# Is it always possible to extend a closed (1, 1)-form on a divisor to a closed (1, 1)-form on a tubular neighbourhood?

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 $U$?

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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 $\omega| Z=\omega_0$.
No, it's not obvious, in fact, it can be false - the existence of a Kaehler class which restricts to $[\omega_0]$ is not guaranteed. However, the proof of the above theorem works to extend the form to a tubular neighbourhood. –  Misha Verbitsky Mar 19 '11 at 22:13