Toric principal bundles, piecewise linear maps and Tits buildings

Abstract

We define the notion of a piecewise linear map from a fan $\Sigma$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_{\Sigma }$ be a toric variety with fan $\Sigma$. We show that when $G$ is reductive the set of integral piecewise linear maps from $\Sigma$ to $\tilde{\mathfrak{B}}(G)$ classifies the isomorphism classes of (framed) toric principal $G$-bundles on $X_{\Sigma }$. This in particular recovers Klyachko's classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.

Final version. 26 pages

References in corpus (1)

• Moduli of toric vector bundles