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

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 $\mathbb{P}<sup>{m(X)}$-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 $X\cong \mathbb{P}<sup>2$. In the example-driven Appendix B, we demonstrate that extending this strategy to the case $m(X)&gt;2$ requires either a substantially more detailed analysis of the combinatorics of primitive collections or a fundamentally new approach.

• The paper confirms that toric $2$-Fano manifolds are limited to projective spaces, solidifying previous conjectures.
• Utilizes combinatorial invariants and toric geometry to transform manifolds into projective bundles, assessing Chern character positivity.
• Highlights the complexity and requirements for $m(X)\ge 3$, paving pathways for classification in higher dimensions.

Authoritative Summary of "On the classification of toric $2$-Fano manifolds: generic $\mathbb{P}^2$-bundles" (2604.18054)

Introduction and Problem Statement

The paper addresses the classification problem for toric $2$-Fano manifolds, leveraging both algebraic and combinatorial methods intrinsic to toric geometry. The $2$-Fano condition, introduced by de Jong and Starr, specifies that for a Fano manifold $X$, the second Chern character must be positive on all surfaces: $\mathrm{ch}_2(X) \cdot S > 0$ for every surface $S \subset X$. This condition significantly strengthens the classical Fano property, narrowing the class of admissible manifolds and motivating an exhaustive classification.

Projective spaces are currently the only established examples of toric $2$-Fano manifolds. The conjecture—originally proposed in [SanoSatoSuyama2020]—asserts that these are the only possibilities. The paper builds on a program, developed in prior work [team2023], centered on the combinatorial invariant $m(X)$, the minimal projective bundle dimension, which quantifies the minimal degree of dominating rational curves in a toric manifold and is directly tied to primitive relations of the form $x_0+\dots+x_m=0$.

Combinatorial and Geometric Framework

The investigation relies on the correspondence between toric varieties and lattice fans, translating geometric data into combinatorial invariants. Primitive collections and their associated primitive relations are crucial, as they encode both the existence of rational curves and the structure of projective bundle fibrations on open subsets. The invariant $\mathbb{P}^2$0 is defined as the minimal $\mathbb{P}^2$1 such that $\mathbb{P}^2$2 admits a centered primitive relation of length $\mathbb{P}^2$3, i.e., $\mathbb{P}^2$4. This corresponds to a $\mathbb{P}^2$5-bundle structure on a big open subset of $\mathbb{P}^2$6, with the projective space $\mathbb{P}^2$7 arising when $\mathbb{P}^2$8.

The authors utilize contractible curve classes, birational maps (blowdowns, flips), and explicit fan combinatorics to reshape $\mathbb{P}^2$9 into a related toric manifold $2$0 that admits a $2$1-bundle structure, thereby enabling comparative Chern character calculations.

Main Classification Results

The central theorem of the paper proves that for toric Fano manifolds $2$2 with $2$3 and $2$4, the $2$5-Fano condition fails; namely, $2$6 cannot be $2$7-Fano unless $2$8. The proof constructs a birational sequence $2$9 (blowdowns and flips), after which $2$0 supports a $2$1-bundle structure on an open subset whose complement has codimension at least $2$2. Surfaces within this subset can be shown to violate the second Chern character positivity condition.

A corollary follows: apart from projective spaces, any toric $2$3-Fano manifold must have $2$4 and satisfy stringent constraints on dimension and Picard number, specifically,

$2$5

Technical Construction and Strategy

The strategy operates in three steps:

• Step 1: Using blowdowns and toric flips, $2$6 is transformed into $2$7 so that all divisorial obstructions to the $2$8-bundle structure are eliminated (codimension $2$9), and relevant primitive relations are contractible.
• Step 2: Rational surfaces $X$0 are constructed in the $X$1-bundle locus, showing $X$2, exploiting the geometry of toric projective bundles and intersections with torus-invariant divisors.
• Step 3: The proper transform $X$3 of $X$4 is analyzed, with precise tracking of second Chern character contributions across blowdowns and flips, yielding $X$5.

The authors provide a refined analysis distinguishing exceptional cases—where primitive relations intertwine—requiring separate combinatorial scrutiny. The appendix illustrates concrete such cases via explicit Macaulay2 examples.

Numerical and Structural Claims

Key numerical claim: For $X$6 with $X$7, and $X$8 as constructed, the intersection $X$9 is strictly non-positive, confirming the failure of the $\mathrm{ch}_2(X) \cdot S > 0$0-Fano property unless $\mathrm{ch}_2(X) \cdot S > 0$1 is a projective space.

The paper also examines the extension of this approach to $\mathrm{ch}_2(X) \cdot S > 0$2, observing that the combinatorial complexity escalates: more types of relevant primitive relations, auxiliary contractions, and possible singularities appear, complicating a straightforward generalization.

Implications and Future Directions

Practically, this strengthens the combinatorial machinery available for the classification of high-index Fano manifolds, pushing toward resolution of the toric $\mathrm{ch}_2(X) \cdot S > 0$3-Fano conjecture. Theoretically, the analysis illustrates the rigidity of higher Chern character conditions under rich symmetries (toric), aligning with broader expectations from rational connectedness and arithmetic geometry.

Future development will likely target $\mathrm{ch}_2(X) \cdot S > 0$4, needing deeper combinatorial classification of primitive relations, refined contraction techniques, and possibly extension to singular toric varieties. The explicit fan-combinatorial tools may also inform broader studies in mirror symmetry and birational geometry, where Chern character positivity interacts with stability and moduli spaces.

Conclusion

The authors rigorously establish that no toric $\mathrm{ch}_2(X) \cdot S > 0$5-Fano manifold with minimal projective bundle dimension $\mathrm{ch}_2(X) \cdot S > 0$6 exists beyond the projective plane $\mathrm{ch}_2(X) \cdot S > 0$7, consolidating the conjecture that only projective spaces are valid examples. The combinatorial and geometric techniques developed here set a foundation for ongoing classification work in higher toric dimensions, with significant implications for the structure theory of algebraic varieties endowed with toric symmetries.