Locally rigid implies globally rigid in Kahler geometry

1. Problem Statement and Context

The paper addresses a fundamental question in complex geometry regarding the rigidity of compact Kähler manifolds under deformation.

• Definitions:
  • Locally Rigid: A manifold $X$ is locally rigid if any smooth family $\pi: \mathcal{X} \to B$ containing $X$ as a fiber is locally trivial near that fiber (i.e., all nearby fibers are isomorphic to $X$).
  • Globally Rigid: A manifold $X$ is globally rigid if any manifold $X'$ deformation-equivalent to $X$ is actually isomorphic to $X$.
  • Deformation Equivalence: Two manifolds are deformation equivalent if they can be connected by a finite sequence of direct deformations (smooth families over a disk).
• The Core Question: Does local rigidity imply global rigidity for compact non-uniruled Kähler manifolds?
  • It is known that infinitesimal rigidity ($H^1(X, T_X)=0$) implies local rigidity (Kodaira-Spencer-Kuranishi theory).
  • For minimal general surfaces, local rigidity implies global rigidity (proven by authors in [1]).
  • Question 1.1: If $X$ is a compact non-uniruled Kähler manifold and is locally rigid, is it globally rigid?

2. Methodology

The author employs a combination of Variation of Hodge Structures (VHS), Barlet Cycle Spaces, and Moduli Theory to bridge the gap between local and global properties.

A. Construction of a Polarized Family

The central technical challenge is to extend a local isomorphism to the entire family. The author constructs a specific cohomology class to "polarize" the family.

1. Local Triviality: Given a family $\pi: \mathcal{X} \to \Delta$ where fibers are Kähler and locally trivial near $t_1$, the author uses the diffeomorphism $T: \mathcal{X} \cong X_{t_1} \times \Delta$ to transport cohomology classes.
2. Hodge Filtration: Using the fact that the family is locally trivial near $t_1$, the Kodaira-Spencer map vanishes locally. By the properties of the period map (holomorphic map to a Grassmannian), the Hodge filtration $F^p$ is shown to be constant over the entire disk $\Delta$.
3. Global Class Construction (Lemma 3.1): The author constructs a global real cohomology class $\alpha \in H^2(\mathcal{X}, \mathbb{R})$ such that its restriction $\alpha_t$ lies in $H^{1,1}(X_t, \mathbb{C})$ for all $t$.
4. Kähler Morphism Construction (Lemma 3.2 & 3.3): Using the Demailly-Păun theorem and sheaf cohomology arguments, the author lifts $\alpha$ to a section $\Theta$ of a quotient sheaf, allowing the construction of a relative Kähler form. This establishes that the family is a Kähler morphism (specifically, a family of polarized Kähler manifolds) over a dense open set.

B. Moduli Space Argument

The proof relies heavily on the properties of the coarse moduli space $\mathcal{M}$ of polarized non-uniruled Kähler manifolds.

• Hausdorff Property: The moduli space $\mathcal{M}$ for polarized non-uniruled Kähler manifolds is known to be a separated (Hausdorff) complex space.
• Analytic Continuation: The family induces a holomorphic map $f: W \to \mathcal{M}$ from a dense open subset $W \subset \Delta$.
• Constancy: Since the family is locally trivial near $t_1$, the map $f$ is constant on a neighborhood. Because $\mathcal{M}$ is Hausdorff and the map is holomorphic, if the image is a point on an open set, the map must be constant on the connected component (or the whole domain if the domain is irreducible/connected).

C. Handling the "Non-Uniruled" Condition

The condition that the fiber $X_{t_1}$ is non-uniruled is crucial.

• Uniruled manifolds (those covered by rational curves) often have non-separated moduli spaces or allow for "jumping" phenomena in deformations.
• For non-uniruled manifolds, the moduli space is well-behaved (separated). The author cites [8, Proposition 11] to ensure that if the central fiber is non-uniruled, nearby fibers remain non-uniruled, allowing the map to the moduli space to be well-defined globally.

3. Key Results and Theorems

Theorem 1.3 (Main Theorem):
Let $\pi: \mathcal{X} \to \Delta$ be a proper smooth morphism with Kähler fibers. If the family is locally trivial at a point $t_1$ (i.e., $X_t \cong X_{t_1}$ for $t$ near $t_1$) and the fiber $X_{t_1}$ is a non-uniruled Kähler manifold, then all fibers are isomorphic ($X_t \cong X_{t_1}$ for all $t \in \Delta$).

Corollary 1.4:
A compact non-uniruled Kähler manifold is globally rigid if it is locally rigid.

• Proof Logic: If $X$ is locally rigid, any deformation equivalent manifold $X'$ can be connected via a chain of families. Applying Theorem 1.3 to each family in the chain proves that every intermediate fiber is isomorphic to the previous one, hence $X \cong X'$.

Corollary 1.5:
Specific to surfaces: A compact non-uniruled Kähler surface is globally rigid if it is locally rigid.

Corollary 1.7 (Global Non-deformability):
If a smooth family of compact Kähler manifolds has fibers $X_t \cong S$ for all $t \in \Delta^* = \Delta \setminus \{0\}$, and $S$ is a non-uniruled Kähler manifold, then the central fiber $X_0$ is also biholomorphic to $S$.

• This resolves the "global non-deformability" problem for non-uniruled manifolds under Kähler morphisms.

4. Significance and Contribution

1. Resolution of a Conjecture: The paper positively answers Question 1.1, establishing that for the broad class of non-uniruled Kähler manifolds, local rigidity is sufficient to guarantee global rigidity. This extends previous results known only for minimal general surfaces.
2. Moduli Space Application: The work demonstrates the power of the Hausdorff property of the moduli space of polarized non-uniruled manifolds in solving deformation problems. It effectively uses the separation of the moduli space to prevent "jumping" of isomorphism classes in a family.
3. Technical Innovation: The construction of a global class $\alpha$ that remains a Kähler class on a dense open set (and extends to the whole family via the non-uniruled condition) is a sophisticated application of the Demailly-Păun theorem and Barlet cycle spaces.
4. Implications for Global Non-deformability: The results provide strong affirmative answers to the global non-deformability problem for non-uniruled manifolds, complementing existing results for projective spaces and rational homogeneous spaces (which are often uniruled or have specific structures).

Summary

Mu-Lin Li proves that in the realm of Kähler geometry, if a compact non-uniruled manifold cannot be deformed locally, it cannot be deformed globally at all. The proof hinges on constructing a global polarization for the family and utilizing the separated nature of the corresponding moduli space to show that a locally constant deformation map must be globally constant.