[Paper Review] Toric manifolds over 3-polytopes

Anton Ayzenberg|arXiv (Cornell University)|Jul 12, 2016

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper establishes that every 3-dimensional simple polytope admits a quasitoric manifold, but not all support smooth projective toric varieties. The key result is that a 3-polytope admits a smooth projective toric variety only if it has a triangular or quadrangular face, which excludes fullerenes—polytopes with only pentagonal and hexagonal faces—from supporting such varieties, despite admitting quasitoric manifolds.

ABSTRACT

In this note we gather and review some facts about existence of toric spaces over 3-dimensional simple polytopes. First, over every combinatorial 3-polytope there exists a quasitoric manifold. Second, there exist combinatorial 3-polytopes, that do not correspond to any smooth projective toric variety. We restate the proof of the second claim which does not refer to complicated algebro-geometrical technique. If follows from these results that any fullerene supports quasitoric manifolds but does not support smooth projective toric varieties.

Motivation & Objective

• To clarify the combinatorial conditions under which a 3-dimensional simple polytope supports a smooth projective toric variety.
• To restate the non-existence result for smooth projective toric varieties over fullerenes using combinatorial-topological methods, avoiding advanced algebro-geometric techniques.
• To demonstrate that quasitoric manifolds exist over all 3-polytopes, including fullerenes, via a construction relying on the four color theorem.
• To highlight the distinction between quasitoric manifolds and smooth projective toric varieties in terms of cohomological and geometric constraints.
• To provide a transparent, combinatorial proof of the absence of smooth projective toric structures on fullerenes, emphasizing the role of extremal cohomology classes.

Proposed method

• Use of the characteristic function assigning primitive integral vectors in ℤ³ to each facet of a 3-polytope to define the torus action on the quasitoric manifold.
• Application of the four color theorem to construct a consistent assignment of stabilizer subgroups (via characteristic vectors) ensuring the existence of quasitoric manifolds over any 3-polytope.
• Use of the Stanley–Reisner ring and cohomology ring relations to analyze the structure of the orbit space and detect extremal classes.
• Proof by contradiction: assuming a vertex in the dual simplicial complex has more than four neighbors leads to a contradiction in the cohomology ring due to non-proportional non-vanishing products.
• Use of linear functionals and determinants (e.g., a₁ = det(λ(i′), λ(i₂), λ(i))) to analyze dihedral angles and cone configurations in the fan structure.
• Duality between the polytope and its simplicial complex: the existence of a vertex with three or four neighbors in the dual implies a triangular or quadrangular face in the original polytope.

Experimental results

Research questions

• RQ1 Does every 3-dimensional simple polytope admit a quasitoric manifold?
• RQ2 What combinatorial conditions must a 3-polytope satisfy to support a smooth projective toric variety?
• RQ3 Why do fullerenes, which are 3-polytopes with only pentagonal and hexagonal faces, not support smooth projective toric varieties?
• RQ4 Can the non-existence of smooth projective toric structures on fullerenes be proven without advanced algebro-geometric tools?
• RQ5 How do extremal cohomology classes in the cohomology ring distinguish smooth projective toric varieties from general quasitoric manifolds?

Key findings

• Every 3-dimensional simple polytope admits a quasitoric manifold, as established via the four color theorem and characteristic vector assignment.
• A 3-polytope admits a smooth projective toric variety only if it has at least one triangular or quadrangular face, as proven through cohomological constraints and extremal class analysis.
• Fullerenes, which are 3-polytopes with only pentagonal and hexagonal faces, do not support smooth projective toric varieties due to the absence of triangular or quadrangular faces.
• The proof of non-existence of smooth projective toric structures on fullerenes is re-expressed in combinatorial-topological terms, avoiding reliance on Mori’s minimality theory or advanced algebraic geometry.
• The existence of a strictly convex effective cone in the cohomology ring—characteristic of smooth projective toric varieties—is absent in general quasitoric manifolds, explaining the distinction in geometric structures.
• In the cohomology ring of a quasitoric manifold, the absence of extremal classes with positive linear combinations implies that the structure is not projective, which explains the failure of the toric variety construction on fullerenes.