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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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