On the Infinitesimal Rigidity of Homogeneous Varieties

Abstract

Let X⊂P be a variety (respectively an open subset of an analytic submanifold) and let x∈X be a point where all integer valued differential invariants are locally constant. 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 P× P, n,m≥2, a Grassmaniann G(2,n+2), n≥4, or the Cayley plane OP2, then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P2×P2 had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v2(P) and the Fubini cubic form of X at x is zero, then X=v2 (P) (resp. an open subset of v2(P)). 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(P ⊔P)⊂ S2C, I2(P1× P)⊂ S2C and I2(S5)⊂ S2C16 are stable in the sense that if A ⊂S* is an analytic family such that for t≠0,A≃A, then A0≃A. We also make some observations related to the Fulton–:Hansen connectedness theorem.