Real affine varieties of nonnegative curvature

TL;DR

This paper proves that smooth real affine varieties with compact real points and a conjugation-invariant nonnegative curvature metric are diffeomorphic to the normal bundle of their real points, linking curvature conditions to topological structure.

Contribution

It establishes a new topological classification of real affine varieties under nonnegative curvature conditions with conjugation symmetry.

Findings

Real affine varieties with the specified conditions are diffeomorphic to the normal bundle of their real points.

The result connects geometric curvature properties with the topological structure of algebraic varieties.

Provides a criterion for the topological type of real affine varieties based on curvature and symmetry.

Abstract

Let $X_{\mathbb{C}}$ be a smooth real affine variety with compact real points $X_{\mathbb{R}}$. We show that $X_{\mathbb{C}}$ is diffeomorphic to the normal bundle of $X_{\mathbb{R}}$ provided that $X_{\mathbb{C}}$ admits a complete Riemannian metric of nonnegative sectional curvature which is also invariant under the conjugation.