On weaker notions for Kähler-Ricci solitons
We recall that a (gradient shrinking) Kaehler Ricci soliton on a compact complex manifold \((M, J)\) is a pair \((g, \xi)\) given by a Kaehler metric \(g\) and a vector field \(\xi\) satisfying the following three conditions: (a) \(g - \operatorname{Ric}_g = {\mathcal L}_\xi g\), (b) \( {\mathcal L}_\xi J = 0\) (i.e. \(\xi\) is \(J\)-holomorphic) and (c) \( {\mathcal L}_{J \xi} \omega = 0\) with \(\omega\) Kaehler form of \((g, J)\). In this paper the author proves that if there exists a pair \((g, \xi)\) satisfying (a) and (b), then there exists also a (unique) vector field \(\widehat \xi\) such that the pair \((g, \widehat \xi)\) is a Kaehler Ricci soliton and, in particular, \((M, J)\) is Fano.
scientific article; zbMATH DE number 7942508
scientific article