Singular fibers of very general Lagrangian fibrations

Let [Formula: see text] be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibers in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence.

P = W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

LAGRANGIAN FIBRATIONS OF HYPERKÄHLER FOURFOLDS

The base surface $B$ of a Lagrangian fibration of a projective, irreducible symplectic fourfold $X$ is shown to be isomorphic to $\mathbb{P}^{2}$ .

Pullbacks of hyperplane sections for Lagrangian fibrations are primitive

Let [Formula: see text] be a Lagrangian fibration on a hyperkähler manifold of maximal holonomy (also known as IHS), and [Formula: see text] be the generator of the Picard group of [Formula: see text]. Assume that [Formula: see text] has no multiple fibers in codimension 1. We prove that [Formula: see text] is a primitive class on [Formula: see text].

Lagrangian fibrations on symplectic fourfolds

We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold.

Lagrangian Constant Cycle Subvarieties in Lagrangian Fibrations

We show that the image of a dominant meromorphic map from an irreducible compact Calabi–Yau manifold X whose general fiber is of dimension strictly between 0 and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety ΣH in X which depends on a divisor class H whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on ΣH in CH0(X) depend neither on H nor on the Lagrangian fibration (provided b2(X) ≥ 8).

On Lagrangian fibrations by Jacobians, II

Let Y → ℙn be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian [Formula: see text] is a Lagrangian fibration. We prove that X is a Beauville–Mukai integrable system if n = 3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville–Mukai system if n = 3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curves.

MULTIPLE FIBERS OF HOLOMORPHIC LAGRANGIAN FIBRATIONS

We determine all possible multiplicities of general singular fibers of a holomorphic Lagrangian fibration, under the assumption that all components of the fibers are of Fujiki class. The multiplicities are at most 6 and the possible values are intricately related to the Kodaira type of the characteristic cycle.