Asymptotically cylindrical steady K\"ahler-Ricci solitons

Abstract

Let $D$ be a compact K\"ahler manifold with trivial canonical bundle and $\Gamma$ be a finite cyclical group of order $m$ acting on $\mathbb{C} \times D$ by biholomorphisms, where the action on the first factor is generated by rotation of angle $2\pi /m$. Furthermore, suppose that $\Omega_D$ is a trivialisation of the canonical bundle such that $\Gamma$ preserves the holomorphic form $dz \wedge \Omega_D$ on $\mathbb C \times D$, with $z$ denoting the coordinate on $\mathbb{C}$. The main result of this article is the construction of new examples of gradient steady K\"ahler-Ricci solitons on certain crepant resolutions of the orbifolds $\left( \mathbb{C}\times D \right) / \Gamma$. These new solitons converge exponentially to a Ricci-flat cylinder $\mathbb{R} \times(\mathbb{S}^1 \times D) / \Gamma$.