There are 160,839<1> + 160,650<-1> 3-planes in a 7-dimensional cubic hypersurface

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 A1-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.

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