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 $c_{1}(K_{D}\otimes(L^{*})^{\otimes (n+1)})>0$. 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.