zbMATH — the first resource for mathematics

Remarks on $$\mathbb{A}^1$$-homotopy groups of smooth toric models. (English) Zbl 1256.14019
In this short note, the author continues his important longstanding project of the classification of homotopy types in the $$\mathbb{A}^1$$-homotopy theory of smooth varieties over a field. Let $$k$$ be a field of characteristic $$0$$, $$T$$ be a torus over $$k$$, and let $$X$$ be a smooth proper variety that is an equivariant compactification of $$T$$ over $$k$$. To this data, there is an associated Neron-Severi torus $$T_{NS(X)}$$ and a $$T_{NS(X)}$$-torsor $$f:U \to X$$. The main result of the paper is that this $$T_{NS(X)}$$-torsor $$f:U \to X$$ is an $$\mathbb{A}^1$$-cover. This implies in particular, that the $$\mathbb{A}^1$$-fundamental groups of $$U$$ and $$X$$ sit in a short exact sequence $1 \to \pi_1^{\mathbb{A}^1}(U, \tilde{x}) \to \pi_1^{\mathbb{A}^1}(X, x) \to T_{NS(X)} \to 1$ where $$x$$ is a given $$k$$-rational point on $$X$$ and $$\tilde{x}$$ is any lift of $$x$$ to $$U$$. Moreover, for $$i>1$$, there are isomorphisms $$\pi_i^{\mathbb{A}^1}(U, \tilde{x}) \cong \pi_i^{\mathbb{A}^1}(X, x)$$. This extends previous results on torsors isomorphic to $$\mathbb{G}_m^{\times n}$$ to the case of general $$k$$-torsors.
MSC:
 14F99 (Co)homology theory in algebraic geometry 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: