K-energy maps integrating Futaki invariants
Let V denote a compact complex connected manifold with \(C_ 1(V)>0\). Fix a Kähler form \(\omega_ 0=\sqrt{-1} \sum \quad g_{\alpha {\bar \beta}} dz^{\alpha} dz^{{\bar \beta}}\) on V. It is assumed for simplicity that \(\omega_ 0\) represents \(2\pi c_ 1(V)_ R\). Let K denote the set of all Kähler forms on V cohomologous to \(\omega_ 0\). The aim of the paper is to prove the following Theorem: There exists a mapping \(\mu\) : \(K\to R\) satisfying the following conditions: (1) An element \(\omega\) of K is a critical point of \(\mu\) if and only if \(\omega\) is an Einstein Kähler form. (ii) Let Y be a holomorphic vector field on V, and \(\omega\) be an element of K. Put \(Y_ R:=Y+\bar Y\) and \(y_ t: \exp / tY_ R\) for \(t\in R\). Then \(\mu (y^*_ t\omega)\) is a linear function in t. Namely for every t, \[ (d/dt) \mu (y^*_ t \omega)=\int_{V}(Y_ Rf_ 0)\omega^ n_ 0/\int_{V} \omega^ n_ 0, \] where the right-hand side is the Futaki invariant of V corresponding to the holomorphic vector field Y. (iii) If \(\omega\) is a critical point of \(\mu\), then the inequality \[ (d^ 2/dt^ 2) \mu (\theta_ t)|_{t=0}\quad >0 \] holds for every smooth path \(\{\theta_ t| \quad -\epsilon \leq t\leq \epsilon \}\) in K such that \(\theta_ 0=\omega\). This map \(\mu\) : \(K\to R\) is called the K-energy map of the Kähler manifold \((V,\omega_ 0)\). Several consequences of this theorem are investigated.
scientific article; zbMATH DE number 4005094
scientific article