2
$\begingroup$

let $(X,g,\omega)$ be a non compact complete K\"ahler manifold of dimension $m\geq3$. Let $\nabla$ the covariant derivative wrt $g$ and $Riem$ the curvature tensor of $g$. Suppose that \begin{equation} |\nabla^{n}Riem|_{g}\leq C_{n} \qquad n\in\mathbb{N} \textrm{ and }C_{n}>0 \end{equation} i.e. the curvature tensor and all of its derivatives are uniformly bounded. Let moreover $\mathcal{L}$ be a holomorphic line bundle over $X$. Let $K\subset X$ be a compact analytic subvariety of dimension $m-1$ and $\sigma$ be a section of $\mathcal{L}|_{X\setminus K}$ such that $\sigma\neq 0$ everywhere.

Is $\sigma$ extendable to a holomorphic section of $\mathcal{L}$?

$\endgroup$
  • $\begingroup$ This already fails if $(X,g,\omega)$ is, say, $\mathbb{C}$ with its flat metric (complete and Kaehler), $\mathcal{L}$ is the trivial holomorphic line bundle $\mathbb{C}\times \mathbb{C}$, $K$ is a single point, say $\{0\}$, and $\sigma$ is a rational function like $1/z$. I realize that you write $m\geq 3$, but you can always repeat this with $X=\mathbb{C}\times T$, where $T$ is a complex torus of dimension $g\geq 2$, $K$ is $\{0\}\times T$, and $\sigma$ is $\text{pr}_{\mathbb{C}}^*(1/z)$. $\endgroup$ – Jason Starr May 4 '15 at 13:06
  • $\begingroup$ Thank you for the answer, but i (implicitly, sorry for being imprecise) assume that $X$ is not a product. The case i have in mind it is indeed a resolution of a singularity $\mathbb{C}^{m}/G$ with $G$ finite subgroup of $U(m)$ and $G$ acting freely outside the origin of $\mathbb{C}^{m}$. $\endgroup$ – student May 4 '15 at 13:18
  • 1
    $\begingroup$ "... I (implicitly, sorry for being imprecise) assume that $X$ is not a product." In that case, I think you should change the statement of your question. The basic issue persists: if you define $\mathcal{L}$ to be the invertible sheaf associated to the complex hypersurface $K$, then the tautological section $1$ on $X\setminus K$ is polar on $K$. In your special case, if $\mathcal{L}$ is the pullback of a holomorphic line bundle from the singular variety $\mathbb{C}^m/G$, then you should be able to use Hartog's theorem to extend $\sigma$ on $\mathbb{C}^m/G$, and then pull this back. $\endgroup$ – Jason Starr May 4 '15 at 13:49
  • $\begingroup$ You're completely right i need to clear up my mind. Thank you! $\endgroup$ – student May 4 '15 at 13:59
  • $\begingroup$ anyway is it possible that such a section $\sigma$ has both zeroes and poles along $K$? $\endgroup$ – student May 4 '15 at 14:13

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.