Projective Lagrangian Fibration

• Projective Lagrangian fibration is a structure on holomorphic symplectic manifolds where the fibration map has fibres that are Lagrangian subvarieties, typically abelian varieties.
• Matsushita’s theorem forces the base to be a normal projective variety of half the dimension, often isomorphic to complex projective spaces with strong Fano-type properties.
• Research employs deformation theory, birational modifications, and cohomological techniques to link hyperkähler geometry, integrable systems, and Hodge theory.

A projective Lagrangian fibration is a proper surjective morphism from a projective holomorphic symplectic space whose fibers are Lagrangian with respect to the holomorphic symplectic form. In the irreducible holomorphic symplectic or hyperkähler setting, such fibrations are constrained by Matsushita-type structure theorems: the base has half the dimension of the total space, the smooth fibers are abelian varieties, and the base is a normal projective variety with strong Fano-type properties. The subject sits at the intersection of hyperkähler geometry, algebraic integrable systems, deformation theory, birational geometry, Hodge theory, and the geometry of degenerations (Greb et al., 2013, Ou, 2014).

1. Foundational definitions and structural constraints

A symplectic variety is a complex variety $M$ of dimension $2n$ with a nowhere degenerate closed holomorphic $2$-form $\Omega$. An irreducible symplectic manifold is a simply-connected complex projective manifold $M$ with $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$, where $\Omega$ is a symplectic form. In parallel language, a hyperkähler manifold is a compact, simply-connected Kähler manifold with $H^0(X,\Omega_X^2)$ spanned by a holomorphic symplectic form $\sigma$ (Ou, 2014, Greb et al., 2011).

A fibration $f:X\to B$ is a proper surjective holomorphic map with $2n$0, $2n$1, and connected fibres. A Lagrangian fibration is a holomorphic map $2n$2 with connected fibres such that every irreducible component of every reduced fibre is Lagrangian; equivalently, in the smooth case the fibres have dimension $2n$3 and the symplectic form restricts to zero on them. For projective irreducible symplectic manifolds, Matsushita’s theorem implies that every nontrivial fibration is automatically Lagrangian, and that the base is a $2n$4-factorial klt Fano variety of dimension $2n$5 with Picard number $2n$6 (Ou, 2014).

This framework is not limited to the projective total-space case. If $2n$7 is hyperkähler and $2n$8 is a fibration onto a smooth base, then $2n$9; Greb and Lehn established this in the Kähler case, extending Hwang’s projective result by combining projectivity of the base, deformation theory, flatness, and the global deformation rigidity of projective space (Greb et al., 2013). Campana later showed that if $2$0 is a Lagrangian fibration from a compact connected Kähler hyperkähler manifold onto a projective normal variety $2$1, then $2$2 is locally projective (Campana, 2017).

2. Existence mechanisms and birational realization

A central existence problem, motivated by the theory of integrable systems and mirror symmetry, asks when a hyperkähler manifold admits a Lagrangian fibration. Beauville’s question asks whether a hyperkähler manifold containing a Lagrangian torus $2$3 admits a possibly meromorphic Lagrangian fibration with fibre $2$4 (Greb et al., 2011).

In the non-projective case, Greb, Lehn, and Rollenske proved: if $2$5 is a non-projective hyperkähler manifold of dimension $2$6 containing a Lagrangian subtorus $2$7, then the algebraic dimension of $2$8 is $2$9, and there exists an algebraic reduction $\Omega$0, which is a holomorphic Lagrangian fibration with fibre $\Omega$1. In the projective case they gave equivalent criteria for the existence of an almost holomorphic Lagrangian fibration with strong fibre $\Omega$2: the pair $\Omega$3 admits a small deformation $\Omega$4 with non-projective $\Omega$5, or there exists an effective divisor $\Omega$6 on $\Omega$7 with

$\Omega$8

They also showed that any almost holomorphic Lagrangian fibration on a projective hyperkähler manifold can be transformed, via birational modification, into a holomorphic Lagrangian fibration on a smooth birational hyperkähler model (Greb et al., 2011).

