# Applications

Let ${\displaystyle X\to C}$ be a surjective projective map, with ${\displaystyle X}$ a complex smooth projective variety and ${\displaystyle C}$ curve, and denote ${\displaystyle C-\{{\mbox{critical values}}\}}$ by ${\displaystyle C^{*}}$. Then, ${\displaystyle H^{k}(X_{t})^{\pi _{1}(C^{*})}}$ contains the image of

${\displaystyle H^{k}(X-({\mbox{critical fibers}}))\to H^{k}(X_{t}),}$
where ${\displaystyle X_{t}}$ 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 ${\displaystyle H^{k}(X)}$. 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

${\displaystyle H^{k}(X_{t})^{\pi _{1}(C^{*})}=\operatorname {im} (H^{k}(X)\to H^{k}(X_{t})).}$

Two of the main ingredients of the proof are the degeneration on the first page of the Leray spectral sequence for ${\displaystyle f}$ 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 ${\displaystyle X}$ allows us to prove something of topological nature.