[Paper Review] Algebraic vector bundles on the 2-sphere and smooth rational varieties with infinitely many real forms

Adrien Dubouloz, Gene Freudenburg+1 more|arXiv (Cornell University)|Jul 16, 2018

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper constructs smooth rational real affine fourfolds with infinitely many pairwise non-isomorphic real forms by algebraizing topological vector bundles on the 2-sphere. It proves that $\mathbb{S}^2 \times \mathbb{A}^2_{\mathbb{R}}$ admits countably infinitely many real forms via algebraic models of complexified rank-2 vector bundles, resolving a gap in the classification of rational varieties with infinitely many real forms.

ABSTRACT

We construct smooth rational real algebraic varieties of every dimension $\\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.

Motivation & Objective

• To address the open question of whether smooth rational real algebraic varieties can have infinitely many pairwise non-isomorphic real forms.
• To construct explicit examples of such varieties in dimension ≥4, filling a gap left by prior results on non-rational varieties.
• To demonstrate that the product $\mathbb{S}^2 \times \mathbb{A}^2_{\mathbb{R}}$ has infinitely many real forms despite $\mathbb{S}^2$ and $\mathbb{A}^2_{\mathbb{R}}$ each having only finitely many.
• To extend the construction to higher dimensions using varieties of log-general type, preserving the infinite real form property.

Proposed method

• Construct algebraic vector bundles $p_n: V_n \to \mathbb{S}^2$ of rank 2 over the real 2-sphere $\mathbb{S}^2$, modeled on topological real vector bundles classified by $\mathcal{O}_{\mathbb{CP}^1}(n)$.
• Show that the complexifications $V_{n,\mathbb{C}}$ are all isomorphic to the trivial bundle $\mathbb{S}_{\mathbb{C}}^2 \times \mathbb{A}^2_{\mathbb{C}}$, ensuring complex isomorphism of total spaces.
• Use the fact that the $V_n$ are pairwise non-isomorphic as real algebraic varieties despite being locally isomorphic over $\mathbb{S}^2$, to generate distinct real forms of $\mathbb{S}^2 \times \mathbb{A}^2_{\mathbb{R}}$.
• Apply the Iitaka-Fujita strong cancellation theorem to show that any isomorphism between $V_n \times X$ and $V_m \times X$ must be fiberwise over $X$, forcing $n = m$ when $X_{\mathbb{C}}$ has trivial automorphism group.
• Adapt the construction by taking products with smooth rational real affine varieties $X$ of log-general type and trivial $\mathrm{Aut}(X_{\mathbb{C}})$, ensuring infinite distinctness of real forms.

Experimental results

Research questions

• RQ1 Can smooth rational real algebraic varieties admit infinitely many pairwise non-isomorphic real forms?
• RQ2 Do real forms of $\mathbb{S}^2 \times \mathbb{A}^2_{\mathbb{R}}$ arise from algebraic models of topological vector bundles on $S^2$?
• RQ3 Can the construction of infinitely many real forms be extended beyond dimension 4 using log-general type varieties?
• RQ4 What role does the automorphism group of the complexification play in distinguishing real forms of products?
• RQ5 Is the property of having infinitely many real forms preserved under products with rational varieties of log-general type?

Key findings

• The smooth rational real affine fourfold $\mathbb{S}^2 \times \mathbb{A}^2_{\mathbb{R}}$ admits at least countably infinitely many pairwise non-isomorphic real forms.
• The real forms are realized as total spaces $V_n$ of algebraic vector bundles of rank 2 over $\mathbb{S}^2$, with $V_{n,\mathbb{C}} \cong \mathbb{S}_{\mathbb{C}}^2 \times \mathbb{A}^2_{\mathbb{C}}$ for all $n \geq 0$.
• The varieties $V_n$ are pairwise non-isomorphic as real algebraic varieties, despite being locally isomorphic over $\mathbb{S}^2$, due to non-trivial topological classification of their underlying real vector bundles.
• For every $d \geq 4$, there exist smooth rational real affine varieties of dimension $d$ with at least countably infinitely many pairwise non-isomorphic real forms.
• The construction extends to $\mathbb{S}^2 \times \mathbb{A}^2_{\mathbb{R}} \times X$ for smooth rational real affine $X$ with $X_{\mathbb{C}}$ of log-general type and trivial $\mathrm{Aut}(X_{\mathbb{C}})$, yielding infinitely many real forms.
• The real structure on $\mathbb{S}_{\mathbb{C}}^2 \times \mathbb{A}_{\mathbb{C}}^2$ corresponding to $V_n$ is not equivalent to that of $V_m$ for $n \neq m$, confirming non-isomorphism of the real forms.