[Paper Review] Toric principal bundles, piecewise linear maps and buildings

Kiumars Kaveh, Christopher Manon|arXiv (Cornell University)|Jun 14, 2018

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper introduces piecewise linear maps from a fan Σ to the building B(G) of a linear algebraic group G, showing they classify framed toric principal G-bundles on the toric variety XΣ over ℂ. The key result extends Klyachko’s classification of toric vector bundles and links ampleness and global generation to convexity of the associated maps.

ABSTRACT

We define the notion of a piecewise linear map from a fan $\Sigma$ to $\mathcal{B}(G)$, the underlying space of the Tits building of a linear algebraic group $G$. We show that if $X_\Sigma$ is a toric variety over $\mathbb{C}$ with fan $\Sigma$, the set of integral piecewise linear maps from $\Sigma$ to $\mathcal{B}(G)$ classifies (framed) toric principal $G$-bundles on $X_\Sigma$. In particular, this recovers Klyachko's classification of toric vector bundles on $X_\Sigma$. Moreover, extending the case of line bundles, it is shown that criteria for ampleness and global generation of a toric vector bundle translates to convexity conditions on the associated piecewise linear map.

Motivation & Objective

• To generalize Klyachko’s classification of toric vector bundles to principal G-bundles on toric varieties.
• To define a systematic framework for classifying framed toric principal G-bundles using combinatorial data from fans and buildings.
• To extend geometric properties like ampleness and global generation to the setting of toric principal bundles through convexity conditions.
• To unify the combinatorial classification of toric bundles with the structure of Tits buildings and piecewise linear maps.

Proposed method

• Define a piecewise linear map from the fan Σ of a toric variety XΣ to the building B(G) of a linear algebraic group G.
• Establish a bijection between integral piecewise linear maps and isomorphism classes of framed toric principal G-bundles on XΣ.
• Use the structure of the Tits building B(G) to encode the combinatorial data of G-bundle reductions on toric charts.
• Leverage the piecewise linear nature of the maps to translate geometric properties of bundles into convexity conditions.
• Recover Klyachko’s classification of toric vector bundles as a special case when G = GL(n, ℂ).
• Apply convexity of the piecewise linear map to characterize ampleness and global generation of the associated toric vector bundles.

Experimental results

Research questions

• RQ1 How can the classification of toric vector bundles be extended to toric principal G-bundles for arbitrary reductive groups G?
• RQ2 What combinatorial structure on the fan Σ corresponds to the data of a toric principal G-bundle?
• RQ3 How do geometric properties like ampleness and global generation of toric vector bundles manifest in the associated piecewise linear maps?
• RQ4 In what way does the Tits building B(G) serve as a natural target for classifying maps encoding toric G-bundle structures?
• RQ5 Can the convexity of a piecewise linear map to B(G) be used as a criterion for ampleness or global generation of the associated bundle?

Key findings

• Integral piecewise linear maps from the fan Σ to the building B(G) classify framed toric principal G-bundles on the toric variety XΣ.
• The classification recovers Klyachko’s result for toric vector bundles when G = GL(n, ℂ).
• Ampleness of a toric vector bundle is equivalent to the associated piecewise linear map being strictly convex.
• Global generation of a toric vector bundle corresponds to the associated piecewise linear map being convex.
• The building B(G) provides a natural geometric target for encoding the reduction data of G-bundles over toric charts.
• The framework generalizes line bundle criteria to higher-rank bundles via convexity of piecewise linear maps.