Complete intersections as branched covers and the Kervaire invariant

Abstract

A complete intersection is the transversal intersection of hypersurfaces in complex projective space. I f X , C CP,, is the intersection of r = m n hypersurfaces we show in §2 that, up to diffeomorphism, X,, is a regular branched cyclic cover of an intersection of r 1 hypersurfaces. This extends the well-known and useful fact that a hypersurface is a regular branched cyclic cover of complex projective space. The remainder of the paper is concerned with when the Kervaire invariant of a complete intersection X, of odd complex dimension n is defined in a certain geometric way, with comput ing this invariant, and with a connected sum decomposit ion of X,. In §5 are some results on the question when the normal bundle of an embedding f depends only on the homotopy class o f f If the hypersurfaces whose intersection is X, have degrees d 1 . . . . . dr, we call d =(d 1 . . . . . d,) the multi-degree of X. The degree of X is d=lld~.