Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties

Details Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties

2011-03-09

arxiv.org/abs/1103.1723v2

Mathematics

Algebraic Geometry

44 pages; Minor changes, comments still welcome

Scientific paper

We study aspects of the A^1-homotopy classification problem in dimensions >= and, to this end, we investigate the A^1-homotopy groups of some A^1-connected smooth varieties of dimension >=. We construct pairs of A^1-connected smooth proper varieties all of whose A^1-homotopy groups are abstractly isomorphic, yet which are not A^1-weakly equivalent. The examples come from pairs of Zariski locally trivial projective space bundles over projective spaces and are of the smallest possible dimension. Projectivizations of vector bundles give rise to A^1-fiber sequences, and when the base of the fibration is an A^1-connected smooth variety, the associated long exact sequence of A^1-homotopy groups can be analyzed in detail. In the case of the projectivization of a rank 2 vector bundle, the structure of the A^1-fundamental group depends on the splitting behavior of the vector bundle via a certain obstruction class. For projective bundles of vector bundles of rank >=, the A^1-fundamental group is insensitive to the splitting behavior of the vector bundle, but the structure of higher A^1-homotopy groups is influenced by an appropriately defined higher obstruction class.

Affiliated with

Also associated with

No associations

Rating

Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Splitting vector bundles and A^1-fundamental groups of higher dimensional varieties will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-643965