Lagrangian Fibrations

We review the theory of Lagrangian fibrations of hyperkähler manifolds as initiated by Matsushita. We also discuss more recent work of Shen–Yin and Harder–Li–Shen–Yin. Occasionally, we give alternative arguments and complement the discussion by additional observations.

Throughout, X denotes a compact hyperkähler manifold of complex dimension 2n. A fibration of X is a surjective morphism f : X / / / / B with connected fibers onto a normal variety B with 0 < dim(B) < 2n. A submanifold T ⊂ X of dimension n is Lagrangian if the restriction σ| T ∈ H 0 (T, Ω 2 T ) of the holomorphic two-form σ ∈ H 0 (X, Ω 2 X ) is zero.

Basics on Lagrangian Fibrations
We first discuss Lagrangian submanifolds and in particular Lagrangian tori. Then we study the cohomology and the singularities of the base B. Next we show that the fibers, smooth ones as well as singular ones, of any fibration are Lagrangian and conclude that fibrations of hyperkähler manifolds over a smooth base are flat. At the end, we mention further results and directions without proof: Matshushita's description of the higher direct image sheaves R i f * O X , Beauville's question whether Lagrangian tori are always Lagrangian fibers, smoothness of the base, etc.
There does not seem to be a direct proof of this fact. However, using that the rank of the restriction map H 4 (X, Q) / / H 4 (T, Q) is one, see Theorem 2.1, it can be shown as follows. The classesq and c 2 in H 4 (X, Q) both have the distinguished property that the homogenous forms Xq · β 2n−2 and c 2 (X) · β 2n−2 defined on H 2 (X, Z) are non-trivial scalar multiples of q(β) n−1 and, therefore, of each other. 3 If [T ] ∈ H 2n (X, Z) is the class of a fiber f −1 (t), then up to scaling [T ] = f * α n for some α ∈ H 2 (B, Q). Hence, for a Kähler class ω on X we find (up to a non-trivial scalar factor) Tq | T · ω| n−2 T = Xq · f * α n · ω n−2 = X c 2 (X) · f * α n · ω n−2 = T c 2 (X)| T · ω| n−2 T = 0.
(ii) For other types of Lagrangian submanifolds, the restrictions of the Chern classes of X are not trivial. For example, for a Lagrangian plane P 2 ⊂ X one easily computes P 2 c 2 (X)| P 2 = 15.
As remarked before, the normal bundle of a Lagrangian torus is trivial. The next observation can be seen as a converse, it applies in particular to the smooth fibers of any fibration f : X / / B. above by exploiting the structure of the subring of SH 2 (X, Q) ⊂ H * (X, Q), the proof of the identities H k (B, Q) H k (P n , Q) for k > 2 uses deeper information about the hyperkähler structure.
(i) The first proof for B smooth and X projective was given by Matsushita [51], as a consequence of the isomorphisms R i f * O X Ω i B , see Sect. 1.5.1. Combining this isomorphism with the splitting [41], one finds , which proves the claim. 6 (ii) Another one, which also works for singular B and non-projective X, was given in [66] and roughly relies on the fact that H * (B, C) can be deformed into O B is excluded, because it would imply H n,0 (B) = 0, which was excluded above. Remark 1.9. It turns out that as soon as the base B is smooth, then B P n . This result is due to Hwang [34] and its proof relies on the theory of minimal rational tangents. The results by Matsushita and more recently by Shen and Yin, see Remark 1.7 and Sect. 2, can be seen as strong evidence for the result. In dimension two, the result is immediate: Any smooth projective surface B with ω * B ample and H 2 (B, Q) Q is isomorphic to P 2 .
It is tempting to try to find a more direct argument in higher dimension, but all attempts have failed so far. For example, according to Hirzebruch-Kodaira [29] it suffices to show that H * (B, Z) H * (P n , Z) such that the first Chern class of a line bundle L corresponding to a generator of H 2 (B, Z) satisfies h 0 (B, L k ) = h 0 (P n , O(k)), see [47] for a survey of further results in this direction.
Alternatively, by Kobayashi-Ochai [39], it is enough to show that ω B is divisible by n + 1, i.e. the Fano manifold B has index n + 1. As a first step, one could try to show that f * ω B is divisible by n + 1.

