On rigidity of gradient Kähler-Ricci solitons with harmonic Bochner tensor

Abstract

In this paper, we prove that complete gradient steady Kähler-Ricci solitons with harmonic Bochner tensor are necessarily Kähler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) Kähler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times {\mathbb{C}}^{n-k}$, where $N$ is a Kähler-Einstein manifold with positive (or negative) scalar curvature.

minor errors corrected