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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
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