For projective hyperkähler fourfolds, the existence statement becomes stronger. Greb, Lehn, and Rollenske introduced the L-reduction

$\Omega$9

whose generic fibre through $M$0 is the connected component through $M$1 of

$M$2

They proved that $M$3 admits an almost holomorphic Lagrangian fibration with strong fibre $M$4 if and only if $M$5 is not $M$6-separable, and in dimension four $M$7-separability never occurs. Consequently, if $M$8 is a four-dimensional projective hyperkähler manifold containing a Lagrangian torus $M$9, then $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$0 admits a holomorphic Lagrangian fibration with fibre $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$1 (Greb et al., 2011).

An alternative projective route proceeds through line bundles. For an irreducible holomorphic symplectic manifold of K3$H^0(M,\Omega_M^2)=\mathbb{C}\Omega$2-type, if $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$3 is a nef line bundle such that $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$4 is primitive and isotropic for the Beauville–Bogomolov–Fujiki form, then, under the non-speciality assumption, the linear system of $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$5 is base point free and induces a Lagrangian fibration. In the same framework, generic pairs $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$6 are bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves with pure one-dimensional support on a projective K3 surface (Markman, 2013).

3. Bases of projective Lagrangian fibrations

The geometry of the base is unusually rigid. Hwang’s theorem states that if $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$7 is a Lagrangian fibration of an irreducible holomorphic symplectic manifold $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$8 of dimension $H^0(M,\Omega_M^2)=\mathbb{C}\Omega$9 and $\Omega$0 is smooth, then $\Omega$1. Greb and Lehn extended this to the Kähler case, and later work supplied both Hodge-theoretic and special-Kähler-theoretic proofs of the same statement (Greb et al., 2013, Bakker et al., 2023, Li et al., 2023).

In dimension four, the singular-base problem has a precise history. Ou proved that if $\Omega$2 is a Lagrangian fibration from a complex projective irreducible symplectic fourfold onto a normal surface $\Omega$3, then

$\Omega$4

where $\Omega$5 is the unique Fano surface with a single singular point of type $\Omega$6 and two nodal rational curves in its anticanonical system. The exclusion of other cases uses a cohomological constraint together with the classification of Fano surfaces with klt singularities and Picard number $\Omega$7 (Ou, 2014).

Huybrechts and Xu removed the remaining exceptional surface. If $\Omega$8 is a connected Lagrangian fibration from a projective, irreducible symplectic fourfold over a normal surface $\Omega$9, then $H^0(X,\Omega_X^2)$0. Their key technical step excludes the $H^0(X,\Omega_X^2)$1 case by analyzing semiabelic degenerations, essential skeletons, and actions of the binary icosahedral group (Huybrechts et al., 2019).

A local smoothness theorem sharpens this picture. If $H^0(X,\Omega_X^2)$2 is a germ of a projective Lagrangian fibration from a holomorphic symplectic manifold onto a normal analytic variety $H^0(X,\Omega_X^2)$3 with only isolated quotient singularities, then $H^0(X,\Omega_X^2)$4 is smooth. In particular, if $H^0(X,\Omega_X^2)$5 is a Lagrangian fibration from a hyper-Kähler fourfold $H^0(X,\Omega_X^2)$6 onto a normal surface $H^0(X,\Omega_X^2)$7, then $H^0(X,\Omega_X^2)$8, recovering the fourfold classification by a local argument using étale covers, the singular Lefschetz–Riemann–Roch theorem, and the determinant condition

$H^0(X,\Omega_X^2)$9

for nontrivial stabilizer elements (Müller et al., 6 Aug 2025).

This collection of results suggests that singular bases are far more constrained than the general Fano-with-Picard-number-one conclusion alone would indicate, and in low dimension the projective plane is the only base that actually occurs.

