On the Infinitesimal Rigidity of Homogeneous Varieties
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Let X ⊂ double struck P signN be a variety (respectively an open subset of an analytic submanifold) and let cursive Greek chi ∈ 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 cursive Greek chi is isomorphic to the second fundamental form of a point of a Segre double struck P signn × double struck P signm, n, m ≥ 2, a Grassmaniann G(2, n + 2), n ≥ 4, or the Cayley plane double struck O signdouble struck P sign2, then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre double struck P sign2 × double struck P sign2 had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at cursive Greek chi is isomorphic to the second fundamental form of a point of a Veronese v2(double struck P signn) and the Fubini cubic form of X at cursive Greek chi is zero, then X = v2(double struck P signn) (resp. an open subset of v2(double struck P signn)). 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 cursive Greek chi. As a byproduct, we show that the systems of quadrics I2(double struck P signm-1 (Square cup) double struck P signn-1) ⊂ S2double struck C signm+n, I2(double struck P sign1 × double struck P signn-1) ⊂ S2double struck C sign2n and I2(double struck S sign5) ⊂ S2double struck C sign16 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.