The complex shade of a real space and its applications

A natural oriented $(2k+2)$-chain in $\mathbb{C}P^{2k+1}$ with boundary twice $\mathbb{R}P^{2k+1}$, the complex shade of $\mathbb{R}P^{2k+1}$, is constructed. The intersection numbers with the shade make it possible to introduce a new invariant, the shade number, of a $k$-dimensional subvariety $W$ with a normal vector field $n$ along the real set. If $W$ is an even-dimensional real variety, then the shade number and the Euler number of the complement of $n$ in the real normal bundle of its real part agree. If $W$ is an odd-dimensional orientable real variety, a linear combination of the shade number and the wrapping number (self-linking number) of its real part does not depend on $n$ and equals the encomplexed writhe as defined by Viro [V]. The shade numbers of varieties without real points and the encomplexed writhes of odd-dimensional real varieties are, in a sense, Vassiliev invariants of degree 1.
The complex shades of odd-dimensional spheres are constructed. The shade numbers of real subvarieties in spheres have properties similar to those of their projective counterparts.