# Holomorphic extension of a section of a line bundle

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 $$|\nabla^{n}Riem|_{g}\leq C_{n} \qquad n\in\mathbb{N} \textrm{ and }C_{n}>0$$ 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 $\mathcal{L}$?

• 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 $\text{pr}_{\mathbb{C}}^*(1/z)$. May 4, 2015 at 13:06
• 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 $\mathbb{C}^{m}$. May 4, 2015 at 13:18
• "... 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. May 4, 2015 at 13:49
• You're completely right i need to clear up my mind. Thank you! May 4, 2015 at 13:59
• anyway is it possible that such a section $\sigma$ has both zeroes and poles along $K$? May 4, 2015 at 14:13