4. Cohomological, Hodge-theoretic, and foliation structures

Projective Lagrangian fibrations carry strong cohomological identities. For a Lagrangian fibration $\sigma$0 from a projective irreducible symplectic manifold of dimension $\sigma$1, and any $\sigma$2-ample divisor $\sigma$3 on $\sigma$4,

$\sigma$5

In the fourfold case this becomes

$\sigma$6

Ou also proved that for a Weil divisor $\sigma$7 on $\sigma$8, with $\sigma$9,

$f:X\to B$0

and that these higher direct images are reflexive; moreover,

$f:X\to B$1

These statements are established by combining the Beauville–Bogomolov form, reflexivity, reduction to positive characteristic, Frobenius splitting, and base change (Ou, 2014).

The geometry of vertical hypersurfaces supplies a distinct dynamical feature. Let $f:X\to B$2 be a projective irreducible holomorphic symplectic manifold with a Lagrangian fibration $f:X\to B$3, let $f:X\to B$4 be a hypersurface, and let $f:X\to B$5 be a smooth irreducible hypersurface. The characteristic foliation

$f:X\to B$6

has leaves contained in the fibres of $f:X\to B$7, but a generic leaf is not algebraically integrable: the Zariski closure of a generic leaf is a fibre of $f:X\to B$8, hence an abelian variety of dimension $f:X\to B$9. A key input is that for a smooth fibre $2n$00, the image of

$2n$01

has rank $2n$02 (Abugaliev, 2019).

Away from the discriminant locus, the base carries a special Kähler metric. For a holomorphic Lagrangian fibration $2n$03, on $2n$04 there is a special Kähler metric $2n$05 with curvature

$2n$06

so $2n$07 has nonnegative bisectional curvature. If $2n$08, minimal rational curves on the base yield a nontrivial splitting

$2n$09

over the universal family of such curves, and this splitting is parallel for the pullback of the Chern connection of $2n$10. Combining this with results of Voisin, Hwang, and Bakker–Schnell forces $2n$11 (Li et al., 2023).

The same rigidity can be phrased in Hodge-theoretic terms. A Hodge-theoretic proof of Hwang’s theorem uses the variation of Hodge structure on $2n$12, Matsushita’s identification of the tangent bundle with a Hodge bundle, semi-positivity via Hodge modules, and Voisin’s irreducibility theorem. Along a covering family of rational curves one obtains

$2n$13

and the trivial summands correspond exactly to the fixed part of the variation of Hodge structure; irreducibility rules this out unless $2n$14, hence $2n$15 (Bakker et al., 2023).

5. Deformations, twists, and sections

A projective Lagrangian fibration $2n$16 has a deformation theory organized by the Shafarevich-Tate group

$2n$17

where $2n$18 is the sheaf of vertical automorphisms of $2n$19 over $2n$20. Its connected component of identity satisfies

$2n$21

with $2n$22 a finitely generated subgroup. The Shafarevich-Tate family is a holomorphic family whose fibre over $2n$23 is the corresponding Shafarevich-Tate twist $2n$24, and this family coincides with Verbitsky’s degenerate twistor family: if $2n$25, then the twist $2n$26 and the degenerate twistor deformation determined by the same cohomology class are isomorphic as fibrations over $2n$27 (Abasheva et al., 2021).

Projectivity is sparse in this family. For a very general $2n$28, the projective members of the Shafarevich-Tate family are exactly the torsion points of $2n$29, and a sufficient geometric condition for projectivity is the existence of a holomorphic section. Under additional cohomological hypotheses—reduced irreducible fibers, vanishing $2n$30, and torsion-free $2n$31—there exists a unique twist in the Shafarevich-Tate family admitting a holomorphic section (Abasheva et al., 2021).

