Kähler-Einstein metrics and the generalized Futaki invariant
The Futaki invariant is defined for any compact Kähler manifold \(X\) with positive first Chern class \(C_ 1(X)\) that has nontrivial holomorphic vector fields. Until now, all known nontrivial obstructions to Kähler-Einstein metrics come from holomorphic vector fields. In this paper, the authors use the jumping of complex structures to produce new obstructions to the existence of Kähler-Einstein (or orbifold) metrics. Kodaira's embedding Theorem implies that \(X\) can be considered as a submanifold in \(\mathbb{C}\mathbb{P}^ N\). Let \(\sigma_ t\) be a one-parameter subgroup in \(\Aut(\mathbb{C}\mathbb{P}^ N) = SL(N + 1,\mathbb{C})\), which is generated by the real part of a holomorphic vector field \(v\). Denote by \(X_ t\) the submanifold \(\sigma_ t(X)\) \((t\in \mathbb{R})\). \(X_ t\) converges to a subvariety \(X_ \infty\) in \(\mathbb{C}\mathbb{P}^ N\) as \(t\) tends to \(\infty\). The authors introduce the generalized Futaki invariant for a class of singular varieties. They prove that the real part of the generalized Futaki invariant \(F_{X_ \infty}(v)\) is nonnegative if \(X\) is normal and \(X\) admits a Kähler-Einstein metric. The authors find many interesting examples of Kähler orbifolds with positive first Chern class and without any holomorphic vector fields, which do not have Kähler-Einstein orbifold metrics, either. Finally, some problems are given.
scientific article; zbMATH DE number 167397