Complete toric varieties with semisimple automorphism group

TL;DR

This paper provides a criterion based on the fan's 1-skeleton to determine if a complete toric variety decomposes into a product of simpler varieties, linking automorphism groups to geometric structure.

Contribution

It establishes a connection between the 1-skeleton of the fan and the decomposition of the toric variety, and characterizes varieties with semisimple automorphism groups as products of projective spaces.

Findings

Decomposition of the 1-skeleton induces a decomposition of the fan.

If the automorphism group's identity component is semisimple, then the variety is a product of projective spaces.

Provides a criterion for decomposing complete toric varieties based on fan analysis.

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.