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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.

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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