach to complex Monge-Ampère equations

Given a domain Ω in ℂn, the Bergman kernel is the kernel of the projection operator from L 2 (Ω) to the Hardy space A 2 (Ω). When the boundary of Ω is strictly pseudoconvex and smooth, Fefferman [2] gave a complete description of the asymptotic behavior of K(z, z) as z approaches the boundary. This work was then extended by Boutet de Monvel and Sjostrand [1] who showed that, for the same domains, a similar asymptotic expansion for K(z, w) holds off the diagonal. Moreover, they showed that the Bergman kernel is a Fourier integral operator with a complex phase function.

The Bergman Kernel and a Theorem of Tian

Subelliptic estimates for the $\overline\partial$-Neumann problem on pseudoconvex domains

Estimates of invariant metrics on pseudoconvex domains of dimension two

Let a be a a-closed form of type (p, q) with L2-coefficients on a smoothly bounded domain Q in C'. The problem of determining both the existence and regularity properties of the solution u of au = a, where u is orthogonal to the null space of a on (p, q l)-forms, is known as the a-Neumann problem. One of the principal methods used in the investigation of this problem is the proof of certain a priori subelliptic estimates. Let U be a neighborhood of a point z0 in the boundary of Q. A subelliptic estimate is said to hold in U if the estimate

Necessary conditions for subellipticity of the $\overline\partial$-Neumann problem

The purpose of this paper is to describe an invariant AX(z) which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in Cn. The motivation for defining the invariant AX(z), referred to as the multitype of z, comes from the study of the boundary regularity properties of solutions of the d-Neumann problem on domains of finite type. By a domain of finite type, we mean one that satisfies the definition given by D'Angelo [5], namely a domain such that at any boundary point the maximum order of contact of one-dimensional complex-analytic varieties with the boundary is bounded. In [4] the author showed how one can give a proof of global regularity of the a-Neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper. In a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains. Before giving the definition of the multitype AX(z), we must introduce some

David W. Catlin

Papers

The Bergman Kernel and a Theorem of Tian

Subelliptic estimates for the $\overline\partial$-Neumann problem on pseudoconvex domains

Estimates of invariant metrics on pseudoconvex domains of dimension two

Necessary conditions for subellipticity of the $\overline\partial$-Neumann problem

Boundary invariants of pseudoconvex domains