Singularities of the Base
It is generally expected that the base manifold B is smooth, but at the moment this is only known for n ≤ 2, see [7,35,61]. The expectation is corroborated by the following computations of invariants of the singularities of B.
Denote by IH * (B, Q) the intersection cohomology of the complex variety B with middle perversity and rational coefficients. It is the hypercohomology of the Proof. For (i) and (ii) one only needs that f : X / / B is a connected and equidimensional morphism from a smooth variety X, while in the proof of (iii) one also needs ω X trivial.
For any t ∈ B, choose a chart ϕ : U x ⊂ X / / C 2n , centered at x, and the analytic subset S:=ϕ −1 (Λ), where Λ ⊆ C 2n is an n-dimensional affine subspace intersecting the fiber ϕ(f −1 (t)) transversely. Since f is equidimensional, the restriction f | S : S / / B is finite over an analytic neighbourhood U of t. Therefore, U is Q-factorial by [38,Lem. 5.16].
Denote S • :=S ∩ f −1 (U ). By the decomposition theorem [4]. 8    This is known for n = 2 by [10, Lem. 2.6], but it is open in higher dimension. One of the main issue is that f itself need not be a quotient map, not even locally.  14. Let f : M / / N be a surjective holomorphic map between compact complex manifolds, with M Kähler. By [70,Lem. 7.28], the pullback f * : H * (N, Q) / / H * (M, Q) is injective. However, this may fail if N is singular, e.g. if f is a normalization of a nodal cubic, even if N has Q-factorial log terminal singularities, see for instance [52,Thm. 5.11]. 7 Are the singularities of B actually factorial? 8 Alternatively, note that the trace map R(f | S • ) * Q S • / / ICU splits the natural morphism  [50].

The Fibers of a Fibration
Next we present Matsushita's result that any fibration of a compact hyperkähler manifold is a Lagrangian fibration. Proof. Comparing the coefficients of x n−2 y n in the polynomial (in x and y) the equation which for a Kähler class ω and using that σ ∧σ is semi-positive implies σ| T = 0. Then conclude by Lemma 1.5. says that R 2 f * ω X is torsion free. Since in our case ω X O X , this shows that R 2 f * O X is torsion free. Letσ ∈ H 2 (X, O X ) be the conjugate of the symplectic form, and ρ be its image in H 0 (B, R 2 f * O X ). Since the general fiber is Lagrangian, ρ must be torsion and hence zero. If T / / T is a resolution of T , then the image of and hence trivial. This implies that the image of σ in H 0 ( T , Ω 2 T ) is trivial, i.e. σ| T = 0. By semi-continuity of the dimension of the fibers, dim T ≥ n, and so T is Lagrangian.
The flatness follows from the smoothness of X and B, see [25, Exer. III.10.9].
Remark 1.18. Note that the conclusion that f is flat really needs the base to be smooth. In fact, by miracle flatness, f is flat if and only if B is smooth.

Further Results
We summarize a few further results without proof.

Higher Direct Images.
The first one is the main result of [51].
is a fibration of a projective 9 hyperkähler manifold over a smooth base. Then which holds because the smooth fibers of f are Lagrangian. A relative polarization is used to show that saying that R i f * ω X are torsion free, which for X hyperkähler translates into R i f * O X being torsion free.
As mentioned in Remark 1.7, the theorem implies H * (B, Q) H * (P n , Q).

Lagrangian Tori are Lagrangian Fibers.
In [6] Beauville asked whether every Lagrangian torus T ⊂ X is the fiber of a Lagrangian fibration X / / B. The question has been answered affirmatively: (i) Greb-Lehn-Rollenske in [20] first dealt with the case of non-projective X and later showed in [21] the existence of an almost 10 holomorphic Lagrangian fibration in dimension four. (ii) A different approach to the existence of an almost holomorphic Lagrangian fibration with T as a fiber was provided by Amerik-Campana [1]. The fourdimensional case had been discussed before by Amerik [2]. (iii) Hwang-Weiss [33] deal with the projective case and proved the existence of an almost Lagrangian fibration with fiber T . Combined with techniques of [20] this resulted in a complete answer.

