On the classification of toric $2$-Fano manifolds: generic $\mathbb{P}^2$-bundles

TL;DR

This paper advances the classification of toric 2-Fano manifolds by analyzing their minimal projective bundle dimension and relating it to their geometric and combinatorial properties.

Contribution

It introduces a new approach connecting toric Fano manifolds via blowdowns and flips, and characterizes the case when the minimal projective bundle dimension is 2.

Findings

The only toric 2-Fano manifold with minimal projective bundle dimension 2 is the projective plane.

Extending the classification to higher dimensions requires more detailed combinatorial analysis.

The approach relates positivity of second Chern characters to the geometric structure of the manifolds.

Abstract

In this paper, we advance the classification of toric 2-Fano manifolds by continuing the investigation of the minimal projective bundle dimension $m(X) \in \{1,\dots,\dim(X)\}$ introduced in our previous work. This invariant captures the minimal degree of a dominating family of rational curves on $X$ and admits a natural combinatorial interpretation in terms of centered primitive collections. We develop an approach that relates, via toric blowdowns and flips, a toric Fano manifold $X$ to a toric manifold $Y$ that admits a $\mathbb{P}^{m(X)}$-bundle structure on a big open subset. We then compare positivity of the second Chern characters of $X$ and $Y$, and show that the only toric 2-Fano manifold $X$ with $m(X) = 2$ is $X\cong \mathbb{P}^2$. In the example-driven Appendix B, we demonstrate that extending this strategy to the case $m(X)>2$ requires either a substantially more detailed analysis of the combinatorics of primitive collections or a fundamentally new approach.