Local isomorphisms for families of projective non-unruled manifolds

Technical Summary: Local Isomorphisms for Families of Projective Non-Unruled Manifolds

Problem Statement
The paper addresses a question posed by Wehler in 1977 regarding the relationship between pointwise isomorphism and local isomorphism in families of compact complex manifolds. Specifically, given two smooth families of compact complex manifolds, $\pi: X \to S$ and $\pi': Y \to S$, over a complex manifold $S$, if the fibers are pointwise isomorphic (i.e., there exists a bi-holomorphism $f_s: Y_s \to X_s$ for every $s \in S$), are the families locally isomorphic?

Previous work established this for specific cases, such as complex tori and manifolds with negative holomorphic curvature. However, a general proof for arbitrary families was attempted by Meersseman but found to contain gaps, highlighted by a counterexample constructed by Kirschner. This led to a weakened conjecture (Conjecture 1.2): if the dimension of the space of holomorphic vector fields, $s \to \dim H^0(Y_s, TY_s)$, is constant, does there exist an open dense subset $U \subset S$ where the restricted families are locally isomorphic?

Methodology
The author, Mu-Lin Li, focuses specifically on families of projective non-uniruled manifolds. The proof strategy relies on the theory of moduli spaces for polarized Kähler manifolds, utilizing constructions by Fujiki and Schumacher. The methodology proceeds through the following logical steps:

1. Topological Trivialization and Cohomology:
   For a smooth family over a disk $\Delta$, the paper utilizes the fact that such a family forms a smooth fiber bundle. This allows for the construction of diffeomorphisms $T: X \cong X_{t_1} \times \Delta$. The author analyzes the behavior of cohomology classes under these diffeomorphisms, noting that diffeomorphisms in the same connected component of the diffeomorphism group induce identical morphisms on cohomology groups (Proposition 2.4).

2. Existence of Relative Polarizations:
   A critical technical hurdle is equipping the families with polarizations (Kähler classes) that vary holomorphically.

   • Lemma 3.1 and 3.2: The author proves that if a cohomology class restricts to a $(1,1)$-class (or an integral class) on an uncountable subset of the base, one can construct a global line bundle over the total space whose first Chern class restricts to the desired class on the fibers. This relies on the Lefschetz theorem on $(1,1)$-classes and the fact that analytic subsets of a disk cannot be uncountable unless they are the whole disk.
   • This allows the construction of relatively ample line bundles $L$ on $X$ and $\tilde{L}$ on $Y$ over a Zariski open dense subset of the base.

3. Moduli Space Mapping:
   With the families now equipped with relative polarizations, they define two holomorphic maps, $f, f': U \to \mathcal{M}$, from the base to the coarse moduli space $\mathcal{M}$ of polarized projective non-uniruled manifolds.

   • The pointwise isomorphism of the fibers implies that the induced maps on the moduli space agree on an uncountable set of points (derived from the connected components of the diffeomorphism group).
   • Lemma 3.3: A standard result in complex analysis is invoked: if two holomorphic maps from a holomorphic convex domain to a complex space agree on an uncountable set, they are identical everywhere. Thus, the maps to the moduli space coincide on the entire open set $U$.

4. Local Completeness and Kuranishi Families:
   The final step utilizes the property that the moduli functor for polarized non-uniruled manifolds is locally complete (Proposition 2.8, based on Fujiki and Schumacher).

   • Since the maps to the moduli space are identical, the families are locally pullbacks of the same universal (Kuranishi) family.
   • Because the moduli space is a coarse moduli space, the local uniqueness of the pullback (up to the action of a finite automorphism group) ensures the existence of a bi-holomorphism between the restricted families $X_U$ and $Y_U$ that commutes with the projections.

Key Contributions and Results
The paper establishes Theorem 1.3:

Let $\pi: X \to S$ and $\pi': Y \to S$ be two smooth families of projective non-uniruled manifolds over a Riemann surface $S$. Suppose these two families are pointwise isomorphic. Then there exists an open dense subset $U \subset S$ such that the restricted families are locally isomorphic at every point $s \in U$.

This result provides a partial affirmative answer to Wehler's question, specifically for the class of projective non-uniruled manifolds. The proof does not require the assumption of constant $\dim H^0(Y_s, TY_s)$ explicitly in the final theorem statement for this specific class, as the properties of non-uniruled manifolds and the moduli space construction suffice to establish the local isomorphism on a dense open set.

Significance
The paper claims to partially resolve a long-standing problem in the deformation theory of complex manifolds. By restricting the scope to projective non-uniruled manifolds, the author successfully bypasses the obstructions found in the general case (as demonstrated by Kirschner's counterexample). The work demonstrates that the existence of a coarse moduli space for polarized non-uniruled manifolds, combined with the local completeness of the Kuranishi family, is sufficient to upgrade pointwise isomorphism to local isomorphism on a dense open set. The result is framed as a step toward understanding the rigidity of families of compact complex manifolds.