Classification of gradient Kahler-Ricci solitons with vanishing B-tensor

Inspired by the Bach tensor on Riemannian manifolds, we introduce the B-tensor (B-(ij) over bar:=n+2/n del((k) over bar)del W-l((ij) over bark (l) over bar)-W(ij) over bark (l) over barR(k) over barl) on Kahler manifolds. We prove that a compact gradient kl Kahler-Ricci soliton with vanishing B-tensor is Kahler-Einstein. Moreover, we show that a complete non-compact extremal gradient shrinking Kahler-Ricci soliton with vanishing B-tensor is Kahler-Einstein. (C) 2019 Elsevier B.V. All rights reserved.