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Let $S$ be a compact connected complex submanifold of $\mathbb{CP}^N$ (i.e. it is a smooth algberaic variety). Let $f: S \longrightarrow \mathcal{O}(1)|_S$ be a holomorphic section of the Hyperplane bundle restricted to $S$. Does $f$ extend to a holomorphic section on the whole of $\mathbb{CP}^N$, i.e. is there a globally defined holomorphic section $f: \mathbb{CP}^N \longrightarrow \mathcal{O}(1)$ whose restriction to $S$ is the given $f$?

$\textbf{Remark}:$ The above statement is not true if $S$ is not connected. For example, take $\mathbb{CP}^1$ and $S$ to be the union of the three points $[1,0], [0,1]$ and $[1,1]$. Now consider the section, induced by the homogeneous function of two variables given by $$ f(1,0) = 1, \qquad f(0,1) = 1, \qquad f(1,1) = 3. $$ This $f$ does induce a section on $S$. But since there is no homogeneous (degree one) function in two variables $f(X,Y)$ that satisfies the above conditions, there is no globally defined $f$ that restricts to the given $f$ (on $S$).

However, here $S$ is not connected.

Ritwik
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  • You may want to have a look at Exercise 5.14(d) of Chapter II, section 5 in Hartshorne's Algebraic Geometry. – random123 Jun 10 '18 at 10:47
  • One of the simplest example where this fails is for a projective line embedded as a quartic curve in 3-space. – Mohan Jun 10 '18 at 15:54
  • @Mohan: In that case, what is the example of f? Could you give me a reference so that I can study this counter example? – Ritwik Jun 10 '18 at 18:19
  • Sections as you describe are just elements of $V=H^0(S,\mathcal{O}S(1))$. So, you just want to make sure that the natural map $H^0(\mathbb{P}^3,\mathcal{O}{\mathbb{P}^3}(1))\to V$ is not onto. The former is four dimensional, the latter five dimensional in the case I describe. – Mohan Jun 10 '18 at 18:40
  • What's relevant here is the notion of projective normality. The usual way of understanding this (e.g., here) is in terms of the short exact sequence of sheaves $0\to\mathscr I_S \to \mathscr O_{\Bbb P^n}\to\mathscr O_S\to 0$. – Ted Shifrin Jun 11 '18 at 16:59

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