Non-upper-semicontinuity of algebraic dimension for families of compact complex manifolds

Abstract

We show that in a certain subfamily of the Kuranishi family of any half Inoue surface the algebraic dimensions of the fibers jump downwards at special points of the parameter space showing that the upper semi-continuity of algebraic dimensions in any sense does not hold in general for families of compact non-Kaehler manifolds. In the Kaehler case, the upper semi-continuity always holds true in a certain weak sense.