Twisted Moduli Spaces and Duistermaat-Heckman Measures

Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in Kähler geometry. These functionals are defined on the space of test configurations, and the Donaldson-Futaki invariant is in…

偏 Okounkov 体与非 Archimedees 度量的 Duistermaat–Heckman 测度

Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics

本文构造光滑射影簇上 Hermite 伪有效线丛的偏 Okounkov 体，证明其为度量奇性的普适不变量，并据此推广 Boucksom–Chen 定理，建立非 Archimedees 有限能量度量的 Duistermaat–Heckman 测度。

Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N x N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard "supersymmetry" approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard-Stratonovich transformation in the "bosonic" sector. The method, suggested recently by one of us, is shown to be capable of calculation when reinforced with a generalization of the Itzykson-Zuber integral to a non…

费米子Dyson展开与环空间上的随机Duistermaat-Heckman局部化

Fermionic Dyson expansions and stochastic Duistermaat-Heckman localization on loop spaces

本文利用费米子演算构造扩大希尔伯特空间及解析半群，系统研究非交换几何中出现的迭代算子积分，并由此导出协变拉普拉斯情形下的随机表示，最终得到自旋流形环空间上Duistermaat-Heckman局部化公式的随机精细化。

Cobordism theory and localization formulas for Hamiltonian group actions

We announce the following result and give several applications: A Hamiltonian $T$-space (for $T$ a torus) with isolated fixed points is cobordant to a disjoint union of weighted projective spaces which are constructed from its fixed point data. The applications concern the Duistermaat-Heckman formula, the topological Jeffrey-Kirwan localization theorem, and geometric quantization.

Measures on differentiable stacks

We introduce and study measures and densities (= geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger's approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, a Stokes formula which provides reinterpretations in terms of (Ruelle-Sullivan type) algebroid currents, and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space, we provide explicit (Weyl-type) formulas that shed light on Weinstein's notion of volumes of differentiable sta…

奇异辛约化空间的动量映射可构造性与线性变化

Constructibility of momentum maps and linear variation for singular symplectic reduced spaces

本文证明紧群真动量映射横截像的积分仿射分层与局部平凡纤维化，据此把 Duistermaat–Heckman 线性变化定理推广至奇异约化值，并扩展到真拟辛群胚作用。

The Equivariant cohomology theory of twisted generalized complex manifolds

It has been shown recently by Kapustin and Tomasiello that the mathematical notion of Hamiltonian actions on twisted generalized Kähler manifolds is in perfect agreement with the physical notion of general $(2,2)$ gauged sigma models with three-form fluxes. In this article, we study the twisted equivariant cohomology theory of Hamiltonian actions on $H$-twisted generalized complex manifolds. If the manifold satisfies the $\bar{\partial}\partial$-lemma, we establish the equivariant formality theorem. If in addition, the manifold satisfies the generalized Kähler condition, we prove the Kirwan in…

The symplectic geometry of closed equilateral random walks in 3-space

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^3$. We study closed random walks using the symplectic geometry of the $(2n-6)$-dimensional quotient of the manifold of polygons by the action of the rotation group $\operatorname {SO}(3)$. The basic objects of study are the moment maps on equilateral random polygon space given by t…