[Paper Review] Gromov-Hausdorff limits of Kähler manifolds with bisectional curvature lower bound I

Gang Liu|arXiv (Cornell University)|May 28, 2015

Geometry and complex manifolds37 references3 citations

TL;DR

This paper establishes that the pointed Gromov-Hausdorff limit of a sequence of complete Kähler manifolds with a lower bound on bisectional curvature and uniformly positive volume is homeomorphic to a normal complex analytic space. The complex structure on the limit is induced by the limit of holomorphic functions, resolving a key issue in the degeneration of complex structures under curvature bounds.

ABSTRACT

Given a sequence of complete(compact or noncompact) Kähler manifolds $M^n_i$ with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of complex structure of $M_i$.

Motivation & Objective

• To understand the structure of Gromov-Hausdorff limits of Kähler manifolds under a bisectional curvature lower bound.
• To determine whether the limit space inherits a complex analytic structure despite potential metric singularities.
• To resolve the discrepancy between metric singularities (codimension 2) and complex analytic singularities (codimension ≥4) in the limit.
• To prove that the limit space is locally contractible and homeomorphic to a normal complex analytic variety.
• To lay foundational tools for studying the uniformization conjecture for noncompact Kähler manifolds with positive bisectional curvature.

Proposed method

• Adapting Cheeger-Colding-Gromov convergence theory to Kähler geometry with bisectional curvature bounds.
• Extending Ni-Tam's maximum principle for heat flow to the negatively curved case using a complex Hessian Bochner formula.
• Constructing good holomorphic coordinates near special points using localization and Hörmander's L²-estimate.
• Proving that metric singularities with tangent cones splitting off ℝ²ⁿ⁻² are regular in the complex analytic sense.
• Using a three-circle theorem for negatively curved Kähler manifolds and adapting techniques from [14] on holomorphic sections.
• Establishing a structure sheaf on the limit space via uniform convergence of holomorphic functions and normalization of local rings.

Experimental results

Research questions

• RQ1 Can the Gromov-Hausdorff limit of Kähler manifolds with bisectional curvature bounded below be given a complex analytic structure?
• RQ2 How do metric singularities in the limit relate to complex analytic singularities when bisectional curvature is bounded below?
• RQ3 Is the limit space locally contractible under these curvature and volume conditions?
• RQ4 Can holomorphic functions on the approximating manifolds be used to define a well-behaved complex structure on the limit?
• RQ5 Does the complex analytic structure on the limit space agree with the metric structure at regular points?

Key findings

• The pointed Gromov-Hausdorff limit of complete Kähler manifolds with bisectional curvature ≥−1 and uniformly positive volume is homeomorphic to a normal complex analytic space.
• The complex analytic structure on the limit is induced by the limit of holomorphic functions on small balls of the approximating manifolds.
• Metric singularities that split off ℝ²ⁿ⁻² as a tangent cone are regular in the complex analytic sense, resolving the codimension discrepancy.
• The limit space is locally contractible, extending Perelman’s result to the Kähler setting with bisectional curvature bounds.
• For complete noncompact Kähler surfaces with positive bisectional curvature and maximal volume growth, the manifold is simply connected.
• The structure sheaf on the limit space is well-defined and normal, with local holomorphic coordinates induced from the limit of holomorphic functions.