# Weak Lefschetz theorem and hypersurface complements in projective space

I am reading the chapter on Griffiths Residues in Arapura's book "Algebraic Geometry over the Complex Numbers" and I find a particular statement very puzzling.

Let $$X \subset \mathbb{P}^{n+1}$$ be a hypersurface. Then Arapura claims that "by weak Lefschetz" the Gysin homomorphism $$H^{n-1}(X) \rightarrow H^{n+1}(\mathbb{P}^{n+1})$$ is an isomorphism. Here I mean singular cohomology over $$\mathbb{C}$$. But I thought that the Lefschetz hyperplane theorem (otherwise known as "weak Lefschetz") relates the cohomology of a variety with that of its hyperplane sections. $$X$$ need not be a hyperplane section, if could be a hypersurface of arbitrary degree.

• Have you looked at Arapura's statement of the weak Lefschetz (theorem 14.3.1)? This is exactly what that theorem says. Mar 23 '20 at 18:37
• @KReiser I have done. Theorem 14.3.1 is stated for $Y \subset X$ where $Y=X \cap H$, where $H$ is a hyperplane. I am confused precisely because in my question $X \not = \mathbb{P}^{n+1} \cap H$. Mar 23 '20 at 18:56

Let $$d$$ be the degree of $$X$$, and let $$\mathbb P^N$$ parameterizes degree $$d$$ monomials in $$\mathbb P^{n+1}$$. By Veronese embedding $$i: \mathbb P^{n+1}\hookrightarrow\mathbb P^N,$$ the defining equation of $$X$$ becomes linear in $$\mathbb P^N$$, so $$i(X)$$ is a hyperplane section of $$i(\mathbb P^{n+1})$$.