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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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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.

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Let \alpha be a closed (1, 1)-form on Z. Add a big multiple of a Kahler form \omega' on Z to \alpha, so that k\omega'+\alpha is a positive (1, 1)-form on Z for some large positive real number k. How do I know that there exists a Kahler class [\omega] on M that restricts to the Kahler class [k\omega'+\alpha] on Z? This may be obvious... –  user3566 Mar 16 '11 at 13:24
    
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

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