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A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. (English) Zbl 1036.53053
A Kähler metric $$g$$ on a compact complex manifold $$M$$ is called a Kähler-Ricci soliton if there is a holomorphic vector field $$X$$ on $$M$$ such that the Kähler form $$\omega_g$$ of $$g$$ satisfies $$\text{Ric} (\omega_g)-\omega_g=L_{X} \omega_g$$, where $$L_X$$ denotes the Lie derivative along $$X$$. In particular, if $$X=0$$, $$g$$ is a Kähler-Einstein metric.
In this paper, a new holomorphic invariant $$F_{X}(.)$$ is introduced for any compact Kähler manifold with positive first Chern class and for any holomorphic vector field $$X$$. This invariant generalizes the Futaki invariant; if $$X=0$$, the invariant is the Futaki invariant.
The authors show that $$F_{X}(.)$$ is an obstruction to the existence of Kähler-Ricci solitons just as the Futaki invariant is an obstruction to the existence of Kähler-Einstein metrics. Moreover, they consider a class of the compactifications of $$C^*$$-bundles over compact Kähler-Einstein manifolds, and prove that the vanishing of the holomorphic invariant is equivalent to the existence of Kähler-Ricci solitons on these manifolds.
Furthermore, the authors solve completely the uniqueness problem of Kähler-Ricci solitons proving that there is at most one Kähler-Ricci soliton on a compact complex manifold $$M$$ modulo the identity component $$\text{Aut}^0(M)$$ of the holomorphic automorphisms group $$\text{Aut}(M)$$ of $$M$$. Such a result extends Bando and Mabuchi’s theorem on the uniqueness of Kähler-Einstein metrics on complex manifolds with positive first Chern class.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32J15 Compact complex surfaces 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 58E11 Critical metrics
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