Geometry of anti-self-dual connections and Kuranishi map
type: Item
instance of: scholarly article
title: Geometry of anti-self-dual connections and Kuranishi map
zbMATH Open document ID: 0619.53042
author: Mitsuhiro Itoh
published in: Journal of the Mathematical Society of Japan
publication date: 1988-01-01T00:00:00Z
review text: A Riemannian structure is investigated on the moduli space of anti-self- dual Yang-Mills connections. The moduli space is a finite dimensional subspace of the orbit space of all connections. The gauge invariant \(L_ 2\)-inner product produces a Riemannian metric there. By the crucial aid of the Kuranishi map, a map linearizing the space and the Hodge theory on the Atiyah-Hitchin-Singer complex the Riemannian curvature can be expressed in terms of the \(L_ 2\)-inner product together with the Green operators, the inverses of the Laplacians. As a consequence, over a Kähler surface the moduli space carries a Kähler structure compatible with the canonically defined complex structure. Moreover it is verified that the moduli space of Einstein- Hermitian connections over a Riemannian surface is endowed with a Kähler structure of nonnegative holomorphic curvature and hence of positive scalar curvature. This positivity is consistent with the unirationality of the moduli spaces.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 4005096
zbMATH Keywords: anti-self-dual Yang-Mills connections
zbMATH Keywords: Kuranishi map
zbMATH Keywords: Atiyah-Hitchin- Singer complex
zbMATH Keywords: Kähler surface
zbMATH Keywords: moduli space
zbMATH Keywords: Einstein-Hermitian connections
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04010009
OpenAlex ID: W1975985060
DOI: 10.2969/JMSJ/04010009
description: scientific article; zbMATH DE number 4005096