Lagrangian fibrations on hyperkähler fourfolds
Abstract.
Answering the strong form of a question posed by Beauville, we give a short geometric proof that any hyperkähler fourfold containing a Lagrangian subtorus admits a holomorphic Lagrangian fibration with fibre .
Key words and phrases:
hyperkähler manifold, Lagrangian fibration2010 Mathematics Subject Classification:
53C26, 14D06, 14E30, 32G10, 32G05.1. Introduction
Let be a hyperkähler manifold, that is, a compact, simplyconnected Kähler manifold such that is spanned by a holomorphic symplectic form . By work of Matsushita it is wellknown that the only possible nontrivial holomorphic maps from to a lowerdimensional complex space are Lagrangian fibrations, see section 2. Moreover, a special version of the socalled Hyperkähler SYZconjecture asserts that any hyperkähler manifold can be deformed to a hyperkähler manifold admitting a Lagrangian fibration.
Hence, it is an important problem to find geometric conditions on a given hyperkähler manifold that guarantee the existence of a Lagrangian fibration; here we address a question posed by Beauville [Bea11, Sect. 1.6]: {custom}[Question B] Let be a hyperkähler manifold and a Lagrangian torus in . Is a fibre of a (meromorphic) Lagrangian fibration ? In our previous article [GLR11] it is shown that Question B has a positive answer in case is nonprojective. Moreover, for any hyperkähler manifold that admits an almost holomorphic Lagrangian fibration, a further hyperkähler manfold, birational to the first one, is found, on which the Lagrangian fibration becomes holomorphic.
The approach to the projective case of Beauville’s question pursued here is based on a detailed study of the deformation theory of in . For this, consider the component of the Barlet space that contains together with its universal family and the evaluation map to :
It was shown in [GLR11, Lemma 3.1] that is surjective and generically finite, and that admits an almost holomorphic Lagrangian fibration if and only if .
If the degree of is strictly bigger than one, some deformations of intersect in unexpected ways. In order to deal with this, we introduce the notion of reduction: for each projective hyperkähler manifold containing a Lagrangian torus there exists a projective variety and a rational map , uniquely defined up to birational equivalence, whose fibre through a general point coincides with the connected component of the intersection of all deformations of through . In this situation, we say that is separable if is birational, and prove the following result: {custom}[Theorem 3.2] Let be a projective hyperkähler manifold and a Lagrangian subtorus. Then admits an almost holomorphic fibration with strong fibre if and only if is not separable .
If is a hyperkähler fourfold, then we can exclude the case that is separable by symplectic linear algebra. Moreover, based upon the rather explicit knowledge about the birational geometry of hyperkähler fourfolds we obtain a positive answer to the strongest form of Beauville’s question:
{custom}[Theorem 5]
Let be a fourdimensional hyperkähler manifold containing a Lagrangian torus . Then admits a holomorphic Lagrangian fibration with fibre .
At the Moscow conference ”Geometric structures on complex manifolds” Ekaterina Amerik brought to our attention that she had independently shown a related result, based on an observation from [AC08], to the effect that in dimension four every projective hyperkähler manifold containing a Lagrangian subtorus admits an almost holmorphic Lagrangian fibration with fibre [A11].
Acknowledgements
The authors want to thank Daniel Huybrechts for his interest in our work and for several stimulating discussions. We are grateful to Ekaterina Amerik for communicating to us the observation contained in Lemma 5, which greatly simplified our previous argument. The second author thanks Laurent Manivel for stimulating discussions, in particular, for pointing out Remark 5. The third author thanks Misha Verbitsky for an invitation to Moscow.
The support of the DFG through the SFB/TR 45, Forschergruppe 790, and the third author’s EmmyNoether project was invaluable for the success of the collaboration. The first author gratefully acknowledges the support of the BadenWürttembergStiftung via the “Eliteprogramm für Postdoktorandinnen und Postdoktoranden”. The second author acknowledges the support by the CNRS and the Institut Fourier.
2. Preliminaries and setup of notation
2.1. Lagrangian fibrations
{defin}Let be a hyperkähler manifold. A Lagrangian fibration on is a holomorphic map with connected fibres onto a normal complex space such that every irreducible component of the reduction of every fibre of is a Lagrangian subvariety of . Due to fundamental results of Matsushita it is known that any fibration on a hyperkähler manifold is automatically Lagrangian: {theo}[[Mat99, Mat00, Mat01, Mat03]] Let be a hyperkähler manifold of dimension . If is a morphism with connected fibres to a normal complex space with , then is a Lagrangian fibration. In particular, is equidimensional and . Furthermore, every smooth fibre of is a complex torus.