Cohomology of the Base and Cohomology of the Fiber
The aim of this section is to prove the following result.
Theorem 2.1. Assume X / / B is a fibration and let X t be a smooth fiber. Then The first isomorphism for X projective and B smooth is originally due to Matsushita [51], see Remark 1.7. The proof we give here is a version of the one by Shen and Yin [66] that works without assuming X projective. Note also that we do not assume that the base B is smooth.
The second isomorphism in degree two is essentially due to Oguiso [60], relying on results of Voisin [69]. The paper by Shen and Yin [66] contains two proofs of the general result, one using the sl 2 -representation theory of the perverse filtration and another one, due to Voisin, relying on classical Hodge theory. 9 Again, the projectivity assumption can presumably be dropped by applying results of Saito. The proof we shall give avoids the perverse filtration as well as the various sl 2 × sl 2 -actions central for the arguments in [66]. The discussion below also proves the second result in [66, Thm. 0.2], namely the equality between the classical and perverse Hodge numbers, see Sect. 2.3. How it fits into the setting of P = W is explained in Sect. 3.

Algebraic Preparations
To stress the purely algebraic nature of what follows we shall use the shorthand H * :=H * (X, C) and consider it as a graded C-algebra. Consider a non-trivial, isotropic element β of degree two, i.e. 0 = β ∈ H 2 with q(β) = 0. Then, according to Verbitsky and Bogomolov [9,68], one has β n = 0 and β n+1 = 0.
In particular, multiplication by β defines on H * the structure of a graded All that is needed in the geometric applications is then put into the following statement.

Proposition 2.2. For every two non-zero, isotropic elements
Proof. Consider the complex algebraic group of automorphisms Aut(H * ) of the graded C-algebra H * and its image G under Aut(H * ) / / Gl(H 2 ). Clearly, the assertion holds if β, β ∈ H 2 are contained in the same G-orbit. As any two non-zero isotropic classes β, β are contained in the same orbit of the complex special orthogonal group SO(H 2 , q), it suffices to show that SO(H 2 , q) ⊂ G. This follows from [65,Prop. 3.4], up to taking complex coefficients in loc. cit. Remark 2.3. The arguments can be adapted to prove the following statement: Assume β, β ∈ H 2 satisfy q(β) = q(β ) = 0. Then the induced graded C[x]/(x 2n+1 )algebra structures on H * , given by letting x act by multiplication with β resp. β , are isomorphic.
For 0 = β ∈ H 2 with q(β) = 0 and d ≤ n we let which is called the space of β-primitive forms. Note, however, that β does not satisfy the Hard Lefschetz theorem; otherwise we would have defined primitive classes in H d as elements in the kernel of β 2n−d+1 . We will also need the two spaces

Geometric Realizations
Let us begin by looking at the obvious choice for β provided by the symplectic form σ ∈ H 0 (X, Ω 2 X ) ⊂ H 2 (X, C). Lemma 2.5. For β = σ one has Proof. Concerning the first equality, one inclusion is obvious: For the other direction, use that σ n−p : Ω p X ∼ / / Ω 2n−p X , for p ≤ n, is an isomorphism and that, therefore, for q > 0 the composition For the second part observe that Ker(σ n ) ∩ H p,q (X) = p>0 H p,q (X).
As an immediate consequence of Corollary 2.4 one then finds. Next let us consider a Lagrangian fibration f : X / / B. We consider the class β:=f * α, which is isotropic since α n+1 = 0 for dimension reasons.  We keep the isotropic class β = f * α and observe that the natural inclusion

Proof. The assertion follows from the Lefschetz decomposition
is actually an isomorphism.

