UBC Algebra and Algebraic Geometry Seminar: Evan Marth

Let G be a split reductive group. For X a projective G-homogeneous variety, there is a description of the decomposition of the Chow motive of X (Chernousov-Gille-Merkurjev, Brosnan). It is also known that these varieties satisfy the so-called Rost nilpotence principle. In classical cohomology theories for algebraic varieties (e.g., singular cohomology, ℓ-adic cohomology), there is a strong relation between the cohomology of a variety and its hyperplane sections, most famously in the form of the Lefschetz Hyperplane Theorem. We present some examples where a motivic version of the Lefschetz theorem holds. More precisely, for Milnor hypersurfaces (incidence varieties of dimension 1 and codimension 1 subspaces) and some twisted forms thereof, we show that hyperplane sections corresponding to regular semisimple elements admit a motivic decomposition of the expected form. As a corollary, we see that the Rost nilpotence principle holds for these hyperplane sections.

This talk is based on joint work with Kirill Zaynullin and Rui Xiong.