Let be a surjective projective map, with a complex smooth projective variety and curve, and denote by . Then, contains the image of
is a smooth fiber. What can we say about the converse?
One of the amazing applications of the theory of mixed Hodge structures leads to a proof that any invariant cycle lifts to
Indeed, in The'orie de Hodge II, Deligne proved the following result.
Theorem 11.3 (Deligne's global invariant cycles theorem)
The subspace of monodromy invariants is
Two of the main ingredients of the proof are the degeneration on the first page of the Leray spectral sequence for and the properties of the mixed Hodge structures (in particular, strictness of maps between them). This is a clear example where the existence of this rich structure on the cohomology of allows us to prove something of topological nature.