[Paper Review] Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds

Chi Li, Xiaowei Wang+1 more|arXiv (Cornell University)|Feb 23, 2015

Geometry and complex manifolds44 references15 citations

TL;DR

This paper establishes the quasi-projectivity of the moduli space of smooth Kähler-Einstein Fano manifolds by constructing a canonical continuous Hermitian metric on the CM line bundle over its proper compactification. The curvature of this metric is a positive (1,1)-current extending the Weil-Petersson metric, proving the CM line bundle is nef and big on the compactification and ample on the smooth locus, thus confirming Tian's conjecture on the moduli space's quasi-projectivity.

ABSTRACT

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifolds $\mathcal{M}$. As a consequence, we show that the CM line bundle is nef and big on $\bar{\mathcal{M}}$ and its restriction on $\mathcal{M}$ is ample.

Motivation & Objective

• To resolve Tian's conjecture on the quasi-projectivity of the moduli space of smooth Kähler-Einstein Fano manifolds.
• To construct a canonical continuous Hermitian metric on the CM line bundle over the proper moduli compactification $̅{\mathcal{M}}$.
• To show the curvature of this metric is a positive (1,1)-current extending the Weil-Petersson current from the smooth locus $̅{\mathcal{M}}^\circ$.
• To establish that the CM line bundle is nef and big on $̅{\mathcal{M}}$ and ample on ${\mathcal{M}}$.
• To provide a differential-geometric approach to moduli compactification using singular metrics and Deligne pairings.

Proposed method

• Constructing a canonical continuous Hermitian metric $h_{\rm DP}$ on the Deligne pairing of the relative canonical bundle over the family of Kähler-Einstein Fano varieties.
• Using the partial $C^0$-estimate to control the degeneration of metrics and ensure uniform integrability of volume forms.
• Applying the theory of plurisubharmonic functions on complex spaces to handle singularities in the moduli compactification.
• Defining the Weil-Petersson current $\omega_{\rm WP}$ as the curvature of $h_{\rm DP}$, which extends the smooth Weil-Petersson metric on the smooth locus.
• Proving that the curvature current $\omega_{\rm WP}$ is positive (1,1) and continuous on the entire compactified moduli space $\overline{{\mathcal{M}}}$.
• Using the descent of the CM line bundle $\lambda_{\rm CM}$ to a line bundle $\Lambda_{\rm CM}$ on $\overline{{\mathcal{M}}}$ via equivariant resolution and toric coordinates.

Experimental results

Research questions

• RQ1 Does the moduli space of smooth Kähler-Einstein Fano manifolds admit a canonical Hermitian metric with positive curvature?
• RQ2 Can the Weil-Petersson current on the smooth locus be extended as a positive (1,1)-current to the entire compactified moduli space?
• RQ3 Is the CM line bundle on the compactified moduli space $\overline{{\mathcal{M}}}$ nef and big?
• RQ4 Is the restriction of the CM line bundle to the smooth locus ${\mathcal{M}}$ ample?
• RQ5 Does the existence of such a metric imply the quasi-projectivity of ${\mathcal{M}}$?

Key findings

• The CM line bundle $\lambda_{\rm CM}$ descends to a line bundle $\Lambda_{\rm CM}$ on the proper moduli compactification $\overline{{\mathcal{M}}}$.
• A canonical continuous Hermitian metric $h_{\rm DP}$ exists on $\Lambda_{\rm CM}$ whose curvature is a positive (1,1)-current $\omega_{\rm WP}$ on $\overline{{\mathcal{M}}}$.
• The curvature current $\omega_{\rm WP}$ extends the canonical Weil-Petersson current $\omega_{\rm WP}^\circ$ from the smooth locus ${\mathcal{M}}^\prime \subset {\mathcal{M}}$.
• The CM line bundle $\Lambda_{\rm CM}$ is nef and big on $\overline{{\mathcal{M}}}$, and its restriction to ${\mathcal{M}}$ is ample.
• The moduli space ${\mathcal{M}}$ of smooth Kähler-Einstein Fano manifolds is quasi-projective.
• The integral of the volume form over degenerating fibers converges to zero uniformly as the parameter $t \to 0$, confirming the integrability of the metric in singular settings.