Freeness of the integral middle cohomology of a smooth hypersurface in $\mathbb{C}P^{n+1}$

Question

Take $X$ a hypersurface in $\mathbb{C}P^{n+1}$ of degree $d$, denote $\Lambda_X=H^n(X,\mathbb{Z})$. We know that $\Lambda_X$ is a finitely generated Abelian group. I was wondering whether $\Lambda_X$ is always free (for any $n$, $d$). Anyone know an answer or a refrence of this? Thanks a lot!

Answer 1

If $X$ is smooth, then by the Lefschetz hyperplane theorem, $H_k(X; \mathbb{Z}) \cong H_k(\mathbb{CP}^{n+1}; \mathbb{Z})$ for $k < n$. By the Universal Coefficient Theorem,

$$H^n(X; \mathbb{Z}) \cong \operatorname{Hom}(H_n(X;\mathbb{Z}), \mathbb{Z})\oplus\operatorname{Ext}^1(H_{n-1}(X; \mathbb{Z}), \mathbb{Z}).$$

As $H_{n-1}(X; \mathbb{Z}) \cong H_{n-1}(\mathbb{CP}^{n+1}; \mathbb{Z})$ is either $0$ or $\mathbb{Z}$ depending on the parity of $n$, we see that $\operatorname{Ext}^1(H_{n-1}(X; \mathbb{Z}), \mathbb{Z}) = 0$. Therefore $H^n(X; \mathbb{Z}) \cong \operatorname{Hom}(H_n(X; \mathbb{Z}), \mathbb{Z})$ which is free.