On local Torelli for a class of extremal varieties

We study the local behaviour of the period map for smooth non-degenerate extremal varieties sitting inside scrolls. We conclude with the following theorem:the period map for smooth non-degenerate extremal varieties of dimension k ≠ 2,3in n-projective space with n≥2k+4is generically locally injective for sufficiently high degree d. In particular, the following two conditions on d suffice: (1)d≥3.k.(n−k)+3; (2)d≥(n − k)²+1.