Holomorphic Lagrangian fibrations of toric hyperkähler manifolds

# Holomorphic Lagrangian fibrations of toric hyperkähler manifolds

Craig van Coevering and Wei Zhang
###### Abstract.

For the sake of hyperkähler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperkähler manifolds are real dimension non-compact hyperkähler manifolds which are quaternion analog of toric varieties. The dimensional residue circle action on it admitting a hyperkähler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to , and study the singular fibers.

Mathematics Classification Primary(2000): Primary 53C26, Secondary 53D20.
The second author is supported by Tian Yuan math Fund. and the Fundamental Research Funds for the Central Universities of China
Keywords: toric hyperkähler manifold, holomorphic Lagrangian fibration, moment map

## 1. Introduction

In [SYZ96], Strominger, Yau and Zaslow conjectured that Mirror Symmetry of Calabi-Yau manifolds comes from real Lagrangian fibrations. Let be a compact, Kähler, holomorphic symplectic manifold. By Calabi-Yau theorem, such a manifold admits a hyperkähler metric(see [Bes87], or [Huy99]). A complex Lagrangian subvariety of is special Lagrangian with respect to , which is clear from the linear algebra. The hyperkähler SYZ conjecture asserts the existence of holomorphic Lagrangian fibrations on the compact hyperkähler manifolds.

Although the original version is concerning the compact hyperkähler manifold, finding holomorphic Lagrangian fibration in non-compact hyperkähler manifold is also an interesting problem. For example, Hitchin had constructed holomorphic Lagrangian fibration in the moduli space of rank-2 stable Higgs bundles of odd degree with fixed determinant over a Riemann surface(cf. [Hit87]).

Another important fact must be mentioned is that, in the quest of examples of special Lagrangian fibration, the physicists set up Mirror Symmetry in toric Calabi-Yau manifold(cf. [AKMV05], [Mar10]), and similar topics were also studied by Batyrev(cf. [Bat94], [Bat98]). Thus it is also natural to consider the fibration in some hyperkähler manifold with large symmetry group.

Toric hyperkähler manifolds(some author call them hypertoric varieties ([Pro08])) are another important class of non-compact hyperkähler manifolds, which are quaternion analogue of toric varieties. They can be obtained as symplectic quotients of level sets of the holomorphic moment maps, and themselves admit residue hyperkähler moment maps. Using symplectic quotient technique, in [BD00], Bielawski and Dancer studied their moment maps, cores, cohomologies, etc. While Hausel and Sturmfels study the toric hyperkähler varieties from a more algebraic view point([HS02]). Then Konno study them as GIT quotients in [Kon03] and [Kon08].

We first use the complex residue moment map to find holomorphic Lagrangian fibration in toric hyperkähler manifold, then study the type of generic and singular fibers. Namely, let be a toric hyperkähler manifold, then

###### Theorem.

The map defines a holomorphic Lagrangian fibration, i.e. for any , is a complex Lagrangian subvariety.

To study the detail of the generic fiber and singular fiber, for simplicity, we let . Define a wall structure on the dual Lie algebra (see the precise definition in section 4), there follows

###### Theorem.

The generic fiber of over is diffeomorphic to complex torus .

In the case of , tt is easy to check that the most singular central fiber is the extended core of the toric hyperkähler manifold , constituted by toric varieties intersecting together(cf. [Pro08]). In general, the singular fiber on the discriminant locus is a little complicate, which can be described by the shrinking torus procedure.

###### Theorem.

Consider the singular fiber of . When lies in the generic position of , then diffeomorphic to shrinking the real torus generated by in the complex torus over the real hyperplane . When lies in the intersection of walls , then the singular fiber is given by shrinking due to over respectively, and at the intersection of , , shrinking a torus of real dimension generated by

The structure of the article is as follows. In section 2, we introduce some facts of Calabi-Yau and hyperkähler geometry, the special Lagrangian and holomorphic Lagrangian fibration, and background of Mirror Symmetry.

We present the basic properties of toric hyperkähler manifold in section 3. Mainly focus on the symplectic quotient and the GIT quotient construction. It has close relation with toric variety, namely the extended core of toric hyperkähler manifold are all constituted by toric varieties and compact toric varieties respectively, and the cotangent bundle of toric variety in the extended core is a dense open set of toric hyperkähler manifold.

The essential part of this paper is section 4, where we show that the complex moment map defines a holomorphic Lagrangian fibration. Then we study the type of generic fiber and singular fiber in great detail.

