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The SYZ conjecture for primitive symplectic varieties

A primitive symplectic variety is a normal compact Kähler variety with symplectic singularities whose smooth locus carries a symplectic form and whose reflexive powers of the canonical sheaf are trivial, in the sense used in the paper. Let XX be a primitive symplectic variety, let LL be a line bundle on XX, and let qXq_X denote its Beauville–Bogomolov–Fujiki quadratic form. A morphism f ⁣:X→Bf\colon X\to B is a lagrangian fibration when it is a fibration whose general fibres are lagrangian with respect to the symplectic form. The SYZ conjecture for primitive symplectic varieties. If LL is nef and

qX(L)=0,q_X(L)=0,

then there exists a lagrangian fibration f ⁣:X→Bf\colon X\to B such that

L=f∗OB(1).L=f^*\mathcal{O}_B(1).

This is the singular analogue of the SYZ conjecture for irreducible holomorphic symplectic manifolds, where nef isotropic line bundles are expected to define lagrangian fibrations. The paper proves the conjecture for irreducible symplectic varieties locally trivially deformation equivalent to moduli spaces of sheaves on K3 surfaces.

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The SYZ conjecture for primitive symplectic varieties

    Let XX be a primitive symplectic variety with second Betti number b2(X)≥5b_2(X)\geq 5. A locally trivial deformation is a deformation preserving the local analytic structure of the singularities, and a Lagrangian fibration is a fibration whose general fiber is a Lagrangian subvariety.

    SYZ conjecture. Every such XX admits a locally trivial deformation to a primitive symplectic variety with a Lagrangian fibration.

    The generalized abundance conjecture would imply this statement for b2≥5b_2\geq 5. The conjecture is open in the stated generality.

    source: Philip Engel, Stefano Filipazzi, François Greer, Mirko Mauri and Roberto Svaldi, “Boundedness of some fibered K-trivial varieties”, arXiv:2507.00973 (2025).

  2. SYZ conjecture for primitive symplectic varieties

    Let MM be a projective primitive symplectic variety and let LL be a nef line bundle on MM.

    SYZ Conjecture. The line bundle LL is semiample.

    This is presented as the precise version of the SYZ conjecture and as a special case of the generalized abundance conjecture. The source does not specify a resolution of this statement.

    source: Alessio Bottini, Emanuele Macrì and Paolo Stellari, “Hyper-Kähler varieties: Lagrangian fibrations, atomic sheaves, and categories”, arXiv:2603.23033 (2026).

Sources & referencesView supporting material

Primary source

Claudio Onorati and Ángel David Ríos Ortiz, “The SYZ conjecture for singular moduli spaces of sheaves on K3 surfaces”, arXiv:2510.01005 (2025).

abundancedeformation-theoryk3-surfaceslagrangian-fibrationssymplectic-varieties

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