Lemma 2.8. (Voisin)
Let β = f * α be as before and let X t ⊂ X be a smooth fiber of f . Then Proof. The result is proved in [66, App. B]. The assertion is shown to be equivalent to the statement that the intersection pairing on the fiber is non-degenerate on the image of the restriction map, which in turn is deduced from Deligne's global invariant cycle theorem.
SinceP 0 H * H * (P n , C) by Corollary 2.6, its image in H * (X t , C) is the subring generated by the restriction of a Kähler class. Hence, π is an isomorphism, which proves the second isomorphism in Theorem 2.1. However, it is easier to argue directly, as the equality holds in Lemma 2.8 by (2.3).

Perverse = Hodge
As in Sect. 2.1, we consider the abstract algebraic situation provided by H * :=H * (X, C) and the additional structure induced by the choice of a non-zero isotropic class β ∈ H 2 . The two spaces P 0 H d andP 0 H d defined there, both depending on β, are part of a filtration To see this, one needs to use the Lefschetz decomposition with respect toσ: Note that from this example one can deduce that indeed for any choice of β one has P β k H d = 0 for k < 0 and P β k H d = H d for k ≥ d. 3. P = W P = W for compact hyperkähler manifolds asserts that the perverse filtration associated with a Lagrangian fibration can be realised as the weight filtration of a limit mixed Hodge structure of a degeneration of compact hyperkäher manifolds. It boils down to the observation that the cup product by a semiample not big class and a logarithmic monodromy operator define nilpotent endomorphisms in cohomology which are not equal, but up to renumbering induce the same filtration. Inspired by P = W, we provide some geometric explanation or conjecture concerning the appearance of the cohomology of P n in the introduction and in Theorem 2.1.

The Weight Filtration of a Nilpotent Operator
with the property that (1) NW k ⊆ W k−2 , and denoting again by N the induced endomorphism on graded pieces, (2)  The weight filtration of N on H * can be constructed inductively as follows: first let W 0 :=ImN l , and W 2l−1 := ker N l . We can replace H * with W 2l−1 /W 0 , on which N is still well-defined and N l = 0. Then define Continuing inductively, we obtain the unique (!) filtration on H * satisfying (1) and (2).
By the Jacobson-Morozov theorem, the nilpotent operator N can be extended to an sl 2 -triple with Cartan subalgebra generated by an element H N which is unique up to scaling. By the representation theory of sl 2 -triples, there exists a decomposition called the weight decomposition, with the property that H N (v) = λv for all v ∈ H * λ . In particular, the decomposition splits the weight filtration of N let us apply this to some geometric examples. (i) Any cohomology class ω ∈ H 2 (X, C) defines a nilpotent operator L ω on H * :=H * (X, C) by cup product. If ω is Kähler, then the Hard Lefschetz theorem implies that the weight filtration of L ω on H * centered at 2n is 12 (ii) Consider a Lagrangian fibration f : X / / B and let β be the pull-back of an ample class α ∈ H 2 (B, Q). Up to renumbering, the weight filtration associated with the class β on H * centered at n coincides with the perverse filtration, see Q). Indeed, the action of β gives the morphisms The isomorphism is called the perverse Hard Lefschetz theorem [13,Prop. 5.2.3]. By Proposition 2.2, this corresponds to the isomorphismσ j : H n−j (X, Ω i X ) H n+j (X, Ω i X ). (iii) Let π : X / / Δ be a projective degeneration of hyperkähler manifolds over the unit disk which we assume to be semistable, i.e. the central fiber X 0 is reduced with simple normal crossings. For t ∈ Δ * , let N denote the logarithmic monodromy operator on H * (X t , Q). The weight filtration of N centered at d on Q), is the weight filtration of the limit mixed Hodge structure associated to π, see [62,Thm. 11.40]. The degeneration π : X / / Δ is called of type III if N 2 = 0 and N 3 = 0 on H 2 (X t , Q). In this case, the limit mixed Hodge structure is of Hodge-Tate type by [64,Thm. 3.8], and in particular Gr W 2i+1 H * (X t , Q) = 0. Then the even graded pieces of the weight filtration are used to define the Hodge numbers Q). The Hodge numbers w h 0,j (X ) have a clear geometric description. The dual complex of X 0 = Δ i , denoted by D(X 0 ), is the CW complex whose k-cells are in correspondence with the irreducible components of the intersection of (k + 1) divisors Δ i . The Clemens-Schmid exact sequence then gives In order to show P = W, namely that the filtrations (ii) and (iii) can be identified, we need the notion of hyperkähler triples with their associated so(5, C)action.
The set of all hyperkähler triples on X forms a Zariski-dense subset in In particular, all algebraic relations that can be formulated for triples in D • and which hold for triples of the form (ω I , ω J , ω K ) hold in fact for all (x, y, z) ∈ D • , see [66,Prop. 2.3].