Acknowledgement: The authors want to thank Prof. Bin Xu, Prof. Bailin Song and Dr. Yalong Shi for valuable conversations. The second author is also grateful to prof. Sen Hu whose string theory lecture in the summer school several years ago inspired the author’s interest in Mirror Symmetry.

## 2. Calabi-Yau and hyperkähler geometry

A Calabi-Yau manifold is a Kähler manifold of complex dimension with a covariant constant holomorphic -form called the holomorphic volume form, which satisfies

 (2.1) ωnn!=(−1)n(n−1)/2(√−1/2)nΩ∧¯¯¯¯Ω,

where is the Kähler form. It is a Riemannian manifold with holonomy contained in (n). A (real) Lagrangian subvariety of an -dimensional Calabi-Yau manifold is called special Lagrangian if it is calibrated by , where is a constant. This condition is equivalent to and (cf. [Joy01]).

In 1996, Strominger, Yau and Zaslow [SYZ96] suggested a geometrical interpretation of Mirror Symmetry between Calabi-Yau 3-folds in terms of dual fibrations by special Lagrangian 3-tori, now known as the SYZ Conjecture.

A -dimensional manifold is hyperkähler if it possesses a Riemannian metric which is Kähler with respect to three complex structures ; ; satisfying the quaternionic relations etc, thus has three forms , , corresponding to the three complex structures. It has holonomy group contained in , a prior is Calabi-Yau. With respect to the complex structure the form is a holomorphic symplectic form. If is a complex Lagrangian submanifold ( is a complex submanifold and vanishes on ), then the real and imaginary parts of , namely and , vanish on . Thus vanishes and if is odd (resp. even), the real (resp. imaginary) part of vanishes. Using the complex structure instead of , we see that is special Lagrangian(see also [Hit97]). A fibration of hyperkähler manifold with complex lagrangian fibers is called holomorphic Lagrangian fibration, which is also very important in Mirror Symmetry. This is because of examples of special Lagrangian fibrations are very rare, all known examples are derived from holomorphic Lagrangian fibrations on , torus, or other hyperkähler manifolds. In the weakest form, the hyperkähler SYZ conjecture is stated as follows.

###### Conjecture 2.1.

Let be a hyperkähler manifold. Then can be deformed to a hyperkähler manifold admitting a holomorphic Lagrangian fibration.

For a more precise form of hyperkähler SYZ conjecture, see [Ver10].

## 3. Toric hyperkähler manifold

One of the most powerful technique for constructing hyperkähler manifolds is the hyperkähler quotient method of Hitchin, Karlhede, Lindström and Roček([HKLR87]). We specialized on the class of hyperkähler quotients of flat quaternionic space by subtori of . The geometry of these spaces turns out to be closely connected with the theory of toric varieties.

Since can be identified with , it has three complex structures . The real torus acts on induce a action on keeping the hyperkähler structure,

 (3.1) (z,w)ζ=(zζ,wζ−1).

Denote the -dimensional connected subtorus of whose Lie algebra is generated by integer vectors(which is always taken to be primitive), then we have the following exact sequences

 0→mι→tdπ→n→0,
 0←m∗ι∗←(td)∗π∗←−n∗←0,

where is the Lie algebra of the -dimensional quotient torus and .

Let be the standard basis of and some basis span . Denote and the dual basis. The subtorus action admits a hyperkähler moment map: , given by,

 (3.2) μR(z,w)=12d∑i=1(|zi|2−|wi|2)ι∗e∗i,
 (3.3) μC(z,w)=d∑i=1ziwiι∗e∗i.

The complex moment map is holomorphic with respect to . Bielawski and Dancer introduced the definition of toric hyperkähler varieties, and generally speaking, they are not toric varieties.

###### Definition 3.1 ([Bd00]).

A toric hyperkähler variety is a hyperkähler quotient for .

A smooth part of is a -dimensional hyperkähler manifold, whose hyperkähler structure is denoted by . The quotient torus acts on , preserving its hyperkähler structure. This residue circle action admits a hyperkähler moment map ,

 (3.4) ¯μR([z,w])=12d∑i=1(|zi|2−|wi|2)e∗i,
 (3.5) ¯μC([z,w])=d∑i=1ziwie∗i.

Differs from the toric case, the map to is surjective, never with a bounded image.

In this article, we always assume that is a smooth manifold(readers could consult the regularity argument for [BD00] and [Kon08]).

## 4. (C∗)n≅Tn×Rn fibration over Cn

We focus on the complex moment map of the residue circle action. The first task of this section is to prove the general theorem.

