Example of a variation of Hodge structure such that the gradation does not vary holomorphically The motivation for considering the Hodge filtration as opposed to just the grading is supposed to be that the filtration varies holomorphically in a family, whereas the grading does not always.
Can someone provide an example of a smooth morphism $f \colon X \to S$ such that the fibers are smooth projective complex manifolds where the grading on $ R^k f_* \mathbb{Z}$ does not vary holomorphically over $S$? Ideally, the example would be a family of abelian varieties.
 A: Almost any interesting example that you can write down will satisfy this property.
E.g. consider a non-trivial family of elliptic curves over some base $S$
(the family $y^2 = x(x-1)(x-\lambda)$ over $\mathbb P^1\setminus \{0,1,\infty\}$
will do).
Then you get a two-dimensional variation of Hodge structure $\mathcal V$ which is a rank two
holomorphic bundle over $S$.  The first step in the Hodge filtration gives a rank one subbundle $\mathcal F \subset \mathcal V$.    This interpolates the $(1,0)$
part of the grading.  It is spanned by the holomorphic differential $\omega := dx/2y$.
If we restrict to a small open subset $U$ of $S$, over which we can trivialize the family of elliptic curves in the smooth category, then we can choose a constant basis for the homology of the family, and integrate $dx/2y$ over it,
to get a pair of functions $\omega_1$ and $\omega_2$, which are the periods of the
elliptic curves in the family; they vary holomorphically, which reflects the
fact that $\mathcal F$ is a holomorphic bundle.   But if we integrate a basis
for the $(0,1)$ part of the grading over this homology basis, we get the pair of anti-holomorphic functions $\overline{\omega}_1, \overline{\omega}_2$.  If the family of elliptic curves is non-trivial, then these will really form non-constant holomorphic functions on $U$, implying that the $(0,1)$ part of the grading cannot be interpolated holomorphically.
Summary: The basic point is that the $(q,p)$ part of the grading is obtained from the  $(p,q)$ part by applying complex conjugation, and this is not a process that will preserve the holomorphicity of a bundle in general.  I.e. the Hodge filtration varies holomorphically but the conjugate filtration $\overline{F}$ does not, and hence neither does the intersection $F^p\cap
\overline{F}^q$, in general.
