Explicit complete Ricci-flat metrics and K\"{a}hler-Ricci solitons on direct sum bundles

Abstract

Let $B$ be a K\"ahler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding K\"ahler-Ricci solitons on the total space $M$, ${\rm dim}_{\mathbb{C}} M = n$ of certain vector bundles $E \to B$, composed of direct sums of powers of $L$. We employ the theory of hamiltonian 2-forms [2, 3] as an Ansatz, thus generalizing recent work of the author and Apostolov on $\mathbb{C}^n$ [5], as well as that of Cao, Koiso, Feldman-Ilmanen-Knopf, Futaki-Wang, and Chi Li [10, 26, 23, 24, 30] when $E$ has Calabi symmetry. As a result, we obtain new examples of asymptotically conical K\"ahler shrinkers, Calabi-Yau metrics with ALF-like volume growth, and steady solitons with volume growth $R^{\frac{4n-2}{3}}$.