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[ges ]2, a Grassmaniann G(2,n+2), n[ges ]4, or the Cayley plane OP$^2$, then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P$^2$×P$^2$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 v$_2$(P) and the Fubini cubic form of X at x is zero, then X=v$_2$(P) (resp. an open subset of v$_2$(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 I$_2$(P [sqcup ]P)⊂ S$^2$C, I$_2$(P$^1$× P)⊂ S$^2$C and I$_2$(S$_5$)⊂ S$^2$C$^16$are stable in the sense that if A ⊂S$^*$ is an analytic family such that for t≠0,A≃A, then A$_0$≃A.

We also make some observations related to the Fulton–:Hansen connectedness theorem.