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Adiabatic Limit and Deformations of Complex Structures

Dan Popovici

TL;DR

This work proves that the deformation limit of a holomorphic family whose generic fibres are Moishezon remains Moishezon, advancing the understanding of Moishezon-ness under complex-structure deformations. It introduces two core tools: the Frölicher approximating vector bundle (FAVB), which links d_h-cohomology to Frölicher page data via a holomorphic bundle over ℂ, and a generalized E_r-sG metric framework that controls deformation limits across the Frölicher pages. The authors develop an h-theory for the Frölicher spectral sequence using Kodaira–Spencer theory, constructing Δ_h, ~Δ_h, and ~Δ_h^{(r)} whose kernels realize Er-cohomology, and they extend these to absolute and relative FAVBs. Leveraging these structures, they show that if all nonzero fibres are ∂∂-manifolds, the limit is E_r-sG (even E_r-sGG), which, together with a uniform Barlet-space bound, yields Moishezon-ness of the limit. The results provide both conceptual insight into deformation limits via spectral sequences and practical criteria for preserving Moishezon-ness in degenerations, with potential implications for the study of algebraic dimensions in complex geometry.

Abstract

Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new ingredients, hopefully of independent interest, are introduced. The first one associates with every compact complex manifold $X$, in every degree $k$, a holomorphic vector bundle over $\C$ of rank equal to the $k$-th Betti number of $X$. This vector bundle, previously given an algebraic construction in the literature, shows that the degenerating page of the Frölicher spectral sequence of $X$ is the holomorphic limit, as $h\in\C^\star$ tends to $0$, of the $d_h$-cohomology of $X$, where $d_h=h\partial + \bar\partial$. A relative version of this vector bundle is then associated with every holomorphic family of compact complex manifolds. The second ingredient is a relaxation of the notion of strongly Gauduchon (sG) metric that we introduced in 2009. For a given positive integer $r$, a Gauduchon metric $γ$ on an $n$-dimensional compact complex manifold $X$ is said to be $E_r$-sG if $\partialγ^{n-1}$ represents the zero cohomology class on the $r$-th page of the Frölicher spectral sequence of $X$. Strongly Gauduchon metrics coincide with $E_1$-sG metrics.

Adiabatic Limit and Deformations of Complex Structures

TL;DR

This work proves that the deformation limit of a holomorphic family whose generic fibres are Moishezon remains Moishezon, advancing the understanding of Moishezon-ness under complex-structure deformations. It introduces two core tools: the Frölicher approximating vector bundle (FAVB), which links d_h-cohomology to Frölicher page data via a holomorphic bundle over ℂ, and a generalized E_r-sG metric framework that controls deformation limits across the Frölicher pages. The authors develop an h-theory for the Frölicher spectral sequence using Kodaira–Spencer theory, constructing Δ_h, ~Δ_h, and ~Δ_h^{(r)} whose kernels realize Er-cohomology, and they extend these to absolute and relative FAVBs. Leveraging these structures, they show that if all nonzero fibres are ∂∂-manifolds, the limit is E_r-sG (even E_r-sGG), which, together with a uniform Barlet-space bound, yields Moishezon-ness of the limit. The results provide both conceptual insight into deformation limits via spectral sequences and practical criteria for preserving Moishezon-ness in degenerations, with potential implications for the study of algebraic dimensions in complex geometry.

Abstract

Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new ingredients, hopefully of independent interest, are introduced. The first one associates with every compact complex manifold , in every degree , a holomorphic vector bundle over of rank equal to the -th Betti number of . This vector bundle, previously given an algebraic construction in the literature, shows that the degenerating page of the Frölicher spectral sequence of is the holomorphic limit, as tends to , of the -cohomology of , where . A relative version of this vector bundle is then associated with every holomorphic family of compact complex manifolds. The second ingredient is a relaxation of the notion of strongly Gauduchon (sG) metric that we introduced in 2009. For a given positive integer , a Gauduchon metric on an -dimensional compact complex manifold is said to be -sG if represents the zero cohomology class on the -th page of the Frölicher spectral sequence of . Strongly Gauduchon metrics coincide with -sG metrics.

Paper Structure

This paper contains 19 sections, 23 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Machinery used in the paper
  3. Brief reminder of the main construction in [Pop17]
  4. Constructions introduced in this paper
  5. The Frölicher approximating vector bundle (FAVB)
  6. Generalised strongly Gauduchon metrics adapted to the various pages of the Frölicher spectral sequence
  7. h-theory for the Frölicher spectral sequence
  8. Brief review of the Kodaira-Spencer theory of $C^\infty$ families of elliptic operators
  9. Constructions of approximations of the Laplacians $\Delta"$, $\widetilde{\Delta}$ and $\widetilde{\Delta}^{(r)}$ with $r\geq 3$ whose kernels are isomorphic to the $E_r$-cohomology spaces by $C^\infty$ families of operators $(\Delta_h)_{t\in B}$, $(\widetilde{\Delta}_h)_{t\in B}$ and $(\widetilde{\Delta}^{(r)}_h)_{t\in B}$ whose term for $h=0$ is the original Laplacian
  10. First FSS page: the Laplace-type differential operators $\Delta_h$
  11. Second FSS page: the pseudo-differential Laplacians $\widetilde{\Delta}_h$
  12. Higher FSS pages: the pseudo-differential Laplacians $\widetilde{\Delta}^{(r)}_h$ with $r\geq 3$
  13. The Frölicher approximating vector bundle (FAVB)
  14. The FAVB in the absolute case
  15. The FAVB in the relative case
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $N$ be a positive integer and let $\pi:{\mathcal{X}}\rightarrow B$ be a complex analytic family of compact complex manifolds over an open ball $B\subset\mathbb{C}^N$ about the origin such that the fibre $X_t:=\pi^{-1}(t)$ is a Moishezon manifold for every $t\in B\setminus\{0\}$. Then $X_0:=\pi^{

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Definition 2.5
  • Lemma 2.6
  • Proposition 2.9
  • ...and 19 more