On steady Kähler-Ricci solitons

Abstract

In this thesis we study the existence and uniqueness of steady Kähler-Ricci solitons. We consider two classes of manifolds on which we obtain new examples of steady solitons by using different methods for each class. In the first part we focus on suitable vector bundles over Kähler manifolds whose Ricci curvature has constant eigenvalues. This condition reduces the soliton equation to an ODE, which we then solve to find new examples. Moreover, we show that these new steady Kähler-Ricci solitons are unique if the Kähler class, the vector field and the asymptotic behavior is fixed. In the second part we consider certain crepant resolutions of orbifolds (C × D)/Γ for some finite group Γ which acts by rotation on complex plane C and preserves a holomorphic volume form on the product C × D. To construct new steady Kähler-Ricci solitons on the resolution we use PDE methods for complex Monge-Ampère equations. The solitons obtained this way are asymptotic to a Ricci-flat cylinder.