Symmetric differentials and variations of Hodge structures.

Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X - D X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X , D) (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big. [ABSTRACT FROM AUTHOR]