2.2. Meromorphic maps
Let be a normal complex space, a compact complex space, and a meromorphic map. Let
(1) 
be a resolution of the indeterminacies of . The fibre of over a point is defined to be . This is independent of the chosen resolution.
Recall that a meromorphic map as above is called almost holomorphic if there is a Zariskiopen subset such that the restriction is holomorphic and proper. A strong fibre of an almost holomorphic map is a fibre of .
Let be a normal algebraic variety, a complete algebraic variety, and an almost holomorphic rational map. If is a divisor on , then its pullback via is defined either geometrically as the closure of the pullback on the locus where is holomorphic, or on the level of locally free sheaves as , where is a resolution of indeterminacies as in diagram (1).
2.3. Deformations of Lagrangian subtori
The starting point for our approach to Beauville’s question is the deformation theory of a Lagrangian subtorus in a hyperkähler manifold . We quickly recall the relevant results from [GLR11, Sects. 2 and 3].
The Barlet space of (or Chow scheme in the projective setting) parametrises compact cycles in and it turns out (see \refenumi of Lemma 2.3 below) that there is a unique irreducible component of containing the point . Denoting by the graph of the universal family over and by the discriminant locus of , i.e., the set of points parametrising singular elements in the family , we obtain the following diagram.
(2) 
A detailed analysis of the maps in diagram (2) shows that a small étale or analytic neighbourhood of in fibres over a neighbourhood of in . More precisely, we have the following result. {lem}[[GLR11, Lem. 3.1]] Let be a hyperkähler manifold of dimension and let be a Lagrangian subtorus of . Then, the following holds.

The Barlet space is smooth of dimension near . In particular, is contained in a unique irreducible component of and is smooth of dimension near .

The morphism is finite étale along smooth fibres of . In particular, a sufficiently small deformation of is disjoint from and there are no positivedimensional families of smooth fibres through a general point .

If with smooth , then is a Lagrangian subtorus of .
We remark two simple but useful consequences of Lemma 2.3.

The locus is the locus of points such that there is a singular deformation of passing through . By dimension reasons it is a proper subset of and by Lemma 2.3 \refenumii the map is finite and étale on the preimage of .

Statement \refenumii implies in particular that for any two points the intersection product as cycles in vanishes. It is therefore impossible for members of the family to intersect in a finite number of points.
2.4. Almost holomorphic Lagrangian fibrations and Barlet spaces.
The following result relates the deformation theory of in discussed above to our question about globally defined almost holomorphic Lagrangian fibrations. {lem}[[GLR11, Lem. 3.2]] Let be a hyperkähler manifold containing a Lagrangian subtorus . Then admits an almost holomorphic Lagrangian fibration with strong fibre if and only if the evaluation map in diagram (2) is bimeromorphic. If is birational, then is the desired almost holomorphic fibration (up to normalisation of ). For the other direction one uses the Barlet space of a resolution of indeterminacies.
3. reduction and separable manifolds
Let be a projective hyperkähler manifold containing a Lagrangian subtorus . In this section we start our analysis of the maps in the associated diagram (2). Recall from Lemma 2.4 above that in order to answer Beauville’s question positively we have to show that the evaluation map is birational.
3.1. reduction
Here, we construct a meromorphic map associated with the covering family . Generically, this map is a quotient map for the meromorphic equivalence relation defined by the family , i.e., generically it identifies those points in that cannot be separated by members of .
Construction of the reduction
We work in the setup summarised in diagram (2). We set . Recall from Remark 2.3 that the map
is finite étale; we denote its degree by .
The map induces a morphism . Composing this map with the natural morphism induced by , we construct a morphism . This morphism naturally extends to a rational map . Let
be a resolution of singularities of the indeterminacies of with nonsingular. The Stein factorisation of then yields the following diagram.
Here, is the rational map induced by . Noting that is unique up to birational equivalence, and hence canonically associated with the pair , we call it the reduction of . {rem} For every point there are exactly pairwise distinct smooth tori in the family containg . By construction, is defined at and maps it to the class of in .
