Let $X$ be a compact Kähler manifold and $S \subset X$ a closed complex submanifold. Given a closed $(1, 1)$-form $\alpha$ on $S$, is there always a closed $(1, 1)$-form $\beta$ on a neighborhood of $S$ in $X$ such that $[\beta|_S] = [\alpha]$ in $H^{1, 1}(S, \mathbb{C})$?

**Note.** [This][1] mathoverflow answer seems related, but I don't think it answers the above question. The proof of Theorem 4.1 in [arXiv:math/0609617][2] begins by assuming that we have such an extension and pursue by proving positivity results. I fail to see how any of the ideas in the proof would help with the question above. But I might be wrong, so any help in that direction would be great too.

**Addendum** (In reply to Donu Arapura's answer)**.** There is related work by Griffiths (*The extension problem in complex analysis. II. Embeddings with positive normal bundle.* Amer. J. Math. 88 (1966)), showing the obstruction to extending certain analytic objects from $S$ to a neighborhood of $S$ in $X$. For instance, it is proved that if $F \to S$ is a vector bundle which is the restriction of a vector bundle $E \to U$ on a neighborhood $U$ of $S$ in $X$, and $\alpha \in H^q(S, F)$, then the first order obstruction to extending $\alpha$ to an element of $H^q(U, E)$ (possibly after shrinking $U$) lies in $H^{q+1}(S, F \otimes N_S^*)$, where $N_S^*$ is the conormal bundle of $S$ in $X$, and it does not vanish in general. In our case, we can interpret $\alpha$ as an element of $H^1(S, T^*S)$. But this does not immediately fit into Griffiths work, since $T^*S$ does not generally extend to a vector bundle on a neighborhood of $S$ in $X$.

  [1]: https://mathoverflow.net/a/58505/392184
  [2]: https://arxiv.org/abs/math/0609617