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

Abstract

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