The holomorphic limit of Kahler manifolds

Technical Summary: The Holomorphic Limit of Kähler Manifolds

Problem Statement
The paper addresses the stability of the Kähler property under holomorphic deformations. Specifically, it considers a smooth family $\pi: X \to \Delta$ of compact complex manifolds over the unit disk $\Delta$, where the fibers $X_t$ are Kähler for all $t \in \Delta^* = \Delta \setminus \{0\}$. The central question is whether the central fiber $X_0$ retains the Kähler property or, more broadly, belongs to the Fujiki class $\mathcal{C}$ (the class of manifolds bimeromorphic to Kähler manifolds). While this holds for surfaces (by Kodaira and Siu), Hironaka's counterexample demonstrates it fails in dimension three. The paper investigates the conditions under which the limit $X_0$ remains in the Fujiki class $\mathcal{C}$ or is Moishezon (bimeromorphic to a projective manifold).

Methodology
The author employs a combination of complex analytic geometry, cohomological techniques, and the theory of currents. The methodology proceeds through several key stages:

1. Relative Kähler Morphisms and Cohomology: The paper establishes a sufficient and necessary condition for a smooth family over a pseudoconvex domain to be a "Kähler morphism" (Definition 3.1). Using sheaf cohomology of pluriharmonic functions and the Leray spectral sequence, the author proves that if the fibers are Kähler, the existence of a global cohomology class $\alpha \in H^2(X, \mathbb{R})$ restricting to a Kähler class on each fiber is equivalent to the morphism being Kähler (Proposition 3.2).
2. Openness of Kähler Classes: Leveraging the work of Demailly and Păun on Kähler cones and the theory of variations of Hodge structures, the author demonstrates that if the family is Kähler at a point $t_0 \in \Delta^*$, there exists an open dense subset $U \subset \Delta^*$ where the restriction of a global class $\alpha$ remains a Kähler class (Theorem 3.6). This relies on the properness of relative Barlet cycle spaces and the invariance of the Kähler cone under parallel transport.
3. Currents and Volume Analysis: The core analytical tool involves the study of pseudo-effective classes and their volumes. The author utilizes the non-pluripolar product of currents and the regularization theorems of Demailly and Boucksom. A critical step involves constructing a sequence of Kähler forms on the fibers $X_t$ ($t \in U$) that converges weakly to a current $T$ on the central fiber $X_0$.
4. Volume Inequalities: By analyzing the absolutely continuous part of the limiting current $T$ and applying Fatou's lemma alongside the continuity of the volume function for pseudo-effective classes, the author derives a lower bound for the volume of the limit class.

Key Contributions and Results

• Theorem 1.2 (Theorem 4.3): The primary result states that if $\pi: X \to \Delta$ is a smooth family with Kähler fibers for $t \in \Delta^*$, and if $\pi$ is a Kähler morphism at some point $t_0 \in \Delta^*$, then there exists a positive $d$-closed $(1,1)$-current $T$ on the central fiber $X_0$ such that its volume $\text{vol}_n(\{T\}) > 0$.

  • Significance: This reduces Conjecture 1.1 (that $X_0$ is in Fujiki class $\mathcal{C}$) to the Boucksom-Demailly-Păun conjecture, which posits that the existence of such a current with positive volume implies the manifold is in Fujiki class $\mathcal{C}$.

• Corollary 1.4: Under the additional assumption that the central fiber $X_0$ admits a Hermitian metric with finite upper volume (a condition known to be invariant under bimeromorphisms and satisfied by Fujiki class $\mathcal{C}$ manifolds), the paper proves that $X_0$ is indeed in the Fujiki class $\mathcal{C}$.

• Corollary 1.5 (Corollary 4.6): In the specific case where the fibers $X_t$ for $t \in \Delta^*$ are projective, the paper proves that $X_0$ is Moishezon without requiring any auxiliary conditions (such as deformation invariance of Hodge numbers).

  • Mechanism: The proof constructs a global line bundle $L$ on $X$ such that $L|_{X_t}$ is ample for $t$ in an open dense set, and $L|_{X_0}$ is big. This is achieved by lifting integral cohomology classes via the exponential sequence and utilizing the upper semi-continuity of cohomology dimensions.

Significance and Claims
The paper claims to provide a new pathway toward resolving Conjecture 1.1. By establishing the existence of a positive volume current on the central fiber under the mild condition that the family is Kähler at a nearby point, the work bridges the gap between the deformation of Kähler metrics and the structural classification of the limit manifold.

The author notes that the finiteness of the upper volume $\text{vol}_n(\omega)$ is an open problem, but since no examples of manifolds with infinite upper volume are known, the main theorem offers strong evidence for the validity of the conjecture. Furthermore, the result for projective families (Corollary 1.5) provides a definitive answer to the Moishezon limit problem without the auxiliary conditions previously required by other researchers (e.g., Popovici, Barlet, Rao-Tsai).

The approach relies on synthesizing methods from previous works (specifically [23], [24], [14], and [6]) to construct global cohomological objects and analyze their behavior under degeneration, rather than introducing entirely new geometric frameworks. The paper remains modest in its scope, focusing strictly on the holomorphic limit of Kähler and projective families over a disk.