Variations of Hodge structure of maximal dimension (Q921071)

A general bound on the rank of period mappings \(X\to \Gamma \setminus D\) is given, improving the bound provided by \textit{J. A. Carlson} in Trans. Am. Math. Soc. 294, 45-64 (1986; Zbl 0593.14006); correction ibid. 299, 429 (1987)]and by \textit{J. A. Carlson} and \textit{D. Toledo} in Trans. Am. Math. Soc. (to appear). Here X is any complex manifold and \(\Gamma\setminus D\) is a Griffiths period domain. This bound is shown to be sharp for X a polydisk and, with a certain exception, for X a quasi- projective variety.