Degrees of real Grassmann varieties

Let G=G(m,m+p) be the Plucker embedding of the Grassmannian of m-subspaces in the real (m+p)-dimensional space. We consider a central projection of G into a projective space of the same dimension as G. Although G may be non-orientable, the topological degree of this projection can be properly defined. Its values are unsigned integers. It turns out that when mp is even, the degree is independent of the center of projection, so we call it deg G, the degree of G.

For even mp, we show that deg G=0 if and only if m+p is even, and give an explicit formula for deg G when m+p is odd.

The result has applications to real enumerative geometry and to the problem of pole assignment by static output feedback in control theory. It implies that, for a generic linear system with m inputs, p outputs, and state of dimension mp, such that m+p is odd, one can assign arbitrary symmetric set of mp poles with a real gain matrix.