[Paper Review] On a conjecture of H. Wu

Robert Treger|arXiv (Cornell University)|Mar 3, 2015

Geometry and complex manifolds1 references3 citations

TL;DR

This paper proves a conjecture by H. Wu that the universal cover of a compact Kähler manifold with negative sectional curvature is biholomorphic to a bounded domain in ℂⁿ, under the assumption that the fundamental group is residually finite. Using results from Kähler geometry, hyperbolicity, and Moishezon theory, the author establishes that such manifolds are projective and applies a theorem from [4] to conclude the universal cover is a bounded domain in complex n-space.

ABSTRACT

Let X be a compact Kahler manifold with negative sectional curvature and residually finite fundamental group. Then its universal covering is a bounded domain in an affine space.

Motivation & Objective

• To resolve a conjecture by H. Wu on the universal cover of compact Kähler manifolds with negative sectional curvature.
• To determine under what conditions the universal cover is a bounded domain in ℂⁿ.
• To establish the projectivity of compact Kähler manifolds with negative curvature and residually finite fundamental group.
• To connect geometric properties (curvature, hyperbolicity) with complex-analytic and algebraic properties (Stein, Moishezon, ampleness of canonical bundle).

Proposed method

• Apply the theorem from reference [4] to the universal cover of the manifold under study.
• Use Wu's result that the universal cover of a compact Kähler manifold with negative curvature is Stein, implying the fundamental group is large.
• Leverage Ballmann and Eberlein's result that such fundamental groups are nonamenable.
• Establish that the manifold is Kahler hyperbolic using Gromov's theory.
• Show that the canonical bundle is quasiample, implying the manifold is Moishezon.
• Conclude projectivity via Kodaira's criterion, as negative Ricci curvature implies ampleness of the canonical bundle.

Experimental results

Research questions

• RQ1 Under what conditions is the universal cover of a compact Kähler manifold with negative sectional curvature biholomorphic to a bounded domain in ℂⁿ?
• RQ2 How does residual finiteness of the fundamental group affect the geometry of the universal cover?
• RQ3 Can Kahler hyperbolicity and the ampleness of the canonical bundle be used to deduce projectivity in this setting?
• RQ4 What is the relationship between the Stein property of the universal cover and the largeness of the fundamental group?
• RQ5 How do curvature conditions influence the algebraic structure of compact Kähler manifolds?

Key findings

• The universal cover of a compact Kähler manifold with negative sectional curvature is a bounded domain in ℂⁿ if the fundamental group is residually finite.
• The manifold is Kahler hyperbolic, as established by Gromov's theory.
• The canonical bundle is quasiample, implying the manifold is Moishezon.
• The negative Ricci curvature implies ampleness of the canonical bundle, so the manifold is projective by Kodaira's criterion.
• The fundamental group is nonamenable, as shown by Ballmann and Eberlein.
• The result follows from applying the main theorem in reference [4] to the projective, Kahler hyperbolic manifold.