Splitting vector bundles on smooth real affine varieties

par Samuel Lerbet (ENS)

mercredi 15 avr. 2026, 14:00 → 15:20 Europe/Paris

Salle Pierre Grisvard (Institut Henri Poincaré)

 

Let X be a smooth affine variety over a perfect field k. A natural question regarding vector bundles on X is when they split a trivial rank 1 summand; this question is related to the comparison between vector bundles and K-theory classes. When k is algebraically closed of characteristic 0, the splitting of a trivial tank 1 summand for vector bundles of large rank in the unstable range, namely of corank at most 1 where the corank is the difference between the dimension and the rank, is entirely governed by the vanishing of the top Chern class: this is the confirmation of Murthy's conjecture by Asok–Bachmann–Hopkins, building on ideas of Asok–Fasel. When k has more arithmetic complexity, this result usually fails, but motivic homotopy theory provides a systematic way of approaching the splitting problem in this generality. We will compare this motivic approach to topology over the real locus of the variety under consideration in the specific case where k is the field of real numbers, for which the arithmetic is as simple as possible, in search for a real Murthy's conjecture. This is joint work with Aravind Asok and Jean Fasel.