Complete toric varieties with semisimple automorphism group

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.

💡 Research Summary

The paper investigates the interplay between the combinatorial structure of a complete toric variety and the algebraic properties of its automorphism group. The authors first establish a decomposition theorem for fans: if a complete fan Σ in a real vector space V⊕W has its set of one‑dimensional cones Σ(1) split as a disjoint union Σ(1)|_V ⊔ Σ(1)|_W, then Σ itself decomposes as the direct sum of the induced fans Σ|_V and Σ|_W. The proof proceeds through a series of lemmas. Lemma 2.1 shows that for any cone σ∈Σ, the intersections σ∩V and σ∩W are faces of σ, guaranteeing that each cone can be written as the sum of its V‑part and W‑part. Lemma 2.2 proves that the collections Σ|_V and Σ|_W satisfy the fan axioms and are complete in their respective subspaces. Lemma 2.3 uses the fact that two complete fans with one contained in the other must coincide, yielding Σ = Σ|_V ⊕ Σ|_W. This result demonstrates that the 1‑skeleton alone determines the whole fan when it separates along a direct sum decomposition.

The second part of the work connects this combinatorial insight with the structure of the identity component Aut⁰(X) of the automorphism group of a complete toric variety X. The authors recall Demazure’s notion of roots of a fan, defined purely in terms of Σ(1). They compute the root set for the standard fan of projective space ℙⁿ, showing it consists of the vectors ±e_i^* and e_i^−e_j^. Proposition 3.4 proves that if a complete fan has the same 1‑skeleton as that of ℙⁿ, then the fan coincides with the ℙⁿ fan, and consequently the variety is isomorphic to ℙⁿ. Using Demazure’s theorem (cited as Theorem 3.5) they relate the set of roots to one‑parameter unipotent subgroups of Aut(X). When Aut⁰(X) is semisimple, Demazure’s work implies that Aut⁰(X) is a product of groups of the form PGL_{n_i+1}. Each factor corresponds to a projective space ℙ^{n_i} whose fan’s 1‑skeleton appears as a component of Σ(1). Applying the fan decomposition theorem, the authors conclude that X must be a product of projective spaces: X ≅ ℙ^{n_1} × … × ℙ^{n_k}.

Thus the paper provides an elementary, combinatorial proof that a complete toric variety with semisimple automorphism group is necessarily a product of projective spaces, avoiding the more sophisticated Lie‑theoretic arguments previously known (e.g., Kuroki 2010). The work highlights how the 1‑skeleton of a fan encodes enough information to recover the full fan and, together with Demazure roots, to control the global symmetry group. The results deepen the bridge between toric geometry, algebraic group theory, and polyhedral combinatorics, and they offer practical criteria for detecting product decompositions of toric varieties directly from their fans.