Section of a holomorphic line bundle with given differential at a zero

Let $X$ be a compact Kähler manifold of dimension $n$ with a given Kähler metric $\omega$. Let $L$ be a hermitian holomorphic line bundle on $X$ whose metric is positive. Let $x_0\in X$.

I would like to construct a section $s\in H^0(X,L)$, so that $s(x_0)=0$ and $ds(x_0)=\alpha\neq 0$ for given $\alpha$. Moreover, I want to control the $L^2$ norm of $s$ by some function of $|\alpha|$. What would be the appropriate conditions for such sections to exist?

Here $ds$ is the differential of $s$, since $x_0$ is a zero of $s$, $ds(x_0):T_{x_0}X \rightarrow L_{x_0}$ is intrinsically defined.

The question is similar to a special case of Ohsawa-Takegoshi theorem, where the prescribed information is only $s(x_0)$. If there is a good solution to my question, then I would like to know if it is possible to replace the single point $x_0$ by a more general analytic subset of $X$?

• What is the question? Already on a compact curve of genus $g$, a general holomorphic line bundle of degree $<g$ admits a metric of positive curvature, but no nonzero section. – abx Apr 12 '18 at 19:47
• @abx Actually, there should be some kind of conditions making the section that I want exist. I have in my mind the conditions like in Ohsawa-Takegoshi theorem, namely we need a lower bound on the curvature of $L$. Otherwise, you may simply take very ample $L$. – Mingchen Xia Apr 12 '18 at 20:51
• @abx An interesting special case would be as follows: we may replace $L$ by $L^k$ for all $k>0$, then when $k$ is large enough and $n>1$, a section $s$ that vanishes at $x_0$ and has the given tangent does exist by dimension counting and very ampleness, the only problem is then to control the $L^2$-norm. – Mingchen Xia Apr 12 '18 at 21:05