Complete toric varieties with semisimple automorphism group

TL;DR

The paper addresses when a complete toric variety decomposes as a product by examining the fan's combinatorics, linking the decomposition of the 1-skeleton to a direct-sum decomposition of the fan. It develops and uses Demazure roots and root-data to connect automorphism-group structure with fan geometry, proving that a semisimple $\operatorname{Aut}^\circ(X)$ forces a product decomposition into projective spaces. A key step is showing that the PGL-root data characterizes $\mathbb{P}^n$ and that, in general, a semisimple automorphism group yields $X\simeq\prod_i \mathbb{P}^{n_i}$. The results provide an elementary, combinatorial proof of this structural classification and clarify how semisimple symmetries constrain toric geometry.

Abstract

Let $X$ be a complete toric variety. We give a criterion to decide whether $X$ decomposes as a product of complete toric varieties by analyzing the $1$-skeleton of its fan. More precisely, we prove that any direct-sum decomposition of the 1-skeleton induces a corresponding direct-sum decomposition of the fan itself. As an application, we show that if the identity component of the automorphism group is semisimple, then $X$ must be a product of projective spaces.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Definition 1.1
• Definition 1.2
• Remark 1.3
• Lemma 1.4
• proof
• Lemma 2.1
• proof
• Lemma 2.2