On some integral invariants on complex manifolds. I
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 \(\omega_ 0)\) with respect to \(v\), and \(c_ i(\omega)\), (\(\omega\in {\mathcal K})\) by the i-th Chern form on X. Then Futaki invariants of X are the linear map \(F<c_ 1^{r+1}>: \Gamma(X,{\mathcal O}(TX))\to R\) defined by \(F<c_ 1^{r+1}>(v)=2 Re\int_{X}T_ r(L_ v-\nabla_ v)c_ 1(\omega_ 0)^ r,\) where \(T_ r(L_ v-\nabla_ v)\in C^{\infty}(X)\) is the trace of the \(C^{\infty}\) section \(L_ v-\nabla_ v\) of the vector bundle End (Tx) over X. It can then be easily shown that \(-F<c_ 1^{r+1}>\) is a Lie group homomorphism \(n_ X: G\to R_+,\) and that \({\tilde \omega}\in {\mathcal K}\) is a critical point for the functional \({\mathcal K}\ni \omega \to \nu_ X(\omega^ r)\in R\) if and only if \({\tilde\omega}\) is an Einstein-Kähler form.
integral invariants on complex manifolds
scientific article; zbMATH DE number 3950986