Multiplicity of codimension 1 fibers in Lagrangian fibrations

1. Problem Statement

The paper addresses a specific conjecture regarding the geometry of Lagrangian fibrations on compact hyper-Kähler manifolds.

• Context: Let $\pi: X \to B$ be a Lagrangian fibration, where $X$ is a compact hyper-Kähler manifold and $B$ is the base variety.
• The Conjecture: It was previously conjectured (noted in [Mar96] and explicitly stated in [KV19]) that such fibrations do not admit codimension 1 multiple fibers.
  • A fiber over a prime divisor $D \subset B$ is defined as the pullback $\pi^*(D)$.
  • The fiber is "multiple" if $\pi^*(D) = m \cdot E$ for some integer $m \geq 2$ and a Weil divisor $E$.
• Significance of the Problem: The existence of such multiple fibers acts as an obstruction to several major results in the literature (e.g., regarding the primitivity of pullback divisors and the structure of singular fibers). Prior to this work, these results relied on the unproven assumption that codimension 1 multiple fibers do not exist.
• Local vs. Global: In dimension 2 (elliptic fibrations), the non-existence of multiple fibers is a known local consequence of the canonical bundle formula. However, for $\dim X \geq 4$, local arguments fail due to counterexamples in the local setting ([HO11]). Therefore, a proof requires a global topological constraint specific to hyper-Kähler manifolds.

2. Methodology

The author employs a global topological approach utilizing cyclic covering constructions and the simply connectedness of hyper-Kähler manifolds. Notably, the proof does not rely on the symplectic structure of $X$ but rather on its topological properties.

Key Steps in the Proof:

1. Generalization: The author first proves a more general theorem (Theorem 2.1) for proper equidimensional fibrations between normal connected varieties, rather than restricting immediately to hyper-Kähler manifolds.
2. Cyclic Covering Construction:
   • Assume a codimension 1 fiber $\pi^*(D)$ has multiplicity $m$.
   • Construct a cyclic covering $\tilde{X} \to X$ of degree $m$ branched along the divisor $mE$ (where $\pi^*(D) = mE$).
   • Normalize this covering to obtain a variety $Y$.
3. Étale Property:
   • The author proves that the composition map $q: Y \to X$ is finite and étale over the regular locus $X_{\text{reg}}$.
   • This is verified locally: where $E$ is Cartier (defined by a function $f$), the cyclic covering splits into $m$ disjoint isomorphic components locally, making the map étale.
4. Connectedness Argument:
   • The core of the proof relies on showing that $Y$ is connected.
   • Using the fact that $B$ is normal, the author shrinks the base to a smooth affine neighborhood of a regular point on $D$.
   • By constructing a commutative diagram involving the cyclic covering of the base $\tilde{B} \to B$ and applying Lemma 2.5 (relating cyclic coverings of the base and total space), the author shows that $Y$ is a fibration over the connected base $\tilde{B}$.
   • Since the base $\tilde{B}$ is connected and the fibration is proper and equidimensional, $Y$ must be connected.
5. Topological Contradiction:
   • Since $Y$ is a connected, normal variety, $q: Y \to X$ represents a degree $m$ finite étale covering of $X_{\text{reg}}$.
   • This induces a surjective homomorphism from the étale fundamental group: $\pi_1^{\text{et}}(X_{\text{reg}}) \to \mathbb{Z}/m\mathbb{Z}$.
   • Crucial Constraint: For a compact hyper-Kähler manifold, $X$ is simply connected ($\pi_1(X) = 0$). Consequently, $\pi_1^{\text{et}}(X_{\text{reg}})$ cannot admit a non-trivial finite quotient if $m > 1$.
   • Therefore, $m$ must be 1.

3. Key Contributions and Results

Main Theorem (Theorem 1.1):
Let $X$ be a compact hyper-Kähler manifold and $\pi: X \to B$ a Lagrangian fibration. Then $\pi$ has no codimension 1 multiple fibers.

General Theorem (Theorem 2.1):
For a proper equidimensional fibration $\pi: X \to B$ between normal connected varieties, if a codimension 1 fiber has multiplicity $m$, there exists a surjective homomorphism $\pi_1^{\text{et}}(X_{\text{reg}}) \to \mathbb{Z}/m\mathbb{Z}$.

• Implication: If $X_{\text{reg}}$ is simply connected, no multiple fibers exist.

Corollaries and Strengthened Results:
The paper demonstrates that the main theorem renders several existing results in the literature unconditional (removing the need for the conjecture as a hypothesis):

1. Primitivity of Pullbacks (Theorem 3.2): If $H_B$ generates the Weil divisor class group of $B$, then $H = \pi^* H_B$ is primitive in $\text{Pic}(X)$. This was previously conditional on the non-existence of multiple fibers.
2. Reduced Singular Fibers (Theorem 3.4): If $X$ is projective with Picard number 2, a general singular fiber is reduced.
3. Structure of $R^1\pi_*\mathbb{Z}$ (Theorem 3.5): Confirms the behavior of the higher direct image sheaf over the smooth locus.
4. Torsor Structure (Theorem 3.6): Confirms that the smooth part of the fibration over a specific open set is a torsor under a smooth commutative group scheme.

Generalization to Irreducible Symplectic Varieties (Theorem 3.1):
The result extends to projective irreducible symplectic varieties. The multiplicity of codimension 1 fibers is bounded by the order of the étale fundamental group $|\pi_1^{\text{et}}(X_{\text{reg}})|$. If this group is trivial, no multiple fibers exist.

4. Significance

• Resolution of a Long-standing Conjecture: The paper provides a rigorous proof for a conjecture that has been a necessary assumption in the study of Lagrangian fibrations for decades.
• Unification of Results: By proving the conjecture, the author removes a major "conditional" barrier in the field. Results by Kamenova, Verbitsky, Voisin, and others regarding the geometry of the base $B$ and the Picard group of $X$ are now established as theorems.
• Methodological Insight: The proof highlights the power of global topological constraints (simple connectivity) over local geometric analysis in the context of hyper-Kähler manifolds. It demonstrates that the global topology of $X$ strictly controls the local branching behavior of the fibration.
• Independence: The author notes that Kamenova and Verbitsky independently proved the same result using a different method, reinforcing the robustness of the conclusion.

In summary, Yoon-Joo Kim's paper establishes that Lagrangian fibrations on compact hyper-Kähler manifolds are "clean" in codimension 1, possessing no multiple fibers, thereby solidifying the foundational geometric properties of these manifolds.