Splitting vector bundles outside the stable range and 𝔸¹-homotopy sheaves of punctured affine spaces

Asok, Aravind¹ ; Fasel, Jean²

¹ Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
² Fakultät Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann-Strasse 9, D-45127 Essen, Germany

Journal of the American Mathematical Society, Tome 28 (2015) no. 4, pp. 1031-1062

Résumé

We discuss the relationship between the ${\mathbb A}^1$-homotopy sheaves of ${\mathbb A}^n {\setminus } 0$ and the problem of splitting off a trivial rank $1$ summand from a rank $n$ vector bundle. We begin by computing $\boldsymbol {\pi }_3^{{\mathbb A}^1}({\mathbb A}^3 {\setminus } 0)$ and providing a host of related computations of “non-stable” ${\mathbb A}^1$-homotopy sheaves. We then use our computation to deduce that a rank $3$ vector bundle on a smooth affine $4$-fold over an algebraically closed field having characteristic unequal to $2$ splits off a trivial rank $1$ summand if and only if its third Chern class (in Chow theory) is trivial. This result provides a positive answer to a case of a conjecture of M.P. Murthy.

Détail

DOI : 10.1090/S0894-0347-2014-00818-3

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