I'm studying

M. A. A. de Cataldo, L. Migliorini -

The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 435(2002) 759-772.

The premises are the following.

Let $f:X\to Y$ be a proper holomorphic (non constant) map of irreducible, complex, projective varieties of dimension $n$. For every $k\in\{-\infty,0,\dotsc,\dim X\}$ defined $Y^k=\{y\in Y\mid\dim f^{-1}(y)=k\}$ with the convection $\dim\emptyset=-\infty$. These spaces $Y^k$ are locally closed analytic subvarieties of $Y$ which (disjoint) union is $Y$ as well.

**Definition 1.** A proper holomorphic map $f:X\to Y$ is called *semismall* if $\dim Y^k+2k\leq\dim Y$ for any $k$.

From now on, I assume that all semismall maps are proper and surjective.

**Definition 2.** A line bundle $L$ over $X$ is *Lef* (*Lefschetz effettivamente funziona*) if a positive multiple of $L$ is generated by its global sections and the corresponding morphism (onto the image) is semismall; in other words there exist $d,N\gg0,\,f:X\to\mathbb{P}^d$ semismall (onto the image) such that $L^{\otimes N}\cong~f^{*}\mathcal{O}_{\mathbb{P}^d}(1)$.

The authors stated the following *Weak Lefschetz theorem for Lef line bundles*

Let $L$ be a Lef line bundle over a smooth, complex, projective variety $X$. Assume that $L$ admits a global section $s$ which reduced locus $Y$ is a smooth divisor, and denoted by $i:Y\hookrightarrow X$ the relevant inclusion. The restriction maps $i^{*}:H^k(X)\to H^k(Y)$ are isomorphisms for $k\in\{0,\dotsc,\dim X-2\}$ and a monomorphism for $i=\dim X-1$.

Proof.The proof can be obtained by a use of Leray spectral sequence coupled with the theorem on the cohomological dimension of constructible sheaves on affine varieties. [...] $\Box$

Ignore whether this proof is a *standard* application of some ideas\techniques, indeed I have no idea on how to explicit it: can someone give me advice, hint, "*roadmap*", bibliographical sources?

Thanks in advance.