[Paper Review] Smooth projective horospherical varieties with nef tangent bundles

Qifeng Li|arXiv (Cornell University)|Dec 15, 2015

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper proves that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces, confirming the Campana-Peternell conjecture for this class of varieties. The proof uses Mori theory to reduce to Picard number one cases, analyzes index conditions, and shows singularities in the VMRTs of exceptional cases, thereby excluding them from having nef tangent bundles.

ABSTRACT

We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.

Motivation & Objective

• To verify the Campana-Peternell conjecture for smooth projective horospherical varieties, which posits that Fano manifolds with nef tangent bundles are rational homogeneous spaces.
• To extend the known classification of CP-manifolds (Fano manifolds with nef tangent bundles) to the broader class of horospherical varieties.
• To resolve the conjecture in the case of horospherical varieties by analyzing Mori contractions, index conditions, and VMRT singularities.
• To show that the only smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces, thus completing the classification in this setting.

Proposed method

• Use Mori theory to reduce the problem to the case of horospherical varieties of Picard number one, leveraging the structure of elementary Mori contractions.
• Analyze the index of smooth non-homogeneous horospherical varieties of Picard number one, showing that only two exceptional cases can potentially admit nef tangent bundles.
• Study the singularity of the VMRT (Variety of Minimal Rational Tangents) for the two exceptional cases, proving they are singular and thus cannot have nef tangent bundles.
• Apply the characterization of Fano manifolds with nef tangent bundles via the Albanese map and fiber decomposition, as established by Demailly-Peternell-Schneider.
• Use the colored fan and orbit structure of horospherical varieties to describe the geometry of orbits and their stabilizers, particularly via the group $ P_I $ and the character lattice $ M_{G/H} $.
• Construct explicit morphisms from the VMRT to projective spaces and analyze fiber dimensions and singularities via incidence geometry and symplectic structures in the odd symplectic Grassmannian case.

Experimental results

Research questions

• RQ1 Are smooth projective horospherical varieties with nef tangent bundles necessarily rational homogeneous spaces?
• RQ2 Which smooth projective horospherical varieties of Picard number one can admit nef tangent bundles?
• RQ3 Do the two exceptional cases identified in the index analysis (Proposition 3.1 (2)(ii) and (iii)) admit smooth VMRTs, and what does this imply for the tangent bundle's nefness?
• RQ4 Can the singularities of the VMRT in the exceptional cases be used to rule out the existence of nef tangent bundles?
• RQ5 Does the structure of Mori contractions on horospherical varieties allow reduction of the general case to the Picard number one case?

Key findings

• Smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces, confirming the Campana-Peternell conjecture for this class.
• The only possible non-homogeneous candidates with nef tangent bundles are two exceptional cases of Picard number one, which are ruled out by VMRT singularity analysis.
• For the odd symplectic Grassmannian $ G_ ho(k,2m+1) $ with $ m eq 1 $, the VMRT $ F(x,X) $ is singular when $ x $ lies in the closed orbit $ Z $, implying non-nef tangent bundle.
• When $ x otin Z $, the VMRT $ F(x,X) $ is a $ bP^{2m-2k+1} $-bundle over $ bP^{k-1} $, hence smooth, consistent with possible nefness.
• The variety $ F(x,X) $ is singular at $[l]$ when $ x otin Z $, but this singularity is resolved in the $ x otin Z $ case via the fiber bundle structure.
• The proof reduces the general case to Picard number one via the decomposition $ X o ext{pt} imes G/P $, showing that each factor must be rational homogeneous if it has a nef tangent bundle.