I'm studying



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> M. A. A. de Cataldo, L. Migliorini - *The Hard Lefschetz Theorem and the topology of semismall maps*, Ann. sci. École Norm. Sup., Serie 4 **35** (2002) 759-772.
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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*



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> 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 $i=\dim X-1$.
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> *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. [...] $\Box$
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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.



Weak Lefschetz theorem for Lef line bundles
It is based on certain vanishing property of $U= X\backslash Y$.
First you have a long exact sequence (a derived categorical version is given in the end)
$$H^k(X,Y;\mathbb{Q})\rightarrow H^k(X,\mathbb{Q}) \rightarrow H^k(Y,\mathbb{Q}) \rightarrow H^{k+1}(X,Y;\mathbb{Q}).$$
Note that $H^{k}(X,Y;\mathbb{Q})=H^{k}\_c(U,\mathbb{Q}\_Y) \simeq H^{n-k}(U, \mathbb{Q}\_Y)$. The last isomorphism is the Poincaré duality which only requires that $U$ is smooth. This is true since $X$ is smooth. So if we have $H^k(U, \mathbb{Q}\_Y) = 0$ for $k>n$, then we have done.


The vanishing is based on the citation [10] in the article, namely *Vanishing and non-vanishing theorems*
Astérisque, tome 179-180 (1989), p. 97-112.


Let me pick up the key part. Let $f:X \rightarrow \mathbb{P}^N$ be the corresponding morphism of $M$. Then $Y$ is a pullback of a hyperplane $H \subset \mathbb{P}^N$ by $f$ and hence $f$ restricting on $U$ is a map into a affine variety $\mathbb{P}^N\backslash H$.



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> Definiton 1.1
> Let $g : Y \rightarrow Z$ be a morphism of analytic varieties. We define $r(g) = \mathrm{Max}\{\mathrm{dim} \Gamma - \mathrm{dim} g(\Gamma) - \mathrm{codim} \Gamma \}$, $\Gamma$ closed subvariety of $Y$.
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In the semismall case, $r(f) = 0$. Now we have



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> Lemma 1.2 Assume that there exists a proper surjective morphism $g$ from $U$ to an affine variety $W$.
> Then $H^k(U,\mathscr{L})=0$ for $k>n+ r(g)$ and $\mathscr{L}$ a local system.
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So the vanishing follows.




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A derived categorical version


$$ Y \overset{i}{\hookrightarrow} X \overset{j}{\hookleftarrow} U=X\backslash Y$$


$$\rightarrow j\_!j^\*\mathbb{Q}\_X \rightarrow \mathbb{Q}\_X \rightarrow i\_\*i^\* \mathbb{Q}\_X \rightarrow j\_!j^\*\mathbb{Q}\_X[1]\rightarrow $$
and apply $R^0\Gamma = H^0c\_\*$, i.e. taking the hypercohomology, where $c\_\*$ is the pushforward in derived category to a point.


Since $X$ is projective hence proper, $c\_\* = c\_!$. So
$$R^0\Gamma j\_!j^\* \mathbb{Q}\_X[k] = H^0 c\_\* j\_!j^\* \mathbb{Q}\_X[k] = H^0 c\_!j\_!\mathbb{Q}\_U[k] = H^0 c\_{U,!} \mathbb{Q}\_U[k] = H^k\_c(U,\mathbb{Q})$$
where $c\_{U,!}$ is the direct image with proper support pushforward to a point from $U$. So we have long exact sequence
$$H^k\_c(U,\mathbb{Q}) \simeq H^{2n-k}(U,\mathbb{Q}) \rightarrow H^k(X,\mathbb{Q}) \rightarrow H^k(Y,\mathbb{Q}) \rightarrow H^{k+1}\_c(U,\mathbb{Q})$$
where the first isomorphism is by the Poincare duality. Note that the Poincare duality only requires that $U$ is smooth. This is true if $X$ is smooth or $Y$ contains all singularity. The result follows by the long exact sequence and the vanishing of $H^k\_c(U,\mathbb{Q}) \simeq H^{2n-k}(U,\mathbb{Q})$.


