  According to the decomposition and relative hard Lefschetz theorems, given a
projective map of complex quasi projective algebraic varieties and a relatively
ample line bundle, the rational intersection cohomology groups of the domain of
the map split into various direct summands. While the summands are canonical,
the splitting is certainly not, as the choice of the line bundle yields at
least three different splittings by means of three mechanisms in a triangulated
category introduced by Deligne. It is known that these three choices yield
splittings of mixed Hodge structures. In this paper, we use the relative hard
Lefschetz theorem and elementary linear algebra to construct five distinct
splittings, two of which seem to be new, and to prove that they are splittings
of mixed Hodge structures.