First properties of the reduction
The following settheoretical assertion is an immediate consequence of the construction of . {lem} The fibre of through a point coincides with the connected component of
containing .
Proof.
If , then is étale in every point of the preimage . Thus the image consists of the points in that parametrise the pairwise distinct subtori in through . In particular, the meromorphic map is defined at and its fibre is
(3) 
After taking the Stein factorisation, the fibre of is the component of (3) through , as claimed. ∎
Let be a projective hyperkähler manifold containing a Lagrangian subtorus . Then the reduction is almost holomorphic.
Proof.
Let be the domain of definition of , and let be the locus where is not defined. We have to show that the general fibre of does not intersect .
Aiming for a contradiction, suppose that for a general the fibre of through intersects nontrivially. Recall from item \refenumi of Remark 2.3 that is the locus swept out by singular deformations of and from Remark 3.1.1 that is holomorphic on . Take a point . Consider the graph of with projections and . As explained for example in [Deb01, Sect. 1.39], the closed subset can be described as
(4) 
As is normal and is birational, has connected fibres. Thus, the variety is connected. We list some further properties of :

, because and is the graph of ,

as ,

the point is contained in all fibres over points in .
Suppose for the moment that we had a diagram
such that is connected, and is a local isomorphism at the point . We claim that this would produce a contradiction. Namely, let be the pairwise distinct tori in the family containing . Since is connected and by item \refenumi above, Lemma 3.1.2 implies that there exists such that . By Lemma 2.3, small deformations of constitute a fibration in an analytic neighbourhood of . Thus, for all points sufficiently close to there is a small deformation of with and
(5) 
On the other hand, item \refenumiii above and Lemma 3.1.2 imply that
which in view of (5) is absurd.
It remains to find the variety . We observe that it suffices to construct in an Euclidean open neighbourhood of . Invoking the generality assumption on and the implicit function theorem we find a small neighbourhood such that the restriction is a trivial holomorphic fiber bundle. In particular, is open and there is a section for the subvariety . The only remaining property to be fulfilled is connectedness of and . This may be achieved by shrinking and , and so we conclude the proof. ∎
A projective hyperkähler manifold containing a Lagrangian subtorus is called separable if its reduction is birational.
3.2. Lagrangian fibrations on nonseparable manifolds
{theo}Let be a projective hyperkähler manifold and a Lagrangian subtorus. Then admits an almost holomorphic fibration with strong fibre if and only if is not separable. As a consequence of this result we can reformulate Beauville’s question in the following way. {custom}[Question B’’] Does there exist a projective hyperkähler manifold together with a Lagrangian subtorus such that is separable?
Proof of Theorem 3.2.
If is not separable, the reduction is an almost holomorphic map (Lemma 3.1.2) such that . Thus by [GLR11, Thm. 6.7], the map is an almost holomorphic Lagrangian fibration on . By the description of the general fibre of the reduction (Lemma 3.1.2), the torus is a strong fibre of .
If conversely is an almost holomorphic Lagrangian fibration with strong fibre , then through the general point there is a unique Lagrangian subtorus in and the reduction coincides with the rational map . In particular, is not separable. ∎
4. Intersections of Lagrangian subtori
As before, let be a projective hyperkähler manifold containing a Lagrangian subtorus . In this section we study a neighbourhood of in more closely, which leads to several results about the geometry of intersections of different members in the family of deformations of . We are going to use the notation and the results of Section 2.3 throughout.
By Lemma 2.3, is smooth at and we can find a neighbourhood of such that the restriction of the evaluation map to the preimage embeds into . We may thus consider as an open subset of . The intersection of with a submanifold is depicted in Figure 1.
Let be a smooth and proper submanifold, and a smooth Lagrangian torus that intersects nontrivially. Then a generic small deformation of has smooth intersection with .
Proof.
We continue to use the notation introduced above. Since is open in , the intersection is smooth. Furthermore, the map is proper, because is proper and is compact. We can therefore apply the theorem on generic smoothness to which proves the result. ∎
Let be a compact submanifold and be a general Lagrangian subtorus, such that . Then is trivial. If is a complex torus, then is a disjoint union of tori.
Proof.
