Sections of Lagrangian fibrations on holomorphic symplectic manifolds

Technical Summary: Sections of Lagrangian Fibrations

Problem Statement
The paper addresses the existence of sections for Lagrangian fibrations $\pi: M \to X$ on compact hyperkähler manifolds $M$ of maximal holonomy (irreducible holomorphically symplectic manifolds). While the base $X$ is known to be biholomorphic to $\mathbb{CP}^n$ (under smoothness assumptions), the existence of a global holomorphic or even meromorphic section is a non-trivial topological and geometric obstruction.

Previous attempts to construct such sections using purely topological arguments (e.g., representing homology classes by submanifolds) failed, particularly for K3 surfaces and elliptic fibrations. The central problem is to determine if, under specific conditions, one can deform the complex structure of $M$ to admit a meromorphic section without altering the deformational type of the manifold.

Methodology
The authors employ a multi-stage approach combining algebraic geometry, homological algebra, and complex differential geometry:

1. Degenerate Twistor Deformations: The core mechanism is the use of "degenerate twistor deformations" (or Tate-Shafarevich twists). Given a Lagrangian fibration $\pi: M \to X$ and a closed $(2,0)+(1,1)$-form $\eta$ on the base $X$, one can deform the complex structure of $M$ to a new structure $I_\eta$ such that the form $\Omega + \pi^*\eta$ becomes holomorphic symplectic. Crucially, if a smooth section $S$ exists, the form $\eta = -S^*\Omega$ renders $S$ holomorphic in the deformed structure.
2. Local Construction via Averaging: The authors first construct a smooth section over a general line $C \cong \mathbb{CP}^1 \subset X$. This relies on:
   • Thom's Theorem: Representing a specific homology class in the preimage $\pi^{-1}(C)$ by a smooth 2-dimensional submanifold.
   • Néron Models and Torsor Structures: Utilizing the fiberwise abelian group structure (or torsor structure) of the smooth fibers. By "averaging" the intersection points of the submanifold with the fibers (using the group law), they construct a smooth section $\sigma: C \to M$.
   • Primitivity Assumption: The construction requires the general fiber to be primitive in integer homology to ensure the existence of a multisection with total multiplicity 1.
3. Extension via Dolbeault Currents: The primary obstruction to extending the local deformation from a neighborhood of $C$ to the entire base $X = \mathbb{CP}^n$ lies in the cohomology group $H^1(UC, \mathcal{O})$. The authors define a "Dolbeault class" associated with the section. They prove this class vanishes by realizing it as a current and showing it is exact, thereby allowing the extension of the closed form $\eta$ to the whole base.
4. Holography Principle: Once a holomorphic section exists over a neighborhood of an ample rational curve $C$, the authors apply a "holography principle" (Theorem 5.12). This principle states that any holomorphic map from a neighborhood of an ample rational curve in $\mathbb{CP}^n$ to a Kähler manifold extends to a meromorphic map on the entire $\mathbb{CP}^n$.

Key Contributions and Results

• Theorem 1.1 (Main Result): Let $\pi: M \to \mathbb{CP}^n$ be a Lagrangian fibration on a compact hyperkähler manifold of maximal holonomy. Assume the general fiber is primitive in integer homology and the fibers are reduced outside a codimension 2 subset. Then, there exists a parameter $t_0 \in H^2(\mathbb{CP}^n, \mathbb{C})$ such that the degenerate twistor deformation $(M, I_{t_0})$ admits a meromorphic section $\pi: (M, I_{t_0}) \to \mathbb{CP}^n$.
• Vanishing of the Obstruction (Theorem 6.19): The paper proves that the Dolbeault class associated with the smooth section constructed over a line vanishes. This is the critical step that allows the local degenerate twistor deformation to be extended globally.
• Refinement of Homology Arguments (Section 4): The authors address subtleties regarding the homology class of the constructed section. They demonstrate that while the "averaging" procedure modifies the homology class of the initial multisection, the resulting class is uniquely determined and compatible with the global structure, resolving concerns raised by J. Kollár regarding the preservation of homology classes.
• Comparison with Tate-Shafarevich Twists: The paper distinguishes its approach from J. Kollár's recent work on Tate-Shafarevich twists. While Kollár constructs rational sections for abelian fibrations via twists that may not preserve the deformational type of the manifold, the authors' use of degenerate twistor deformations ensures the resulting manifold remains deformationally equivalent to the original $M$.

Significance and Claims
The paper claims to solve the topological problem of constructing a smooth section for Lagrangian fibrations by developing a robust algebro-geometric approach that overcomes the limitations of previous topological methods.

• Deformational Equivalence: A significant claim is that the resulting manifold with a meromorphic section is a degenerate twistor deformation of the original manifold. Unlike other constructions (such as certain Tate-Shafarevich twists), this deformation preserves the deformational type of the hyperkähler manifold.
• Primitivity Necessity: The authors explicitly argue that the primitivity of the general fiber is a necessary condition for the existence of a bimeromorphic section, citing examples (Hellmann's Mukai system) where non-primitive fibers prevent such sections.
• Generality: The result applies to all Lagrangian fibrations on hyperkähler manifolds satisfying the primitivity and reduced fiber conditions, providing a unified framework for the existence of sections in this context.

The paper does not claim to construct sections for all Lagrangian fibrations without the primitivity assumption, nor does it claim the section is holomorphic on the original complex structure; rather, it establishes the existence of a section within a specific, well-defined family of deformations of the complex structure.