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 $4(n-2)=\binom{d+3}{3}$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3$ 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.