Note on the semi-continuity of the algebraic dimension

In this short Note we show that the direct image sheaf R 1 $π$ * (O X) associated to an analytic family of compact complex manifolds $π$ : X $\rightarrow$ S parametrized by a reduced complex space S is a locally free (coherent) sheaf of O S --modules. This result allows to improve a semi-continuity type result for the algebraic dimension of compact complex manifolds in an analytic family given in [B.15]. AMS Classification 2010. 32G05-32A20-32J10.

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