Asymptotically cylindrical steady Kähler-Ricci solitons

Abstract

Let $D$ be a compact Kähler 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 Kähler-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 $\mathbb{R}\times ({\mathbb{S}}^1\times D)/\Gamma$.

64 pages; results unchanged, but details added to computations in Section 5

References in corpus (2)

• Entropy and heat kernel bounds on a Ricci flow background
• Invariant metrics on negatively pinched complete Kähler manifolds