[Paper Review] On a conjecture of Matsushita

Bert van Geemen, Claire Voisin|arXiv (Cornell University)|Mar 25, 2015

Geometry and complex manifolds7 references4 citations

TL;DR

This paper proves a weakened form of Matsushita's conjecture on Lagrangian fibrations of hyper-Kähler manifolds: under the condition that the transcendental lattice has rank at least 5, a very general deformation of the manifold with fixed Néron-Severi group satisfies the conjecture—either the moduli map to the moduli space of polarized abelian varieties is generically of maximal rank or constant. The result relies on the Kuga-Satake construction and the Mumford-Tate group being the full special orthogonal group on the transcendental cohomology.

ABSTRACT

We prove a weak version of a conjecture of Matsushita saying that for a Lagrangian fibration on a hyper-Kaehler manifold $X$, the moduli map for the fibers is either generically of maximal rank or constant. Assuming the base is smooth and $b_2(X)-ρ(X)\geq5$, we prove the conjecture for a very general deformation of $X$ with constant Néron-Severi group.

Motivation & Objective

• To prove a weakened version of Matsushita's conjecture on the moduli map of Lagrangian fibrations on hyper-Kähler manifolds.
• To establish conditions under which the moduli map is either generically of maximal rank or constant.
• To analyze the behavior of the moduli map under deformations preserving the Néron-Severi group.
• To use the Kuga-Satake construction and Mumford-Tate group structure to constrain possible fibrations.

Proposed method

• Use the universal family of marked deformations with fixed Néron-Severi group $\mathcal{M}_P$ to study deformations of the triple $(X,f,B)$.
• Apply the fact that very general points in $\mathcal{M}_P$ realize the full special orthogonal group as the Mumford-Tate group on the transcendental cohomology $H^2(X,\mathbb{Q})_{tr}$.
• Leverage the Kuga-Satake construction to embed the transcendental Hodge structure into a tensor product of weight-1 Hodge structures.
• Use the universal property of the Kuga-Satake construction to show that only finitely many such weight-1 Hodge structures can exist under the full Mumford-Tate group condition.
• Apply a contradiction argument using the Néron-Severi group of the total space to show that non-isotrivial fibrations lead to linearly independent divisor classes, violating $\mathrm{NS}(X_V/V)=\mathbb{Z}$.
• Construct a continuous family of weight-1 Hodge structures $H_1$ for which $H \subset H_1 \otimes H_2$ holds, showing the necessity of the Mumford-Tate group assumption.

Experimental results

Research questions

• RQ1 Under what conditions does the moduli map of a Lagrangian fibration on a hyper-Kähler manifold have maximal rank or remain constant?
• RQ2 How does the Mumford-Tate group of the transcendental cohomology influence the geometry of the fibration and its moduli map?
• RQ3 Can the Kuga-Satake construction be used to constrain the existence of non-isotrivial fibrations?
• RQ4 What role does the rank of the transcendental lattice play in the validity of Matsushita's conjecture?
• RQ5 Is the assumption $b_{2,tr}(X) \geq 5$ essential, or can it be relaxed?

Key findings

• For a projective hyper-Kähler manifold $X$ of dimension $2n$ with a Lagrangian fibration $f:X\to B$ and $b_{2,tr}(X) \geq 5$, the moduli map $m':B'\dashrightarrow \mathcal{A}_{n,\alpha}$ is either constant or generically of maximal rank $n$ for a very general deformation in $\mathcal{M}_P$.
• The condition that the Mumford-Tate group of $H^2(X,\mathbb{Q})_{tr}$ is the full $SO(H^2(X,\mathbb{Q})_{tr}, q)$ implies Matsushita's conjecture holds.
• The set of irreducible weight-1 Hodge structures $H_1$ such that $H \subset H_1 \otimes H_2$ is finite when the Mumford-Tate group is full $SO$, but can be continuous if this condition fails.
• The proof shows that if the moduli map has positive-dimensional fibers, the fibration must be isotrivial, contradicting the existence of two independent divisor classes in $\mathrm{NS}(X_V/V)$.
• A continuous family of weight-1 Hodge structures $H_1$ is constructed for which $H \subset H_1 \otimes H_2$ holds, demonstrating that the Mumford-Tate group condition is essential.
• The result implies that in the deformation space $\mathcal{M}_P$, either a dense Zariski open set satisfies maximal rank moduli maps or all satisfy constant maps.