Variation of Hodge structures for non-K\"ahler manifolds

In this note, we discuss unpolarized, complex variation of Hodge structures for non-K\"ahler manifolds. In particular, given a holomorphic family of compact complex manifolds whose central fiber satisfies: the inclusions $F^{p}A^{p+q+1}(X)\hookrightarrow A^{p+q+1}(X), F^{p}A^{p+q}(X)\hookrightarrow A^{p+q}(X)$ are injective in cohomology, it is shown that the period map is holomorphic and transversal.