[Paper Review] Smooth toric varieties are Oka

Finnur Lárusson|arXiv (Cornell University)|Jul 19, 2011

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper proves that every smooth toric variety over the complex numbers is an Oka manifold, using the fact that such varieties arise as geometric quotients of Zariski-open subsets of complex affine space by free actions of reductive groups. The key result is that the Oka property is preserved under such quotients when the total space and structure group are Oka, establishing smooth toric varieties as Oka manifolds.

ABSTRACT

In this brief note, we show that every smooth toric variety over the field of complex numbers is an Oka manifold.

Motivation & Objective

• To establish that smooth toric varieties over C satisfy the Oka property.
• To extend the class of known Oka manifolds beyond complex Lie groups and their homogeneous spaces.
• To investigate whether the Oka property is preserved under geometric quotients of complex spaces by reductive group actions.
• To clarify the relationship between toric geometry and Oka theory in complex geometry.

Proposed method

• Represent a smooth toric variety X as a geometric quotient X = (C^m \ Z)/G, where G is a reductive subgroup of (C*)^m and Z is a union of coordinate subspaces of codimension at least 2.
• Use the fact that C^m \ Z is Oka because it is the complement of a subvariety of codimension ≥ 2 in C^m.
• Apply the Oka property of G, which is a complex Lie group and thus elliptic and Oka by Gromov's theory.
• Leverage slice theory for reductive group actions to show that the quotient map C^m \ Z → X is a holomorphic fibre bundle.
• Apply Theorem 2 from Oka theory: if the total space and structure group are Oka, then the base space is Oka if and only if the total space is Oka.
• Conclude that X is Oka because C^m \ Z is Oka and the quotient map is a holomorphic fibre bundle with Oka fibres.

Experimental results

Research questions

• RQ1 Are all smooth toric varieties over C Oka manifolds?
• RQ2 Does the Oka property descend through geometric quotients of complex spaces by free reductive group actions?
• RQ3 Can the Oka property be established for toric varieties without assuming ellipticity?
• RQ4 Is the Oka property preserved under the quotient construction used in toric geometry?

Key findings

• Every smooth toric variety over the complex numbers is an Oka manifold.
• The total space C^m \ Z is Oka because it is the complement of a subvariety of codimension at least 2 in C^m.
• The group G acting on C^m \ Z is reductive and isomorphic to a product of a torus and a finite abelian group, hence Oka.
• The quotient map C^m \ Z → X is a holomorphic fibre bundle due to slice theory for reductive group actions.
• The base space X is Oka because the total space and structure group are Oka, and the fibre bundle property ensures the Oka property lifts to the base.
• The result holds even though it is not known whether smooth toric varieties are elliptic, showing the Oka property can hold without ellipticity.