The so(5, C)-Action
Recall the scaling operator By the Jacobson-Morozov theorem, to any ω ∈ H 2 (X, C) of Lefschetz type we can associate a sl 2 -triple (L ω , H, Λ ω ). Let p = (x, y, z) ∈ D • . The sl 2 -triples associated to x, y and z generate the Lie subalgebra g p ⊂ End(H * (X, C)), isomorphic to so(5, C), with Cartan subalgebra There is an associated weight decomposition Vol. 90 (2022) Lagrangian Fibrations 473 such that for all v ∈ H i,j (p) we have The following sl 2 -triples in g p induce the same weight decomposition, since for any v ∈ H i,j (p) we have Remark 3.2. The previous identities for hyperkähler triples are due to Verbitsky. The result for a general triple p = (x, y, z) ∈ D • follows from the density of hyperkähler triples in D • , and the fact that the sl 2 -representation H * (X, C) associated to x, y and z have the same weights, since x, y, and z are all of Lefschetz type, see [66, §2.4].

P = W
The main result of [30] is the following.

Theorem 3.3. (P = W)
For any Lagrangian fibration f : X / / B, there exists a type III projective degeneration of hyperkähler manifolds π : X / / Δ with X t deformation equivalent to X for all t ∈ Δ * , together with a multiplicative isomorphism H * (X, Q) H * (X t , Q), such that Proof. Let β = f * α be the pullback of an ample class α ∈ H 2 (B, Q), and η ∈ H 2 (X, Q) with q(η) > 0. Since β n+1 = 0, we have q(β) = 0. Up to replacing η with η + λβ for some λ ∈ Q, we can suppose that q(η) = 0. Set By scaling a nonzero vector x ∈ H 2 (X, C) perpendicular to y and z with respect to q, we obtain p(f ) = (x, y, z) ∈ D • with Soldatenkov showed that the nilpotent operator E p(f ) is the logarithmic monodromy N of a projective type III degeneration π : X / / Δ of compact hyperkähler manifolds deformation equivalent to X, see [64,Lem. 4.1,Thm. 4.6]. 13 The weight decomposition for the sl 2 -triple (3.4) splits the perverse filtration associated to f , since E p(f ) acts in cohomology via the cup product by β. The weight decomposition for the sl 2 -triple (3.5) splits the weight filtration of the limit The conjecture is inspired by the geometric P = W conjecture for character varieties, see the new version of [53] (to appear soon). Lemma 2.8 and (2.1) give If X 0 has simple normal crossings (or dlt singularities modulo adapting [26,Thm. 3.12]), one obtains that Therefore, Conjecture 3.8 would give a geometric explanation of P = W at the highest weight It is not clear what a geometric formulation of P = W should be that could explain the cohomological statement in all weights. Recent advance in the SYZ conjecture due to Li [48] suggests that profound tori can be made special Lagrangian, modulo a conjecture in non-archimedean geometry. A few months ago, the existence of a single special Lagrangian torus on X t was a complete mystery, see [23, §5, p.152]. Note also that Li's result is compatible with the expectation in symplectic geometry [3,Conj. 7.3]. Profound tori appear as general fibers of the SYZ fibration that Li constructed on an open set which contains an arbitrary large portion of the mass of X t with respect to a Calabi-Yau metric, still modulo the non-archimedean conjecture. It is curious (but maybe not surprising) 476 D. Huybrechts and M. Mauri Vol. 90 (2022) that also the previously quoted results [35] and [8] highly rely on non-archimedean techniques.

