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

TL;DR

This paper demonstrates that algebraic dimension does not always behave semi-continuously in families of compact non-Kaehler manifolds, contrasting with the Kähler case where semi-continuity generally holds.

Contribution

It provides a counterexample showing the failure of upper semi-continuity of algebraic dimension in certain non-Kaehler families, highlighting differences from Kähler geometry.

Findings

Algebraic dimensions jump downwards at special points in non-Kaehler families.

Upper semi-continuity fails in general for non-Kaehler manifolds.

In Kähler cases, semi-continuity of algebraic dimension always holds.

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.