Expanding K\"ahler-Ricci solitons coming out of K\"ahler cones

We give necessary and sufficient conditions for a K"ahler equivariant resolution of a K"ahler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient K"ahler-Ricci soliton. In particular, it follows that for any $n\in {\mathbb{N}}_0$ and for any negative line bundle $L$ over a compact K"ahler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient K"ahler-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 K"ahler-Ricci solitons on ${\mathbb{C}}^n$ with positive curvature operator on $(1,1)$-forms is path-connected.