On some integral invariants on complex manifolds. I
type: Item
instance of: scholarly article
title: On some integral invariants on complex manifolds. I
zbMATH Open document ID: 0592.32022
author: S. Bando
author: Toshiki Mabuchi
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: Let \(X\) be a compact complex \(r\)-dimensional manifold and \(G\) (\(=Aut(X))\) a group of all holomorphic automorphisms of \(X\) with \(G^ 0\) \((=Aut^ 0(X))\) as its identity component. Denoting by \({\mathcal V}_ X\) the set of all normalized volume forms \(\Omega\) on X, we associate to each pair \((\Omega',\Omega'')\in {\mathcal V}_ X\times {\mathcal V}_ X\) the real number \[ N_ X(\Omega',\Omega'')= \int^{b}_{a}dt \int_{X}\{(\sqrt{-1}/2\pi)\partial {\bar\partial} \log(\Omega_ t)\}^ r (\partial\Omega_ t/\partial t)/\Omega_ t, \] where \(\{\Omega_ t:\quad a\leq t\leq b\}\) is an arbitrary piecewise smooth path in \({\mathcal V}_ X\) such that \(\Omega_ a=\Omega'\), \(\Omega_ b=\Omega''\). This number can be proven to be invariant by \(G\) (the proof made in terms of the anti-canonical bundle \(K_ X^{-1}\) of \(X\) is due to \textit{S. K. Donaldson}, Proc. Lond. Math. Soc., III. Ser. 50, 1-26 (1985; Zbl 0529.53018)). Setting \(\nu_ X(\Omega)=N_ X(\Omega_ 0,\Omega),\) if we set \(n_ X(g)=\exp (\nu_ X(g^*\Omega_ 0))\) for \(g\in G\), then \(n_ X: G\to R_+\) is a Lie group homomorphism. Also, let \(\lambda=c_ 1(X)^ r[X]\). Then \(\Omega\in {\mathcal V}_ X\) is a critical point of \(\nu_ X\) if and only if \((\sqrt{-1}/2\pi) \partial{\bar\partial}\log(\Omega)\) is an Einstein form, i.e. \(\{(\sqrt{-1}/2\pi) \partial{\bar\partial} \log(\Omega)\}^ r= \lambda\Omega.\) We now assume that X is a compact connected r-dimensional Kählerian manifold with normalized Kähler form \(\omega_ 0\), and put \({\mathcal K}=\)\{K''ahler forms on X cohomologous to \(\omega_ 0\}\), \(\omega_ 0(\psi)=\omega_ 0+\sqrt{-1} \partial{\bar\partial}\psi\) with \(\psi \in C^{\infty}(X)_ R\), and \({\mathcal H}=\{\psi \in C^{\infty}(X)_ R| \quad \omega_ 0(\psi)\in {\mathcal K}\}.\) For each holomorphic vector field \(v\in \Gamma(X,{\mathcal O}(TX))\) on \(X\), we denote by \(L_ v\) (resp. \(\nabla_ v)\) the Lie derivative (resp. covariant derivative in terms of the Kähler metric \(\omega_ 0)\) w…
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3950986
zbMATH Keywords: complex Lie groups
zbMATH Keywords: automorphism group of complex
zbMATH Keywords: spaces
zbMATH Keywords: integral invariants on complex manifolds
zbMATH Keywords: volume forms
zbMATH Keywords: Kählerian manifold
zbMATH Keywords: critical point
zbMATH Keywords: Einstein-Kähler form
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
full work available at URL: pjaa.62.197
OpenAlex ID: W2024816500
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DOI: 10.3792/PJAA.62.197
description: scientific article; zbMATH DE number 3950986