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The sign of scalar curvature on Kähler blowups

Garrett M. Brown

TL;DR

The paper proves that blowing up a point on any compact Kähler manifold $(M,\omega)$ of complex dimension $n\ge 2$ yields a family of Kähler metrics on $Bl_pM$ whose scalar curvature $S(\omega_i)$ can be made arbitrarily close to $S(\omega)$ and whose cohomology classes converge to the pullback of $[\omega]$. The key method is a gluing construction: join the original metric away from the blown-up point with a scaled Burns–Simanca metric near the exceptional divisor, then solve a scalar-curvature equation by inverting the adjoint of the linearized scalar curvature operator in suitable weighted Hölder spaces, using a contraction mapping. In dimension two this requires a refined modification of the gluing data, yielding uniform estimates that lead to a complete proof. The results imply that if $M$ admits a positively curved Kähler metric, so do all its blowups, and they complete a LeBrun-type classification for positive scalar curvature Kähler surfaces, tying geometric positivity to Kodaira dimension $-\,\infty$ and the blowup structure of $M$.

Abstract

We show that if $(M,ω)$ is any compact Kähler manifold, then the blowup of $M$ at any point furnishes a Kähler metric with scalar curvature globally and arbitrarily $C^0$-close to the scalar curvature of $ω$. It follows that if $M$ admits a positive scalar curvature Kähler metric, then so do all of its blowups. This special case extends a result of N. Hitchin to surfaces and answers a conjecture of C. LeBrun in the affirmative, consequently completing the classification of positive scalar curvature Kähler surfaces as being precisely those of negative Kodaira dimension (i.e. blowups of either the projective plane or a holomorphic bundle of projective lines over a Riemann surface).

The sign of scalar curvature on Kähler blowups

TL;DR

The paper proves that blowing up a point on any compact Kähler manifold of complex dimension yields a family of Kähler metrics on whose scalar curvature can be made arbitrarily close to and whose cohomology classes converge to the pullback of . The key method is a gluing construction: join the original metric away from the blown-up point with a scaled Burns–Simanca metric near the exceptional divisor, then solve a scalar-curvature equation by inverting the adjoint of the linearized scalar curvature operator in suitable weighted Hölder spaces, using a contraction mapping. In dimension two this requires a refined modification of the gluing data, yielding uniform estimates that lead to a complete proof. The results imply that if admits a positively curved Kähler metric, so do all its blowups, and they complete a LeBrun-type classification for positive scalar curvature Kähler surfaces, tying geometric positivity to Kodaira dimension and the blowup structure of .

Abstract

We show that if is any compact Kähler manifold, then the blowup of at any point furnishes a Kähler metric with scalar curvature globally and arbitrarily -close to the scalar curvature of . It follows that if admits a positive scalar curvature Kähler metric, then so do all of its blowups. This special case extends a result of N. Hitchin to surfaces and answers a conjecture of C. LeBrun in the affirmative, consequently completing the classification of positive scalar curvature Kähler surfaces as being precisely those of negative Kodaira dimension (i.e. blowups of either the projective plane or a holomorphic bundle of projective lines over a Riemann surface).
Paper Structure (16 sections, 21 theorems, 165 equations)

This paper contains 16 sections, 21 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Organization
  4. Conventions
  5. Acknowledgements
  6. The Metric $\omega_\epsilon$ on the Blowup
  7. Weighted Hölder Spaces
  8. Estimates on the Adjoint of the Linearized Operator
  9. An Improvement on $\omega_\epsilon$ when $n = 2$
  10. Proof of the Main Theorem
  11. The Equation for $n > 2$
  12. The Equation for $n = 2$
  13. Proof of Theorem \ref{['mainthm']}
  14. Positive Scalar Curvature Kähler Surfaces and Further Discussion
  15. Positive Scalar Curvature Kähler Surfaces
  16. ...and 1 more sections

Key Result

Theorem 1

Let $(M,\omega)$ be a compact Kähler manifold of complex dimension $n \geq 2$ and scalar curvature $S(\omega)$. For any $p \in M$, there exists a sequence $\omega_i$ of Kähler metrics on $Bl_pM$ with whose Kähler classes $[\omega_i] \in H^{1,1}(M)$ converge to $\pi^*[\omega]$ as $i \rightarrow \infty$ (we implicitly pull $S(\omega)$ back along $\pi$).

Theorems & Definitions (40)

  • Theorem 1: Main Theorem
  • Theorem 2
  • Definition 3.1
  • Proposition 3.4
  • Definition 3.5
  • Definition 3.6
  • Lemma 3.7
  • proof
  • Lemma 4.1
  • proof
  • ...and 30 more