Higher-order differentials of the period map and higher Kodaira-Spencer classes

Abstract

In \cite{K} we introduced two variants of higher-order differentials of the period map and showed how to compute them for a variation of Hodge structure that comes from a deformation of a compact K\"ahler manifold. More recently there appeared several works (\cite{BG}, \cite{EV}, \cite{R}) defining higher tangent spaces to the moduli and the corresponding higher Kodaira-Spencer classes of a deformation. The $n^{th}$ such class $\kappa_n$ captures all essential information about the deformation up to $n^{th}$ order. A well-known result of Griffiths states that the (first) differential of the period map depends only on the (first) Kodaira-Spencer class of the deformation. In this paper we show that the second differential of the Archimedean period map associated to a deformation is determined by $\kappa_2$ taken modulo the image of $\kappa_1$, whereas the second differential of the usual period map, as well as the second fundamental form of the VHS, depend only on $\kappa_1$ (Theorems 2, 5, and 6 in Section~3). Presumably, similar statements are valid in higher-order cases (see Section~4).