As is general, the intersection is smooth by Lemma 4. Moreover, both statements can be verified by looking at one connected component of at a time. We invoke the notation introduced in the beginning of this section, and let be a connected component of . If is sufficently small, then the inclusion induces a onetoone correspondence of their respective connected components. Let be the unique component of corresponding to . By generality of we may assume that is a smooth map, thus is smooth of dimension near . Moreover, parametrizes those small deformations of that induce a flat deformation of inside . Corresponding to the family we thus obtain a classyfying map from to the Douadyspace of .
On the level of tangent spaces we have , where the last equality comes from the HilbertChow morphism, compare [GLR11, Lem. 3.1]. The morphism induces a map . But small deformations of inside induced by deformations of are disjoint from by Lemma 2.3 \refenumii. Thus the map is injective, and the image of consists of nowhere vanishing sections. For dimension reasons these sections generate the normal bundle of in , and consequently is trivial, as claimed.
If is a torus as well, then is likewise trivial. So, by the normal bundle sequence
the tangent bundle is trivial, and thus is a complex torus. ∎
Based on the preceeding result we can now refine the observation in Remark 2.3(ii):
Let be a fourdimensional hyperkähler manifold. Let and be two Lagrangian tori intersecting smoothly, and set . Then, is a finite disjoint union of elliptic curves.
Proof.
It remains to exclude the existence of zerodimensional connected components of . By general Lagrangian intersection theory, see for example [BF09, Introduction], we have
However, this already implies the claim, since by Proposition 4 above, any positive dimensional component of is a smooth elliptic curve, contributing zero to the Euler characteristic . ∎
Let be a fourdimensional projective hyperkähler manifold, a Lagrangian subtorus. Assume that is separable. Then, the evaluation map in Diagram (2) has degree at least three.
Proof.
As is assumed to be separable, Theorem 3.2 and Lemma 2.4 imply that is not birational. It remains to exclude the case . By Lemma 3.1.2, the separability means that at a general point the connected component of is just . If there are just two tori in containing , say and . As was general, Lemma 4 tells us that is smooth. Then Lemma 4 contradicts the fact that the connected component of containing is . ∎
5. Hyperkähler fourfolds
Using the results from the last section we can now prove our main result which gives the strongest possible positve answer to Beauville’s question: {theo} Let be a fourdimensional hyperkähler manifold containing a Lagrangian torus . Then admits a holomorphic Lagrangian fibration with fibre . {rem} We are grateful to E. Amerik for communicating the following linear algebra observation to us which serves to exclude separable manifolds in dimension four. This greatly simplified a previous deformationtheoretic argument. {lem} Let be a fourdimensional symplectic vector space with symplectic form , and let be three Lagrangian subspaces satisfying for all . Then .
Proof.
Suppose on the contrary that and consider the span . It is of dimension as . Moreover, we claim that
(6) 
Indeed, otherwise we would have , implying that the intersections are all onedimensional, in contradiction to our assumption that .
Now, again using we write . As is symplectic and is Lagrangian, there is and such that . According to the inclusion (6) we can write with , so that
as and are Lagrangian. Contradiction. ∎
Lemma 5 can also be proven using the following beautiful geometric argument which was explained to us by Laurent Manivel: The Grassmanian of Lagrangian subspaces in is biholomorphic to the (smooth) intersection of the Plücker quadric and the linear subspace defined by vanishing of the symplectic form , and hence is a smooth quadric . The condition means that the line in joining the points is contained in . If the triple intersection , then span a plane . But has degree and thus cannot contain a union of lines.
Let be a fourdimensional projective hyperkähler manifold containing a Lagrangian torus . Then is not separable.
Proof.
Suppose on the contrary that is separable. Given a general point , it follows from Lemma 3.1.2 and Corollary 4 that there exists a natural number , and smooth Lagrangian subtori that locally cut out . The point being general, Lemma 4 and Lemma 4 imply that there exist two such tori, say and , that intersect in an elliptic curve at . Any other torus in passing through either contains or cuts out a zerodimensional subscheme.
As a consequence of separability there exists a lagrangian torus containing the point but not containing , such that the intersection scheme is zerodimensional at the point . Again invoking that was general, we may assume that the intersections , and are smooth at . Consequently, the three Lagrangian subspaces satisfy the assumptions of Lemma 5. It follows that , contradicting our choice of . Therefore, cannot be separable. ∎
Proof of Theorem 5.
