Bando Futaki invariants and Kähler Einstein metric
Narasiham-Seshadri, Donaldson, Uhlenbeck-Yau proved that irreducible holomorphic vector bundles over a Kähler manifold \(X\) are Mumford stable if and only if there exists a Hermitian Einstein metric on \(X\). On a compact Einstein manifold \(M\), the almost Kähler-Einstein metrics are the solutions of: \(\text{Tr}(\omega+ {i\eta\over 2\pi}F)^n= \chi_\eta {\omega^n \over (n-1)!}\), where \(n=\dim M\), \(\eta>0\), \(\omega\) is the Kähler form and \(F\) is the full curvature tensor of \(M\). When \(M\) has positive first Chern class, the existence of almost Kähler-Einstein metrics is shown here to be equivalent to the vanishing of Futaki-Bando invariants [\textit{S. Bando}, ``An obstruction for Chern class forms to be harmonic'', \textit{A. Futaki}, Invent. Math. 73, 437-443 (1983; Zbl 0514.53054)]. At the end, these invariants are related with the supertrace of the heat kernel.
scientific article; zbMATH DE number 1267477