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Let $\pi \colon \mathfrak{X} \rightarrow B$ be a deformation of complex compact manifolds and $E$ be a holomorphic vector bundle on $\mathfrak{X}$ (or a coherent sheaf on $\mathfrak{X}$ that is flat over $B$). The function $b \rightarrow \mathrm{h}^0(X_b, E_{|X_b})$ is known to be upper semi-continuous, but I wonder if it is constructible.

If the deformation is algebraic, that is if there exists a relatively ample line bundle on $\mathfrak{X}$, then there exists a complex $\mathcal{L}^{\bullet}$ of locally free sheaves on ${B}$ such that for any $b$ in $B$ and any nonnegative integer $i$, $$ \mathrm{H}^i(\mathcal{L}^{\bullet}_{|b}) \simeq \mathrm{H}^i(X_b, E_{| X_b}). $$ This is proved for instance in [Voisin, Complex algebraic geometry, Vol. 1, section 9.3.1], and implies constructibility.

Is the same result known in the complex analytic case?

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The constructibility result you ask for is Satz 7.7(1) in Ein Kriterium für die Offenheit der Versalität (Flenner, 1981).

The appropriate generalization of the result in Voisin's book that you mention is the topic of Eine Bemerkung über relative Ext-Garben (Flenner, 1981).

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  • $\begingroup$ Awesome! I spent hours looking for this... If you tell me who you are, I will thank you in my paper for this reference :-) $\endgroup$ – Julien Grivaux Dec 7 '15 at 23:15

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