𝔸-curves on log smooth varieties

Abstract

In this paper, we study A 1 -connected varieties from log geometry point of view, and prove a criterion for A 1 -connectedness. As applications, we provide many interesting examples of 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.

On Smooth Projective D-Affine Varieties

“…The classification of contact varieties with b 2 ≥ 2 in [KPSW00] used Mori's theory and, more precisely, a careful study of contractions and rational curves on contact varieties. So it is natural to approach the classification of contact snc pairs using log contractions and the non-proper analogue of rational curves suggested by S. Keel and J. McKernan in [KM99] (see also [CZ19]). However, many of analogues of usual results are not available in the log case and to prove Theorem 0.1 we need to find appropriate formulations working in the log case.…”

Strong approximation over function fields

“…Since affine spaces satisfies strong approximation [Ros02, Theorem 6.13], it is natural to search geometric substitute for affine spaces that strong approximation holds. As the non-proper generalization of rationally connected varieties, A 1 -connected varieties have been studied in our previous works [CZ14b,CZ14a,CZ15,Zhu15]. On one hand side, this program is influenced by Iitaka's philosphy and the works of by studying the birational geometry of pairs.…”