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Constant Scalar Curvature Kähler Metrics on Resolutions of an Orbifold Singularity of Depth 1

Mehrdad Najafpour

TL;DR

This work extends the construction of constant scalar curvature Kähler metrics to resolutions of orbifold singularities of depth one with type $\mathcal{I}$ along a codimension $k>2$ submanifold $Y$. By gluing a cscK orbifold metric to ALE scalar-flat Kähler model metrics on the exceptional divisor, and working within the Melrose manifolds with corners framework, the authors prove that the resolved space $\widehat{X}$ admits a cscK metric in the class $[\omega_X]-\varepsilon^2[E]$ for small $\varepsilon$. The analysis splits into linear and nonlinear parts: uniform invertibility of a twisted Lichnerowicz operator on weighted Hölder spaces, followed by a Banach fixed-point argument to solve the nonlinear scalar curvature equation, with a precise topological expansion $S(\widetilde{\omega}_\varepsilon)=S(\omega_X)+\lambda\varepsilon^{2k-2}+R_\varepsilon$. The method also yields that, for a sequence of finite weighted blow-ups leading to a smooth model, one can iteratively apply the main result to obtain cscK metrics on the final smooth resolution. Collectively, the paper generalizes Arezzo–Pacard–Singer and Seyyedali–Székelyhidi’s desingularization results to orbifolds with singularities of type $\mathcal{I}$ and demonstrates a robust gluing schema in a singular setting with a detailed topological control of the scalar curvature.

Abstract

We construct new examples of constant scalar curvature Kähler metrics on suitable resolutions of certain constant scalar curvature Kähler orbifolds with type I singularities, in the sense of Apostolov--Rollin, along a suborbifold of complex codimension greater than 2.

Constant Scalar Curvature Kähler Metrics on Resolutions of an Orbifold Singularity of Depth 1

TL;DR

This work extends the construction of constant scalar curvature Kähler metrics to resolutions of orbifold singularities of depth one with type along a codimension submanifold . By gluing a cscK orbifold metric to ALE scalar-flat Kähler model metrics on the exceptional divisor, and working within the Melrose manifolds with corners framework, the authors prove that the resolved space admits a cscK metric in the class for small . The analysis splits into linear and nonlinear parts: uniform invertibility of a twisted Lichnerowicz operator on weighted Hölder spaces, followed by a Banach fixed-point argument to solve the nonlinear scalar curvature equation, with a precise topological expansion . The method also yields that, for a sequence of finite weighted blow-ups leading to a smooth model, one can iteratively apply the main result to obtain cscK metrics on the final smooth resolution. Collectively, the paper generalizes Arezzo–Pacard–Singer and Seyyedali–Székelyhidi’s desingularization results to orbifolds with singularities of type and demonstrates a robust gluing schema in a singular setting with a detailed topological control of the scalar curvature.

Abstract

We construct new examples of constant scalar curvature Kähler metrics on suitable resolutions of certain constant scalar curvature Kähler orbifolds with type I singularities, in the sense of Apostolov--Rollin, along a suborbifold of complex codimension greater than 2.

Paper Structure

This paper contains 9 sections, 45 theorems, 249 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Weighted projective spaces and their singularities
  3. Manifolds with corners and Lie structures at infinity
  4. Blow-up in Melrose Sense
  5. Constant Scalar Curvature Kähler metrics
  6. An example of orbifold with singularities of type $\mathcal{I}$
  7. Gluing Technique
  8. Linear analysis
  9. Nonlinear analysis and the main Theorem

Key Result

Theorem 1

Let $(X,\omega_X)$ be a compact cscK complex manifold with discrete group of automorphisms (in particular, there are no non-trivial holomorphic vector fields on $X$) and $Y\subset X$ be a submanifold of codimension $k$ greater than 2. Then $\mathop{\mathrm{Bl}}\nolimits_Y^X$ admits a cscK metric in

Figures (7)

  • Figure 1: Blowing-up the origin in $\mathbb{R}^2$
  • Figure 2: Blowing-up the origin in $\mathbb{R}\times [0, \infty)_\varepsilon$
  • Figure 3: A manifold with a cylindrical end
  • Figure 4: A manifold with a conical end
  • Figure 5: Orbifold with corner $X\times [0,\infty)_{\varepsilon}$
  • ...and 2 more figures

Theorems & Definitions (137)

  • Theorem 1: seyyedali2020extremal
  • Theorem 2: Theorem \ref{['themain']}
  • Corollary 3
  • Definition 2.1: Compact weighted projective space
  • Definition 2.2: Non-compact weighted projective space
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 127 more