There are $160,839 \langle 1 \rangle + 160,650 \langle -1\rangle$ 3-planes in a 7-dimensional cubic hypersurface

Abstract: It is a result of Debarre--Manivel that the variety of $d$-planes on a generic complete intersection has the expected dimension. When this dimension is 0, the number of such $d$-planes is given by the Euler number of a vector bundle on a Grassmannian. There are several Euler numbers from $A^1$-homotopy theory which take a vector bundle to a bilinear form. We equate some of these, including those of Barge-Morel, Kass-W., Déglise-Jin-Khan, and one suggested by M.J. Hopkins, A. Raksit, and J.-P. Serre using duality of coherent sheaves. We establish integrality results for this Euler class, and use this to compute the Euler classes associated to arithmetic counts of d-planes on complete intersections in projective space in terms of topological Euler numbers over the real and complex numbers. The example in the title uses work of Finashin-Kharlamov. This is joint work with Tom Bachmann.

algebraic geometry

Audience: researchers in the topic

( video )

Stanford algebraic geometry seminar

Series comments: The seminar was online for a significant period of time, but for now is solely in person. More seminar information (including slides and videos, when available): agstanford.com