On rigidity of gradient K\"ahler-Ricci solitons with harmonic Bochner tensor

TL;DR

This paper proves that complete gradient K"ahler-Ricci solitons with harmonic Bochner tensor are either Calabi-Yau or are isometric to quotients involving K"ahler-Einstein manifolds, revealing rigidity properties.

Contribution

It establishes rigidity results for complete gradient K"ahler-Ricci solitons with harmonic Bochner tensor, characterizing their geometric structure.

Findings

Steady solitons are Calabi-Yau.

Shrinking/expanding solitons are quotients of products with K"ahler-Einstein manifolds.

Rigidity results constrain the geometry of such solitons.

Abstract

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