# Deformation limit and bimeromorphic embedding of Moishezon manifolds

## TL;DR

This paper proves that under certain deformation conditions, a fiber of a holomorphic family of compact complex manifolds remains Moishezon, and constructs a bimeromorphic embedding of the total space, extending previous results with new assumptions.

## Contribution

It introduces new conditions under which a fiber remains Moishezon and constructs a bimeromorphic embedding, providing a new algebraic proof of earlier results.

## Key findings

- Fiber remains Moishezon under specified deformation invariance conditions.
- Constructs a bimeromorphic embedding of the total space into projective space times the base.
- Uses solutions to degenerate Monge–Ampère equations to analyze the geometry.

## Abstract

Let $\pi: \mathcal{X}\rightarrow \Delta$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber $\pi^{-1}(t)$ for each nonzero $t$ in an uncountable subset $B$ of $\Delta$ is Moishezon and the reference fiber $X_0$ satisfies the local deformation invariance for Hodge number of type $(0,1)$ or admits a strongly Gauduchon metric introduced by D. Popovici, then $X_0$ is still Moishezon. We also obtain a bimeromorphic embedding $\mathcal{X}\dashrightarrow\mathbb{P}^N\times\Delta$. Our proof can be regarded as a new, algebraic proof of several results in this direction proposed and proved by Popovici in 2009, 2010 and 2013. However, our assumption with $0$ not necessarily being a limit point of $B$ and the bimeromorphic embedding are new. Our strategy of proof lies in constructing a global holomorphic line bundle over the total space of the holomorphic family and studying the bimeromorphic geometry of $\pi:\mathcal{X}\rightarrow \Delta$. S.-T. Yau's solutions to certain degenerate Monge--Amp\`ere equations are used.