Meromorphic Group Actions and the Support Theorem for Lagrangian Fibrations

Technical Summary: Meromorphic Group Actions and the Support Theorem for Lagrangian Fibrations

Problem Statement
The paper addresses the geometric and cohomological structure of Lagrangian fibrations $f: X \to B$ on holomorphic symplectic complex spaces. While the Liouville–Arnold theorem describes the dynamics of such systems over the regular locus (where fibers are compact tori), the behavior over the singular fibers and the entire base $B$ remains complex. Specifically, the authors investigate whether a family of commutative Lie groups can be constructed to act faithfully on the total space $X$ over the entire base $B$, including singular fibers, and how this group action governs the topology of the fibration.

A central goal is to prove a "Support Theorem" for the Decomposition Theorem of the direct image complex $Rf_* \mathbb{R}_X[\dim X]$ (or $Rf_* IC_X$ in the singular case). This theorem relates the possible supports (irreducible subvarieties $Z \subseteq B$) appearing in the decomposition to the geometry of the fibers of the group family, specifically their maximal compact torus quotients.

Methodology
The authors employ a synthesis of complex analytic geometry, Hodge theory, and the theory of Hopf algebras. The methodology proceeds in three main stages:

1. Construction of Meromorphic Group Actions:

   • Starting with the Liouville–Arnold theorem, the authors integrate vertical Hamiltonian vector fields to define a holomorphic action of the cotangent bundle $T^*B$ on the regular locus.
   • They extend this to the entire space $X$ by utilizing the relative Douady space of $X \times_B X$. By analyzing the irreducible component containing the image of the classifying morphism, they construct a family of meromorphic groups $G \to B$ (in the sense of Fujiki).
   • A key technical achievement is proving that this construction yields a smooth family of commutative Lie groups acting meromorphically on $X$, even when $X$ is singular or non-compact, provided $X$ is Kähler and $B$ is a complex manifold. This removes the dependency on previous algebraic assumptions or conditions on the critical locus.

2. Cohomological Freeness via Hopf Modules:

   • The authors establish a "Freeness Theorem" for the cohomology of the fibers. They utilize the structure of meromorphic groups, which admit a decomposition $1 \to L \to G \to T \to 1$ (where $L$ is linear algebraic and $T$ is a compact torus).
   • Using mixed Hodge theory, they show that the cohomology $H^*(X_b)$ of a fiber is a free comodule over the Hopf algebra $H^*(T_b)$.
   • Crucially, they adapt Deligne's splitting construction for the decomposition theorem to show that the perverse filtration on the cohomology of the fibers is compatible with the comodule structure. This allows them to treat the cohomology of the direct image complex as a free module over the cohomology of the maximal compact quotient.

3. Proof of the Support Theorem:

   • Combining the freeness of the cohomology modules with the $\delta$-regularity condition (a bound on the dimension of the linear part of the group over the base), the authors constrain the possible supports in the Decomposition Theorem.
   • They demonstrate that if $Z \subseteq B$ is a support, its codimension is determined by the dimension of the maximal compact torus quotient $T_b$ of the group fiber over a general point of $Z$.

Key Contributions and Results

• Theorem A (Existence of Group Action): For a Lagrangian fibration $f: X \to B$ with $X$ Kähler (possibly singular) and $B$ smooth, there exists a smooth family $G \to B$ of commutative Lie groups acting faithfully and meromorphically on $X$. This generalizes previous results that required algebraic settings or specific conditions on the critical locus.
• Theorem B (Freeness Theorem): The singular cohomology $H^*(X_b, \mathbb{R})$ of a fiber is a free module over the cohomology algebra $H^*(T_b, \mathbb{R})$, where $T_b$ is the maximal compact torus quotient of the identity component of the group fiber. This result holds for constructible complexes under the group action, including the intersection complex $IC_X$ for singular $X$.
• Theorem C (Support Theorem): If $Z \subseteq B$ is a support in the Decomposition Theorem for $Rf_* IC_X$, then $\dim Z = \dim T_b$ for a general $b \in Z$. Furthermore, the direct sum of simple perverse sheaves supported on $Z$ is constructed from the cohomology of the compact torus quotients and a local system with finite monodromy.
• Generalization to Singular Spaces: The results are extended to singular holomorphic symplectic spaces (in the sense of Beauville) by utilizing Poisson geometry and functorial resolutions of singularities, proving that the equidimensionality and flatness properties necessary for the proof hold in this broader context.

Significance and Claims
The paper claims to provide a complete and intrinsic description of the supports in the Decomposition Theorem for Lagrangian fibrations, unifying the algebraic and Kähler settings.

• Elimination of Conditions: The authors emphasize that their construction of the group action eliminates the need for the "condition (1)" (surjectivity of the non-critical locus) used in prior work, and it does not require the fibration to be algebraic.
• Hodge-Theoretic Compatibility: The use of real coefficients is motivated by the need for the module structure to be canonical and compatible with mixed Hodge structures. The authors note that while the results hold over $\mathbb{Q}$, the module structure is not canonical in that setting.
• Geometric Interpretation: The Support Theorem provides a geometric characterization of the decomposition: the "size" of the supports is directly governed by the "size" of the compact torus quotients of the acting group family. This links the topology of the direct image complex directly to the geometry of the group action constructed via the Liouville–Arnold mechanism.
• Applicability: The framework applies to both smooth and singular Lagrangian fibrations, including those arising in the context of Hitchin systems and moduli spaces, offering a robust tool for analyzing the topology of these spaces without relying on specific algebraic models.

The authors position their work as a significant step in understanding the global structure of integrable systems in complex geometry, bridging the gap between the local Liouville–Arnold picture and the global singular behavior of the fibration.