Normalized tangent bundle, varieties with small codegree and pseudoeffective threshold

TL;DR

This paper investigates Fano manifolds with pseudoeffective normalized tangent bundles, relating their properties to small codegree varieties and explicitly determining pseudoeffective thresholds for tangent bundles of rational homogeneous spaces.

Contribution

It proposes a conjectural classification of such Fano manifolds and computes pseudoeffective thresholds using geometric techniques involving VMRTs and Mukai flops.

Findings

Confirmed the conjectural list in various cases.

Explicitly determined pseudoeffective thresholds for tangent bundles.

Established vanishing theorems for twisted symmetric vector fields.

Abstract

We propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalized tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Russo and Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and hence the pseudoeffective cones of the projectivized tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual VMRT and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number $1$.