Adiabatic Limit and Deformations of Complex Structures

We prove that if all the fibres, except possibly one, in a holomorphic family of compact complex manifolds are Moishezon (i.e. bimeromorphically equivalent to projective manifolds), then the remaining, limiting, fibre is again Moishezon. Two new ingredients are introduced for this purpose. The first one is the Frölicher Approximating Vector Bundle (FAVB) that displays the degenerating page of the Fröilicher spectral sequence as the limit, when a complex constant $h$ tends to $0$, of what we call the $d_h$-cohomology, where $d_h=h\partial + \bar\partial$. The second ingredient is the introduction of $E_r$-sG metrics, for $r\geq 1$, that generalise the strongly Gauduchon metrics we introduced in 2009.