A complementary section theorem was established for degenerate twistor deformations. Let $2n$32 be a compact hyperkähler manifold of maximal holonomy equipped with a Lagrangian fibration $2n$33. A degenerate twistor deformation, sometimes also called a Tate-Shafarevich twist, is a family of holomorphically symplectic structures on $2n$34 parametrized by $2n$35, all preserving the holomorphic Lagrangian projection to $2n$36. If the general fibre is primitive in integer homology and $2n$37 has reduced fibers in codimension $2n$38, then $2n$39 has a degenerate twistor deformation $2n$40 such that $2n$41 admits a meromorphic section (Bogomolov et al., 2024).

These deformation theories connect back to existence results. Markman’s analysis of isotropic nef line bundles shows that, for K3$2n$42-type manifolds, Lagrangian fibrations are naturally organized by period domains, monodromy, and Tate-Shafarevich-type twists of relative compactified Jacobians on K3 surfaces (Markman, 2013).

6. Local projectivity, compactification, Néron models, and geometric realizations

Local projectivity is a foundational analytic-algebraic property. Campana proved that if $2n$43 is a Lagrangian fibration from a compact connected Kähler hyperkähler manifold onto a projective normal variety $2n$44, then $2n$45 is locally projective. The proof uses a criterion based on the rationality of a Kähler class on generic smooth fibres, torsion-freeness of $2n$46, the exponential sequence, and the relative Nakai–Moishezon criterion (Campana, 2017).

A more global compactification problem arises for quasi-projective Lagrangian fibrations. A recent framework considers a surjective Lagrangian fibration $2n$47 from a quasi-projective symplectic variety over an open subset $2n$48 containing all codimension-$2n$49 points. If the holomorphic symplectic form on $2n$50 extends to a holomorphic form on a smooth compactification of $2n$51, then there exists a possibly singular projective $2n$52-factorial terminal symplectic variety $2n$53 with a Lagrangian fibration $2n$54 extending $2n$55. The same framework constructs the relative Albanese fibration and compactifies torsors over it, or over smooth commutative group schemes isogenous to it (Saccà, 2024).

For projective Lagrangian fibrations over smooth bases, the higher-dimensional Néron model behaves differently from the elliptic case. Let $2n$56 be a projective Lagrangian fibration of a smooth symplectic variety $2n$57 to a smooth variety $2n$58, and assume that the smooth locus $2n$59 is surjective. Then there exists a Néron model $2n$60 of the abelian fibration over the smooth locus and a Néron model $2n$61 of the translation automorphism abelian scheme $2n$62; moreover, $2n$63 is a $2n$64-torsor. Contrary to elliptic fibrations, $2n$65 need not be the Néron model, precisely because of flops in higher-dimensional symplectic varieties. The Néron model is described as a moduli space of birational translation automorphisms

$2n$66

for étale $2n$67 (Kim, 2024).

Projective Lagrangian fibrations also arise from explicit geometric constructions. A singular $2n$68-dimensional projective symplectic variety $2n$69 can be realized as a relative compactified Prym variety associated to a family of genus $2n$70 curves with involution on a K3 surface double cover of a generic cubic surface. The support morphism

$2n$71

is a Lagrangian fibration whose generic fibre is $2n$72, an abelian $2n$73-fold of polarization type $2n$74; $2n$75 has only finite quotient singularities and no symplectic desingularization (Matteini, 2014).

A smooth projective compactification of a different kind appears for cubic fourfolds. For a general cubic fourfold, the relative intermediate Jacobian fibration over the smooth hyperplane-section locus carries a natural holomorphic symplectic form making the fibration Lagrangian, and there exists a smooth projective compactification

$2n$76

whose total space is an irreducible hyper-Kähler $2n$77-fold deformation equivalent to O’Grady’s OG10 example. This supplies a projective Lagrangian fibration in principally polarized abelian varieties obtained through compactified Prym geometry, descent from the Fano space of lines, and extension of the holomorphic symplectic form (Laza et al., 2016).