# Weak Lefschetz theorem for Lef line bundles

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?

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  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.

A derived categorical version

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

$$\rightarrow j_!j^*\mathbb{Q}_X \rightarrow \mathbb{Q}_X \rightarrow i_*i^* \mathbb{Q}_X \rightarrow j_!j^*\mathbb{Q}_X\rightarrow$$ 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 $$H^k_c(U,\mathbb{Q}) \simeq H^{2n-k}(U,\mathbb{Q})$$.

• You are assuming that $U$ is affine, so this is just the standard Lefschetz theorem.
– abx
Nov 22, 2020 at 7:11
• @abx This is true since a projective variety excising a hyperplane is affine. Nov 22, 2020 at 7:15
• Of course, but this is not what the OP is asking for.
– abx
Nov 22, 2020 at 10:14
• @abx Doesn't the OP want an explicit proof? Sorry for my poor English reading ability. Nov 22, 2020 at 10:16
• @Armandoj18eos Oh sure. Thank you! Nov 23, 2020 at 14:09