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

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arxiv 2410.23645 v2 pith:U7VTXWSK submitted 2024-10-31 math.DG

Charles Cifarelli This is my paper · ORCID
classification math.DG
keywords metricssolitonsbundlescalabi-yaucompletedirectgrowthmathbb
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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 $R^{\frac{4n-2}{3}}$.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Uniqueness of shrinking K\"ahler-Ricci solitons on resolutions of K\"ahler cones

    math.DG 2026-07 conditional novelty 7.0 of 10

    Every complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is asymptotically conical, which yields uniqueness up to pullback by biholomorphism.

  2. Toric geometry of generalized K\"ahler-Ricci solitons

    math.DG 2025-09 conditional novelty 7.0 of 10

    Toric generalized Kähler-Ricci solitons of A-type are locally equivalent to steady toric Kähler-Ricci solitons, yielding a four-dimensional classification and new complete examples.

  3. Compactness and Rigidity of Complete K\"ahler Ricci Shrinkers

    math.DG 2026-08 conditional novelty 6.0 of 10

    A new first-order visibility argument derives compactness, splitting, and Gaussian rigidity for complete Kähler-Ricci shrinkers from the weight structure of their polarized Fano fibrations.

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