Hamiltonian $2$-forms and new explicit Calabi--Yau metrics and gradient steady Kähler--Ricci solitons on ${\mathbb{C}}^n$

For each partition of the positive integer $n=\mathrm{\ell }+{\sum }_{j=1}^{\mathrm{\ell }}{d}_j$, where $\mathrm{\ell }\ge 1$ and $d_j\ge 0$ are integers, we construct a continuous $(\mathrm{\ell }-1)$-parameter family of explicit complete gradient steady Kähler–Ricci solitons on ${\mathbb{C}}^n$ admitting a hamiltonian $2$-form of order $\mathrm{\ell }$ and symmetry group $\mathrm{U}(d_1+1)\times \cdots \times \mathrm{U}(d_{\mathrm{\ell }}+1)$. For $\mathrm{\ell }=1,d_1=n-1$ we obtain Cao’s example [17] whereas for other partitions the metrics are new. Furthermore, when $n=2,\mathrm{\ell }=2,d_1=d_2=0$ we obtain complete gradient steady Kähler–Ricci solitons on ${\mathbb{C}}^2$ which have positive sectional curvature but are not isometric to Cao’s $\mathrm{U}(2)$-invariant example. This disproves a conjecture by Cao. We also present a construction yielding explicit families of complete gradient steady Kähler-Ricci solitons on ${\mathbb{C}}^n$ containing higher dimensional extensions of the Taub-NUT Ricci-flat Kähler metric on ${\mathbb{C}}^2$. When $n\ge 3$, the complete Ricci-flat Kähler metrics, and when $n\ge 2$, their deformations to complete gradient steady Kähler Ricci solitons seem not to have been observed before our work.