Let \(X\) be an algebraic regular threefold with trivial canonical bundle. Suppose \(H^1(X,\theta)\) unobstructed. Then the local moduli space \(\{X_s\}_{s\in S}\) of \(X=X_0\) has dimension \(m=h^{2,1}(X)\) and the period mapping \(\phi: S\to \tilde D\) can be thought of as a holomorphically varying filtration of weight three \(\{F^p(s)\}\) on \(H: H^3(X,C)\). \(dF^p(s)\subset F^{p-1}(s)\) are the infinitesimal period relations: these conditions define a subbundle \(T_h(\tilde D)\subset T(\tilde D)\) and correspondingly a graded sheaf of ideals \(I\subset \Omega^*_D (= \) sheaf of exterior algebra of holomorphic forms). In other words \(I\) defines a differential system. The general theory of differential system applied to this case gives in particular the following results:
(a) Let \(\alpha_0,\ldots,\alpha_m\), \(\beta_0,\ldots,\beta_m\in H^3(X,Z)\) be a canonical homology basis \((\alpha_i\cdot \alpha_j=0=\beta_i\cdot \beta_j\), \(\alpha_i\cdot \beta_j=\delta_{ij})\), \(\omega(s)\in H^{3,0}(X_s)\) a generator and set \(A_i(s)=\int_{\alpha_i} \omega(s)\), \(B_i(s)=\int_{\beta_i} \omega(s):\) the Hodge structure \(H^{p,q}(X_s)\) is completely determined by the functions \(A_i(s)\).
(b) Let \(J(X)\) be the intermediate Jacobian of \(X\). The infinitesimal variation of the Hodge structure associated to \(X\) gives a \(\otimes^2H^{0,3}(X)\)-valued cubic form \(\eqcirc\) on \(T^*_{(0)}(J(X))\). The algebraic invariants of the infinitesimal variation of this Hodge structure are determined by \(\eqcirc\). Moreover the authors conjecture that, in general, this cubic form uniquely determines \(X\).
[For the entire collection see Zbl 0518.00005.]