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Generic Torelli and variational Schottky. (English) Zbl 0603.14010

Topics in transcendental algebraic geometry, Ann. Math. Stud. 106, 239-258 (1984).
[For the entire collection see Zbl 0528.00004.]
Our aim in this work is to give a simple treatment of the generic Torelli theorem for projective hypersurfaces, proved by the author in Compos. Math. 50, 325-353 (1983; Zbl 0598.14007), and of related results. The main result of the cited paper asserts the generic injectivity of the period map for n-dimensional projective hypersurfaces of degree \(d\) with the following possible exceptions: (0) \(n=2\), \(d=3\) (cubic surfaces). (1) d divides \(n+2\). (2) \(d=4\), \(n=4m\) or \(d=6\), \(n=6m+1\) \((m\geq 1)\). The proof of that result uses a melange of methods. Here we focus attention on one of these methods, the use of symmetrizers.
In the last section we discuss Schottky’s problem of determining which Hodge structures arise from geometry. We formulate the variational version of this problem, solve it for hypersurfaces using the symmetrizer construction, and muse on the variational Schottky problem for curves and its relation to classical Schottky.

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K30 Picard schemes, higher Jacobians
32G20 Period matrices, variation of Hodge structure; degenerations