Abundance of 3-planes on real projective hypersurfaces

Abstract

We show that a generic real projective n-dimensional hypersurface of odd degree d , such that 4(n - 2) = (d + 3 3), contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, d3 log d, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.