If is not projective, we are done by [GLR11, Thm. 4.1], so we may assume to be projective. By Proposition 5, is not separable and hence admits an almost holomorphic Lagrangian fibration by Theorem 3.2. It remains to show that the existence of an almost holomorphic Lagrangian fibration implies the existence of a holomorphic one, which will be done in Lemma 5 below. ∎
Let be an almost holomorphic Lagrangian fibration on a projective hyperkähler fourfold. Then there exists a birational modification such that is a holomorphic Lagrangian fibration.
The proof of Lemma 5 rests on the explicit knowledge of the birational geometry of hyperkähler fourfolds. For this we recall the notion of Mukai flop: Assume that a hyperkähler fourfold contains a smooth subvariety . If we blow up , the exceptional divisor is isomorphic to the projective bundle , and it is well known that it can be blown down in the other direction to yield another hyperkähler manifold . The resulting birational transformation is called the Mukai flop at .
Proof of Lemma 5.
By [GLR11, 6.2] there exists a holomorphic model for , that is, a Lagrangian fibration on a possibly different hyperkähler manifold and a diagram
with birational horizontal arrows such that is an isomorphism near the general fibre of .
We claim that the composition is holomorphic and thus a Lagrangian fibration on . To see this first note that by [WW03, Thm. 1.2] the map factors as a finite composition of Mukai flops, so by induction we may assume that is the simultaneous Mukai flop of a disjoint union of embedded projective planes .
As is holomorphic near a general fibre of , none of the ’s can intersect the general fibre. Thus is a proper subset of and hence of dimension at most 1. Since there is no nonconstant map from to a curve, is a single point. In other words, the locus of indeterminacy of is contained in the fibres of , and thus the composition remains holomorphic. ∎
Footnotes
 After this article was written, JunMuk Hwang and Richard Weiss have posted a proof of the projective case of the weak form of Beauville’s question, producing an almost holomorphic Lagrangian fibration on any projective dimensional hyperkähler manifold containing a Lagrangian torus, see [HW12]. Their argument has two parts: one is geometric and one is concerned with abstract group theory. In contrast, our answer to the strong form of Beauville’s question, Theorem 5, is purely geometric, uses global arguments in addtion to local deformation theoretic ones, and uses symplectic linear algebra in place of their grouptheoretic arguments.
References
 Ekaterina Amerik. A remark on a question of Beauville about lagrangian fibrations, arXiv:1110.2852, 2011.
 Ekaterina Amerik and Frédéric Campana. Fibrations méromorphes sur certaines variétés à fibré canonique trivial. Pure Appl. Math. Q., 4(2, part 1):509–545, 2008.
 Arnaud Beauville. Holomorphic symplectic geometry: a problem list. In Complex and Differential Geometry, Conference held at Leibniz Universität Hannover, September 14 – 18, 2009, volume 8 of Springer Proceedings in Mathematics. Springer, 2011, preprint arXiv:1002.4321.
 Kai Behrend and Barbara Fantechi. Gerstenhaber and BatalinVilkovisky structures on Lagrangian intersections. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, volume 269 of Progr. Math., pages 1–47. Birkhäuser Boston Inc., Boston, MA, 2009.
 Olivier Debarre. HigherDimensional Algebraic Geometry. Universitext. SpringerVerlag, New York, 2001.
 Daniel Greb, Christian Lehn, and Sönke Rollenske. Lagrangian fibrations on hyperkähler manifolds — On a question of Beauville. Preprint arXiv:1105.3410, 2011, to appear in Ann. Sci. Éc. Norm. Supér.
 JunMuk Hwang and Richard Weiss. Webs of Lagrangian Tori in Projective Symplectic Manifolds, Invent. Math., Online First, DOI:10.1007/s0022201204072
 Daisuke Matsushita. On fibre space structures of a projective irreducible symplectic manifold. Topology, 38(1):79–83, 1999.
 Daisuke Matsushita. Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds. Math. Res. Lett., 7(4):389–391, 2000.
 Daisuke Matsushita. Addendum: “On fibre space structures of a projective irreducible symplectic manifold” [Topology 38 (1999), no. 1, 79–83]. Topology, 40(2):431–432, 2001.
 Daisuke Matsushita. Holomorphic symplectic manifolds and Lagrangian fibrations. Acta Appl. Math., 75(13):117–123, 2003.
 Jan Wierzba and Jarosław A. Wiśniewski. Small contractions of symplectic 4folds. Duke Math. J., 120(1):65–95, 2003.