13.
Let XP be a variety (respectively an open subset of an analytic submanifold) and let xX 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,m2, a Grassmaniann G(2,n+2), n4, 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 P) S
2C, I
2(P
1× P) S
2C and I
2(S
5) S
2C
16 are stable in the sense that if A S
* is an analytic family such that for t0,AA, then A
0A. We also make some observations related to the Fulton–:Hansen connectedness theorem.
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