[Paper Review] Isotrivial elliptic K3 surfaces and Lagrangian fibrations

Justin Sawon|arXiv (Cornell University)|Jun 4, 2014

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper classifies isotrivial elliptic K3 surfaces using Weierstrass models and constructs new examples of holomorphic symplectic orbifolds with isotrivial Lagrangian fibrations via group actions on abelian varieties. The key contribution is the construction of such orbifolds in dimensions six and higher that admit isotrivial Lagrangian fibrations but do not admit symplectic desingularizations, providing new examples of primitively symplectic V-manifolds with non-trivial singularities.

ABSTRACT

A fibration is said to be isotrivial if all of its smooth fibres are isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that are isotrivial, and use them to construct Lagrangian fibrations that are isotrivial. We then modify the construction to produce new examples of holomorphic symplectic orbifolds, that also admit isotrivial Lagrangian fibrations.

Motivation & Objective

• To classify isotrivial elliptic K3 surfaces using Weierstrass models and group actions on complex tori.
• To construct new examples of holomorphic symplectic orbifolds that admit isotrivial Lagrangian fibrations.
• To demonstrate that such orbifolds in dimensions six and higher do not admit symplectic desingularizations.
• To extend Matsushita’s construction by restricting group actions to subgroups, yielding new singular symplectic varieties with trivial canonical class on the base.

Proposed method

• Classify isotrivial elliptic K3 surfaces by analyzing their j-invariant and using Weierstrass models to describe singular fibres with finite monodromy.
• Construct holomorphic symplectic orbifolds as quotients $E^{2n}/\Gamma$ where $\Gamma$ is a finite subgroup of $\mathrm{Sp}(n,\mathbb{C})$ acting on a product of elliptic curves.
• Use the Hilbert scheme construction on isotrivial K3 surfaces to produce higher-dimensional holomorphic symplectic orbifolds with isotrivial Lagrangian fibrations.
• Modify the group action by restricting to a subgroup $\Gamma' < \Gamma$, such as $\Gamma' = \langle (a_1,\dots,a_n) \in G^n, \tau \in \mathfrak{A}_n \mid a_1\cdots a_n = 1 \rangle$, to generate new examples.
• Prove that the resulting orbifolds do not admit symplectic desingularizations by analyzing local singularities isomorphic to $\mathbb{C}^4/\pm 1$.
• Show that the base of the fibration is a Calabi-Yau $n$-fold by verifying the existence of a unique $\Gamma'$-invariant holomorphic $n$-form on the even factors.

Experimental results

Research questions

• RQ1 Which elliptic K3 surfaces are isotrivial, and how can they be classified using Weierstrass models and monodromy considerations?
• RQ2 Can isotrivial Lagrangian fibrations be constructed on holomorphic symplectic orbifolds in dimensions six and above?
• RQ3 Do the constructed holomorphic symplectic orbifolds admit symplectic desingularizations, and if not, why?
• RQ4 How does restricting the group action to a subgroup $\Gamma'$ affect the symplectic and fibration structure of the quotient orbifold?

Key findings

• All isotrivial elliptic K3 surfaces are classified via Weierstrass models, with singular fibres corresponding to Kodaira types with finite monodromy, such as $I_0^*$, $II^*$, $III^*$, and $IV^*$.
• The construction of holomorphic symplectic orbifolds $E^{2n}/\Gamma$ yields isotrivial Lagrangian fibrations over $\mathbb{P}^n$ for $n \geq 3$, with $\Gamma$ acting via semi-direct products of cyclic groups and symmetric groups.
• The orbifold $E^{2n}/\Gamma'$, constructed by restricting the group action to a subgroup $\Gamma'$, is a primitively symplectic V-manifold with $h^{2,0}(Z) = 1$ and $h^{1,0}(Z) = 0$ for $n \geq 3$, and its base is a Calabi-Yau $n$-fold.
• The orbifold $E^{2n}/\Gamma$ does not admit a symplectic desingularization because it contains singularities locally isomorphic to $\mathbb{C}^4/\pm 1$, which are known to be non-desingularizable in the symplectic category.
• The orbifold $E^{2n}/\Gamma'$ also fails to admit a symplectic desingularization due to similar local singularities of type $\mathbb{C}^4/G$ with $G$ a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$, as per Fu’s Corollary 3.5.
• For $n \geq 3$, the singularities of $E^{2n}/\Gamma'$ are locally of the form $({{\mathbb{C}}}^{4}/G)\times{{\mathbb{C}}}^{2n-4}$, and since $\mathbb{C}^4/G$ does not admit a symplectic desingularization, neither does the full orbifold.