Expanding Kähler-Ricci solitons coming out of Kähler cones

Abstract

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in {\mathbb{N}}_0$ and for any negative line bundle $L$ over a compact Kähler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_1(K_D\otimes (L^{\ast }{)}^{\otimes (n+1)})>0$. This generalises the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler-Ricci solitons on ${\mathbb{C}}^n$ with positive curvature operator on $(1,1)$-forms is path-connected.

arXiv comment44 pages. Corrected several typos. Refined Theorem A and Corollary B