$\mathbb{A}^1$-curves on log smooth varieties

TL;DR

This paper investigates $ ext{A}^1$-connected varieties using log geometry, providing criteria, examples, and a logarithmic version of Hartshorne's conjecture to characterize projective and affine spaces.

Contribution

It introduces a criterion for $ ext{A}^1$-connectedness in log geometry and applies it to classify certain varieties, including complements of divisors and homogeneous spaces.

Findings

Established a criterion for $ ext{A}^1$-connectedness.

Constructed examples of $ ext{A}^1$-connected varieties.

Proved a logarithmic version of Hartshorne's conjecture.

Abstract

In this paper, we study $\mathbb{A}^1$-connected varieties from log geometry point of view, and prove a criterion for $\mathbb{A}^1$-connectedness. As applications, we provide many interesting examples of $\mathbb{A}^1$-connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne's conjecture characterizing projective spaces and affine spaces.