Complexity of intersection of real quadrics and topology of symmetric determinantal varieties

Let W be a linear system of quadrics on the real projective space ℝPn and X be the base locus of that system (i.e. the common zero set of the quadrics in W). We prove a formula relating the topology of X to that of the discriminant locus ∑W (i.e. the set of singular quadrics in W). The set ∑W equals the intersection of W with the discriminant hypersurface for quadrics; its singularities are unavoidable (they might persist after a small perturbation of W) and we let {∑W (r)}r≥1 be its singular point filtration, i.e. ∑W (1) = ∑W and ∑W (r) = Sing(∑W (r-1)). With this notation, for a generic W the above mentioned formula reads b(X) ≤ b(ℝPn) + ∑r≥1b(ℙ∑W (r)). In the general case a similar formula holds, but we have to replace each b(ℙ∑W (r)) with 1/2b(∑ε (r)), where ∑ε equals the intersection of the discriminant hypersurface with the unit sphere on the translation of W in the direction of a small negative definite form. Each ∑ε (r) is a determinantal variety on the sphere Sk-1 defined by equations of degree at most n + 1 (here k denotes the dimension of W); we refine Milnor's bound, proving that for such affine varieties, b(∑ε (r)) ≤ O(n)k-1. Since the sum in the above formulas contains at most O(k)1/2 terms, as a corollary we prove that if X is any intersection of k quadrics in ℝPn then the following sharp estimate holds: b(X) ≤ O(n)k-1. This bound refines Barvinok's style estimates (recall that the best previously known bound, due to Basu, is O(n)2k+2). © European Mathematical Society 2016.