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The monodromy of compact Lagrangian fibrations

Edward Varvak

TL;DR

This work analyzes the monodromy representations arising from compact Lagrangian fibrations of hyperkähler manifolds. Employing Deligne’s structure theorem for complex VHS, foliation theory, and Matsushita’s constraints on bases, it distinguishes two regimes: maximal variation and isotrivial fibrations. In the maximal variation case, the monodromy on $V_\mathbb{C}$ is irreducible, ruling out complex and quaternionic types; in the isotrivial case, the fibers are isogenous to $E^n$ and $V_\mathbb{C}$ splits into two irreducible complex local systems, with a further splitting over the CM-field $K$ as $V_K=U_1\oplus U_2$ and $U_1\simeq U_2$ iff defined over $\mathbb{Q}$. These results extend and clarify the Kim–Laza–Martin structure theorem and yield insights into the possible geometric realizations of the base (e.g., Abelian torsors, K3$^{[n]}$- or Kum$_n$-type fibrations) when Matsushita’s conjecture holds. Overall, the paper provides a detailed monodromy dichotomy for Lagrangian fibrations and a precise description of the isotrivial case via CM structures.

Abstract

We study the monodromy representations underlying compact Lagrangian fibrations. In the case where the associated period map is generically immersive, we prove that the mondromy representation is irreducible over \(\mathbb{C}\). In the alternative case where the fibration is isotrivial, we recover a result of \cite{kim-laza-martin23}, proving that its fibers are isogeneous to a power of an elliptic curve. We show that over \(\mathbb{C}\), the monodromy representation underlying an isotrivial Lagrangian fibration is a direct sum of two irreducible \(\mathbb{C}\)-local systems.

The monodromy of compact Lagrangian fibrations

TL;DR

This work analyzes the monodromy representations arising from compact Lagrangian fibrations of hyperkähler manifolds. Employing Deligne’s structure theorem for complex VHS, foliation theory, and Matsushita’s constraints on bases, it distinguishes two regimes: maximal variation and isotrivial fibrations. In the maximal variation case, the monodromy on is irreducible, ruling out complex and quaternionic types; in the isotrivial case, the fibers are isogenous to and splits into two irreducible complex local systems, with a further splitting over the CM-field as and iff defined over . These results extend and clarify the Kim–Laza–Martin structure theorem and yield insights into the possible geometric realizations of the base (e.g., Abelian torsors, K3- or Kum-type fibrations) when Matsushita’s conjecture holds. Overall, the paper provides a detailed monodromy dichotomy for Lagrangian fibrations and a precise description of the isotrivial case via CM structures.

Abstract

We study the monodromy representations underlying compact Lagrangian fibrations. In the case where the associated period map is generically immersive, we prove that the mondromy representation is irreducible over . In the alternative case where the fibration is isotrivial, we recover a result of \cite{kim-laza-martin23}, proving that its fibers are isogeneous to a power of an elliptic curve. We show that over , the monodromy representation underlying an isotrivial Lagrangian fibration is a direct sum of two irreducible -local systems.
Paper Structure (5 sections, 17 theorems, 31 equations)

This paper contains 5 sections, 17 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Structure Theory of Complex Representations
  3. Review of Algebraic Foliations
  4. Fibrations of Maximal Variation
  5. Isotrivial Fibrations

Key Result

Theorem 1.3

Suppose that $V_\mathbb{Q}$ is the local system associated to the variation of Hodge structures arising from a maximal variation Lagrangian fibration. Then $V_\mathbb{C} = V_\mathbb{Q} \otimes_\mathbb{Q} \mathbb{C}$ is irreducible as a representation of $\pi_1(B^\circ)$.

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: =\ref{['complex-irreducibility']}
  • Theorem 1.4: =\ref{['elliptic-curve-theorem']}
  • Theorem 1.5: =\ref{['isotrivial-rep-classification']}
  • Proposition 1.6: =\ref{['application-isotrivial-real']}
  • Remark 1.7
  • Proposition 2.1: del87, Proposition 1.13
  • Proposition 2.2
  • proof
  • ...and 30 more