HAMILTONIAN 2-FORMS AND NEW EXPLICIT CALABI–YAU METRICS AND GRADIENT STEADY KÄHLER–RICCI SOLITONS ON Cⁿ

For each partition of the positive integer (Formula presented) , where ℓ ≥ 1 and d_j ≥ 0 are integers, we construct a continuous (ℓ − 1)-parameter family of explicit complete gradient steady Kähler–Ricci solitons on Cⁿ admitting a hamiltonian 2-form of order ℓ and symmetry group U(d₁ +1)×···×U(d_ℓ+1). For ℓ = 1, d₁ = n−1 we obtain Cao’s example [17] whereas for other partitions the metrics are new. Furthermore, when n = 2, ℓ = 2, d₁ = d₂ = 0 we obtain complete gradient steady Kähler–Ricci solitons on C² which have positive sectional curvature but are not isometric to Cao’s 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 Cⁿ containing higher dimensional extensions of the Taub-NUT Ricci-flat Kähler metric on C². When n ≥ 3, the complete Ricci-flat Kähler metrics, and when n ≥ 2, their deformations to complete gradient steady Kähler–Ricci solitons seem not to have been observed before our work.