I am studying about the natural map (homomorphism) from the divisors of a complex manifold $X$ to the holomorphic line bundles on $X$, $$Div(X)\longrightarrow\;Pic(X)$$ where each divisor $D$ is sent to its associated line bundle $[D]$. The image of this map is the set of all line bundles which admit a meromoprhic line bundle.

It would help me tremendously if I had a few examples in mind.

(a) Can you give me an example of a line bundle $\mathbb{L}$ over a complex manifold $X$ which does not admit a meromorphic section?

(b) Is there an example where a line bundle admits a meromorphic section but not a holomorphic one? (which means that it is a line bundle $\mathbb{L}=[D]$ where $D$ cannot be effective)

Thank you!

  • 2
    $\begingroup$ For your second question, take $X=\Bbb P^n$ and $ \Bbb L=\omega_X$ it's canonical bundle. $\endgroup$ – Avi Steiner Oct 10 '15 at 4:45
  • 3
    $\begingroup$ Even easier, for the second question, take $X=\Bbb P^n$ and $\Bbb L = \mathscr O(-1)$, the tautological line bundle. $\endgroup$ – Ted Shifrin Oct 10 '15 at 5:03
  • 1
    $\begingroup$ For your first question, you need, for starters, a (compact) complex manifold that is non-algebraic. See this for discussion and references. $\endgroup$ – Ted Shifrin Oct 10 '15 at 5:13
  • $\begingroup$ Related: math.stackexchange.com/questions/1155939/… $\endgroup$ – user99914 Oct 10 '15 at 6:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.