Weak Lefschetz theorem for Lef line bundles

Question

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 4 35 (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.

Answer 1

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 $\mathbb{P}^N\backslash H$.

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 $Y$.

In the semismall case, $r(f) = 0$. Now we have

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.

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 $H^k_c(U,\mathbb{Q}) \simeq H^{2n-k}(U,\mathbb{Q})$.