Let $Q^n$ be a smooth $n$-dimensional quadric in $P^{n+1}(\mathbb{C})$. Does the restriction of $\mathcal{O}(1)$ to $Q^n$ have a square root, for any $n \geq 1$?
If $n = 1$, then $Q^1$ is biholomorphic to $P^1(\mathbb{C})$, and the restriction of $\mathcal{O}(1)$ to $P^1(\mathbb{C})$ is $\mathcal{O}(2)$ (that is, $\mathcal{O}(2)$ with respect to $P^1(\mathbb{C})$) and thus has a square root, namely $\mathcal{O}(1)$ (with respect to $P^1(\mathbb{C})$).
My question is whether this also holds for $n > 1$.
Edit: I don't think so, but could someone please confirm? If $n = 2$, I think that the restriction of $\mathcal{O}(1)$ to $Q^2$, itself biholomorphic to $P^1(\mathbb{C}) \times P^1(\mathbb{C})$, is the line bundle $\mathcal{O}(1,1)$, which does not have a square root. Am I right?
You're correct about $n = 2$. One way to see this is that there is an embedding $\mathbb{CP}^1\times\mathbb{CP}^1 \to \mathbb{CP}^3$ given by $([x_0, x_1], [y_0, y_1]) \mapsto [x_0y_0, x_0y_1, x_1y_0, x_1y_1]$ with image the quadric defined by the equation $z_0z_3 - z_1z_2 = 0$. Restricting the map to $\mathbb{CP}^1\times\{[0, 1]\}$, we see that the image is the intersection of two hyperplanes, so pulling back $\mathcal{O}(1) \to \mathbb{CP}^3$ to $\mathbb{CP}^1$ via this map gives $\mathcal{O}(1)$, and likewise for the other factor.
For $n > 2$, the inclusion $i : Q^n \hookrightarrow \mathbb{CP}^{n+1}$ induces an isomorphism on cohomology $i^* : H^2(\mathbb{CP}^{n+1}; \mathbb{Z}) \to H^2(Q^n; \mathbb{Z})$ by the Lefschetz Hyperplane Theorem. As $c_1(\mathcal{O}(1))$ is a generator of $H^2(\mathbb{CP}^{n+1}; \mathbb{Z})$, the element $i^*c_1(\mathcal{O}(1)) = c_1(i^*\mathcal{O}(1))$ is a generator of $H^2(Q^n ; \mathbb{Z})$. It follows that $i^*\mathcal{O}(1)$ does not have a square root.
To apply the Lefschetz Hyperplane Theorem, note that a degree $d$ hypersurface of $\mathbb{CP}^{n+1}$ is a hyperplane section of $\nu_d(\mathbb{CP}^{n+1}) \subset \mathbb{CP}^N$ where $\nu_d : \mathbb{CP}^{n+1} \to \mathbb{CP}^N$ denotes the degree $d$ Veronese embedding.