Let $X$ be a compact Kahler manifold, let $D$ be a smooth divisor in $X$, and let $U$ be a tubular neighbourhood of $D$ in $X$. Suppose that $D$ is Fano. Is it possible to extend every closed (1, 1)-form on $D$ to a closed (1, 1)-form on $U$? 



Is it always possible to extend a closed (1, 1)-form on a divisor to a closed (1, 1)-form on a tubular neighbourhood?
Yes. For the proof, see e.g. <http://arxiv.org/abs/math/0609617>
Theorem 4.1. Here a stronger result is actually proven:


Theorem: Let $(M, \omega)$ be a compact Kahler manifold,
and $Z\subset M$ a closed complex submanifold.
Denote by $[\omega]\in H^2(M)$ the Kahler class of $M$.
Consider a Kahler form $\omega\_0$ on $Z$ such that
its Kahler class coinsides with the restriction
$[\omega]| Z$. Then there exists a
Kahler form $\omega'$ on $M$ in the same Kahler
class as $\omega$, such that $\omega|
Z=\omega\_0$.


An additional cohomological assumption is needed, because we build a global extension, and for an extension to a local neighbourhood you don't need it; the positivity (needed in assumption) is achieved by adding a big multiple of a Kaehler form.


