Some examples due to H. Hironaka

TL;DR

This paper explains Hironaka's construction of a holomorphic family of complex manifolds where all fibers except one are projective, illustrating the subtle differences between projective and non-Kählerian manifolds.

Contribution

It clarifies Hironaka's method for constructing such families and discusses why simpler non-Kählerian Moishezon examples cannot be obtained similarly.

Findings

Construction of a family with projective fibers for s ≠ 0 and non-Kählerian fiber at 0

Explanation of why simpler non-Kählerian Moishezon examples are not achievable with the same method

Insight into the relationship between algebraic and complex geometric properties

Abstract

The aim of this paper is to explain the construction by H. Hironaka [H.61] of a holomorphic (in fact "algebraic") family of compact complex manifolds parametrized by $\C$ such for all $s \in \C\setminus \{0\}$ the fiber is projective, but such that the fiber at the origin in non k{\"a}hlerian, to mathematicians which are not algebraic geometers. We also explain why it is not possible to make in the same way such an example with fiber at $0$ a simpler example of non k{\"a}hlerian Moishezon manifold which is also due to H. Hironaka (see section 5).