I am wondering about the following situation. Suppose $X$ is a compact Kahler manifold and $F \subset T_X$ is a holomorphic foliation. Suppose that the ``normal bundle'' $T_X / F$ of the foliation is trivial. Suppose that I have a compact leaf $L \subset X$. Is it true that the holonomy of $L$ is trivial?

My guess is that the holomorphicity of $F$ and $X$ being Kahler is irrelevant but this is the situation I am considering.

Since $L$ has a trivial normal bundle, I want to say that there is a tubular neighborhood of $L$ such that $F$ is taken to the trivial horizontal foliation in $N_L = L \times \mathrm{C}^k$. I do not see how to do this directly.

Choose a basis $\omega_1, \dots, \omega_k$ giving a frame of $(T_X / F)^*$ and choose a ``constant metric'' on $T_X/F$ meaning,

$$ g(X,Y) = \sum_{i = 1}^k \omega_i(X) \omega_i(Y) $$

I claim that this is a bundle-like metric for $F$. Indeed,

$$ (\mathcal{L}_Z g)(X,Y) = Z(g(X,Y)) + g([Z,X], Y) + g(X, [Z,Y]) $$

but the forms $\omega_i$ are harmonic so closed meaning,

$$ Z(\omega_i(X)) - X(\omega_i(Z)) = \omega_i([Z,X]) $$

and if $Z$ is a section of $F$ then $X(\omega_i(Z)) = 0$ so we see that $\mathcal{L}_Z g = 0$ along $F$.

Therefore, since the foliation is Riemannian, every compact leaf should have finite holonomy. Then by Reeb stability, I get a tubular neighborhood $U$ of $L$ which is diffeomorphic to $\tilde{L} \times_H T$ where $\tilde{L} \to L$ is the holonomy cover, $H$ is the holonomy groupoid and $F$ is taken to the standard foliation arising from the fiber bundle structure over $L$. Can I argue that this fiber bundle is trivial if I know $T_X / F$ is trivial?