On the infinitesimal rigidity of homogeneous varieties

Original · EN

Let X⊂ Pⁿ be a variety (respectively a patch of an analytic submanifold) and let x∈ X be a general point. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre Pⁿ× Pᵐ, n,m≥ 2, a Grassmaniann G(2,n+2), n≥ 4, or the Cayley plane OP², then X is the corresponding homogeneous variety (resp. a patch of the corresponding homogeneous variety). If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v₂(Pⁿ) and the Fubini cubic form of X at x is zero, then X=v₂(Pⁿ) (resp. a patch of v₂(Pⁿ)). All these results are valid in the real or complex analytic categories and locally in the C∞ category if one assumes the hypotheses hold in a neighborhood of any point x. As a byproduct, we show that the systems of quadrics I₂(Pᵐ⁻¹ Pⁿ⁻¹), I₂(P¹× Pⁿ⁻¹) and I₂(S₅) are stable in the sense that if Aₜ⊂ S²T* is an analytic family such that for t≠ 0, Aₜ≃ A, then A₀≃ A. We also make some observations related to the Fulton-Hansen connectedness theorem.