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Resolutions of non-regular Ricci-flat Kahler cones

Dario Martelli, James Sparks

TL;DR

This work addresses constructing complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki–Einstein manifolds $Y^{p,k}(V)$. It develops explicit local Kähler metrics admitting a Hamiltonian two-form of order two on a positively curved Kähler–Einstein base $(V,g_V)$, reducing Ricci-flatness to tractable ODE data; the metrics extend to global, complete RC Kähler manifolds on various total spaces and fiberings. The authors obtain explicit RC Kähler metrics on total spaces of $C^2/Z_p$ bundles over $V$, orbifold fibrations over $ ext{WCP}^1$, and canonical line bundles over Fano orbifolds, including special toric cases when $V= ext{CP}^1$; they classify the allowed $(p,k)$ via the relation $ rac{pI}{2}<k<pI$ and construct both smooth and orbifold small, partial, and canonical resolutions. The results significantly extend known RC Kähler cone resolutions, provide a unifying Hamiltonian-two-form framework (generalizing Calabi’s ansatz), and have potential implications for AdS/CFT via explicit geometries asymptotic to non-regular Sasaki–Einstein cones.

Abstract

We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP^1, with generic fibres being the canonical complex cone over V, and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kahler metrics on the total spaces of (a) rank two holomorphic vector bundles over V, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base V. When V=CP^1 our results give Ricci-flat Kahler orbifold metrics on various toric partial resolutions of the cone over the Sasaki-Einstein manifolds Y^{p,q}.

Resolutions of non-regular Ricci-flat Kahler cones

TL;DR

This work addresses constructing complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki–Einstein manifolds . It develops explicit local Kähler metrics admitting a Hamiltonian two-form of order two on a positively curved Kähler–Einstein base , reducing Ricci-flatness to tractable ODE data; the metrics extend to global, complete RC Kähler manifolds on various total spaces and fiberings. The authors obtain explicit RC Kähler metrics on total spaces of bundles over , orbifold fibrations over , and canonical line bundles over Fano orbifolds, including special toric cases when ; they classify the allowed via the relation and construct both smooth and orbifold small, partial, and canonical resolutions. The results significantly extend known RC Kähler cone resolutions, provide a unifying Hamiltonian-two-form framework (generalizing Calabi’s ansatz), and have potential implications for AdS/CFT via explicit geometries asymptotic to non-regular Sasaki–Einstein cones.

Abstract

We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP^1, with generic fibres being the canonical complex cone over V, and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kahler metrics on the total spaces of (a) rank two holomorphic vector bundles over V, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base V. When V=CP^1 our results give Ricci-flat Kahler orbifold metrics on various toric partial resolutions of the cone over the Sasaki-Einstein manifolds Y^{p,q}.

Paper Structure

This paper contains 20 sections, 12 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Introduction
  3. Summary
  4. Local metrics
  5. Kähler metrics with Hamiltonian two-forms
  6. Complex structure
  7. Asymptotic structure
  8. Global analysis: $\pm x > \pm x_{\pm}$
  9. Zeroes of the metric functions
  10. Regularity for $\pm x > \pm x_{\pm}$
  11. Allowed values of $p$ and $k$
  12. Summary
  13. Small resolutions
  14. Partial resolutions I: $x_-=y_1$
  15. Smooth resolutions: $p=1$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(V,g_V)$ be a complete Kähler-Einstein manifold of positive Ricci curvature with canonical line bundle $K_V$ and Fano index $I$. Then for every $p, k\in \mathbb{N}$ positive integers with $pI/2 < k < pI$ there is an explicit complete Ricci-flat Kähler orbifold metric on the total space of a $\m where and $z_1,z_2$ are standard complex coordinates on $\mathbb{C}^2$. The metric asymptotes to a

Figures (1)

  • Figure 1: An illustration of the change of coordinates (\ref{['newnewpolar']}). Curves of constant $x$ and constant $y$ are depicted.

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 2 more