On geometry of steady toric Kähler-Ricci solitons

Abstract

In this paper we study the gradient steady Kähler-Ricci soliton metrics on non-compact toric manifolds. We show that the orbit space of the free locus of such a manifold carries a natural Hessian structure with a nonnegative Bakry-Émery tensor. We generalize Calabi's classical rigidity result and use this to prove that any complete ${\mathbf{T}}^n$-invariant gradient steady Kähler-Ricci soliton with a free torus action must be a flat $({\mathbb{C}}^{\ast }{)}^n$.

arXiv comment20 pages, minor corrections; new references added