Multiplicativity of the Perverse Filtration
Proof. By P=W, it is sufficient to show that the weight filtration is multiplicative.
To this end, endow the tensor product H * (X t , Q) ⊗ H * (X t , Q) with the nilpotent endomorphism N ⊗ :=N ⊗ 1 + 1 ⊗ N , and call W ⊗ the weight filtration of N ⊗ . Since the monodromy operator e N is an algebra homomorphism of H * (X t , Q), N is a derivation, i.e.
As a consequence, the construction of the weight filtration (see Sect. 3.1) gives Together with [16, 1.6.9.(i)] which says that we conclude that the weight filtration is multiplicative. Alternatively see [30, §5]. Remark 3.10. For an arbitrary morphism of projective varieties or Kähler manifolds, the perverse filtration is not always multiplicative [71, Exa. 1.5], but it is so for instance if it coincides with the Leray filtration, or if P = W holds. Indeed, the Leray filtration and the weight filtration of the limit mixed Hodge structure are multiplicative.
It is natural to ask whether the multiplicativity holds at a sheaf theoretic level, for Rf * Q X , or over an affine base. The motivation for this comes from the celebrated P = W conjecture for twisted character varieties [12], which has been proved to be equivalent to the conjectural multiplicativity of the perverse filtration of the Hitchin map that is a proper holomorphic Lagrangian fibration over an affine base, see [14,Thm. 0.6]. From this viewpoint, it is remarkable that Shen and Yin give a proof of the multiplicativity in the compact case [66, Thm. A.1] which uses only the representation theory of sl(2)-triples, with no reference to the weight filtration.

Nagai's Conjecture for Type III Degenerations
Let π : X / / Δ be a projective degeneration of hyperkähler manifolds with unipotent monodromy T d on H d (X t , Q). The index of nilpotence of N d := log T d is nilp(N d ) = max{i | N i d = 0}, and nilp(N d ) ≤ d by [22,Ch. IV]. It is known that H 2 (X t , Q) determines the Hodge structure of H d (X t , Q) by means of the LLV representation, see [65]. Nagai's conjecture investigates to what extent nilp(N 2 ) determines nilp(N d ). The ring structure of the subalgebra generated by H 2 implies the inequality nilp(N 2k ) ≥ k · nilp(N 2 ), see [55,Lem. 2.4], but equality is expected. Conjecture 3.11. (Nagai) nilp(N 2k ) = k · nilp(N 2 ) for k ≤ 2n.
The previous inequalities imply Nagai's conjecture for type III degenerations, i.e. nilp(N 2 ) = 2. Remarkably, P = W explains Nagai's conjecture in terms of the level of the Hodge structure H d (X t , Q), and determines nilp(N d ) even for d odd.
Recall that the level of a Hodge structure H = ⊕H p,q , denoted by level(H), is the largest difference |p − q| for which H p,q = 0, or equivalently the length of the Hodge filtration on H. Proposition 3.12. Let π : X / / Δ be a type III projective degeneration of hyperkähler manifolds with unipotent monodromy. Then nilp(N d ) = level(H d (X t , C)).
Proof. Let l d be half of the length of the weight filtration of N d , i.e. l d := min{i : W 2i H d (X t , Q) = H d (X t , Q)}. By Definition 3.1, we have nilp(N d ) = l d .
For any type III degeneration of Hodge structures of hyperkähler type with unipotent monodromy, we know by the proof of Theorem 3.3 that the logarithmic monodromy N * is of the form E p = [β, Λ x ] for some β and x in H 2 (X, Q) with q(β) = 0. Here, we use the assumption b 2 (X t ) ≥ 5, see [64, §4.1]. Then, by Corollaries 2.9 and 3.4, we have l d = level(H d (X t , C)). Hence, nilp(N d ) = level(H d (X t , C)).