Dolbeault cohomology of a loop space
type: Item
instance of: scholarly article
title: Dolbeault cohomology of a loop space
zbMATH Open document ID: 1087.58005
DOI: 10.1007/BF02392565
author: László Lempert
author: Ning Zhang
published in: Acta Mathematica
publication date: 2005-06-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0403405
full work available at URL: fulltext.htm
review text: This paper studies complex analysis on loop spaces. The idea is that the study of Dolbeault cohomology of the loop space \(LM\) of a complex manifold \(M\) will give information about the manifold. Here the loop spaces \(L\mathbb{P}_1\) of the Riemann sphere \(\mathbb{P}_1\) are considered; with different requirements of smoothness \(C^k\) or Sobolev class \(W^{k,p}\) of the loops. The Dolbeault cohomology groups \(H^{0,1}(L\mathbb{P}_1)\) of these spaces are computed. Let \(\mathcal{F}\) be the space of holomorphic functions \(F:\mathbb{C}\times L\mathbb{C}\to\mathbb{C}\) which satisfy (1) \(F(\frac{\zeta}{\lambda},\lambda^2y)=O(\lambda^2)\) as \(\mathbb{C}\ni\lambda\to 0\); (2) \(F(\zeta,x+y)=F(\zeta,x)+F(\zeta,y)\) if \(\text{supp}(x)\cap \text{supp}(y)=\emptyset\); (3) \(F(\zeta,y+const)=F(\zeta,y)\). Theorem 0.1 states \(H^{0,1}(L\mathbb{P}_1)\approx \mathbb{C}\oplus\mathcal{F}\). The isomorphism is topological, and equivariant with respect to the obvious actions of the \(C^k\) diffeomorphism group of \(S^1\). The topology on the cohomology groups is induced by the compact-\(C^{\infty}\) topology on the space of smooth \((r,p)\)-forms. The cohomology groups \(H^{0,1}(L\mathbb{P}_1)\) are acted on by the group \(G=\text{PSL}(2,\mathbb{C})\) of holomorphic automorphisms of \(\mathbb{P}_1\) and theorems 0.2--0.4 are on the structure of \(H^{0,1}(L\mathbb{P}_1)\) as a \(G\)-module. In section 2, an action of \(G\) on \(\mathcal{F}\) is introduced, making \(\mathcal{F}\) a holomorphic \(G\)-module. The \(n\)-th isotypical subspace \(\mathcal{F}^n\) is the subspace of \((n+1)\)-posthomogeneous functions. Theorem 0.2 calculates \(H^{0,1}(L_{\infty}\mathbb{P}_1)\): If \(n\geq 1\), the \(n\)-th isotypical subspace of \(H^{0,1}(L_{\infty}\mathbb{P}_1)\) is isomorphic to the space \(\mathcal{F}^n\), which is spanned by functions of the form \(F(\zeta,y)=\zeta^{\nu} \langle\Phi, \prod_{j=0}^{n}y^{(d_j)}\rangle\), where \(\Phi\) is a distribution on \(S^1\), \(y^d\) is the …
reviewed by: Lisbeth Fajstrup
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 58B12
Mathematics Subject Classification ID: 32W05
zbMATH DE Number: 2183719
zbMATH Keywords: Dolbeault cohomology
zbMATH Keywords: loop spaces
MaRDI profile type: Publication
arXiv ID: math/0403405
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OpenAlex ID: W1966119856
description: scientific article