###### Theorem 4.1.

The map defines a holomorphic Lagrangian fibration, i.e. for any , is a complex Lagrangian subvariety.

###### Proof.

First of all, is obviously a holomorphic map from to . Let be the tangent vector space of , for , we have . Denote the dimensional space of tangent vectors to the orbit of residue circle action , this is equivalent saying , i.e. and . At a smooth point of , has real dimension , thus is the orthogonal component of , thus must be . By the quaternionic relation, we have and , hence , which means that is a complex Lagrangian subvarieties. ∎

For the study of the singularity of the fibers, we have to investigate the regularity of the complex moment map of toric hyperkähler manifold . We need to reinterpret the dual of Lie algebra and first. For lies in , there is some , such that

 (4.1) α=d∑i=1xiι∗e∗i.

Assume , and , above equation turns to a linear equation system

 (4.2) Ax=α,

where is matrix with entry , and represents the column vector . Its -dimensional solution space is denoted as , a -plane in . We can identify with . A hyperplanes arrangement is defined by , where are the coordinate hyperplanes in . Similarly, we could define the the complex solution space which identifies to , and a complex hyperplanes arrangement by , where is the coordinate hyperplanes in . We call the union a wall structure on .

###### Lemma 4.2.

The set of regular value of the moment map of toric hyperkähler manifold is

 n∗Creg=n∗C∖d⋃i=1Wi.
###### Proof.

Let be a map defined by . We can easily observe that is a regular point of if and only if . If one of equals to zero, constrained by equation , the image of at the point can not span the whole . Thus the point for some is a critical point of . ∎

Immediately, we have

###### Theorem 4.3.

The generic fiber of over is diffeomorphic to complex torus .

###### Proof.

By the regularity of the moment map , is a smooth manifold. We claim that the residue circle action acting on freely. To see this, lift the to . Then has nontrivial isotropy group in if and only if the orbit of a subgroup of through lies in the -orbit. For is the quotient group, the only possibility is that some equals to zero, which can not hold if . At another hand, the real moment map restricted to is still surjective on , moreover since , is also regular, thus must diffeomorphic to . ∎

It is natural to ask what the singular fiber looks like, the picture will be a little bit complicated. Suggested by the above proof, we need to investigate the isotropy group in detail.

We first check the simplest case, where lies in generic position of . Fixing , based on above discussion, the point where but , is with trivial isotropy, thus a smooth point on . And the real torus acts on the complex subvariety of defined by must has a 1 dimensional isotropy subgroup. For the real moment map restricted to is always surjective, this is equivalent to shrinking the torus whose Lie algebra is over the dimensional real subvariety , where is the hyperplane in the real arrangement.

Consider lies in the intersection of walls , the situation becomes more complicated. On the subvariety , we shrink the torus corresponding to over the hyperplane . Some of these subvarieties may intersect, or equivalent saying, the hyperplanes will intersect via the real residue moment map. For simplicity, let first subvariety intersects. Their normals may be linear dependent, thus we denote the dimension of the subspace they spanning. By the Equation (3.4), we know that the image of the real residue moment map is the intersection of , . Over this intersection, we shrink the real dimensional torus generated by .

Finally, we get the conclusion

###### Theorem 4.4.

Consider the singular fiber of . When lies in the generic position of , then diffeomorphic to shrinking the real torus generated by in the complex torus over the real hyperplane . When lies in the intersection of walls , then the singular fiber is given by shrinking due to over respectively, and at the intersection of , , shrinking a torus of real dimension generated by

Theoretically, using the data defining the , checking all the intersections of the singular subvarieties, we can identify the type of the singular fiber. As we know, when grows bigger, the computation becomes overwhelming.

###### Example 4.5.

Let the subgroup generated by and , then , , . If we take and , consider the toric hyperkähler manifold with generic fiber . The complex equation becomes

 (110101)⎛⎜⎝y1y2y3⎞⎟⎠=⎛⎜⎝000⎞⎟⎠,

has solution .

The solution space only intersects the coordinates hyperplanes in the origin, thus the central fiber is the only singular fiber. We investigate it closer. For the residue circle action is -dimensional, the point will be fixed point if it has nontrivial isotropy. By the defining Equation (3.2), (3.3) and (3.5), the central fiber satisfies

 (4.3) |z1|2−|w1|2+|z2|2−|w2|2=12|z1|2−|w1|2+|z3|3−|w3|2=1z1w1=z2w2=z3w3=0.

For 0 is the intersection of 3 walls, consider them respectively: if , then and must be nonzero, similarly, if , then and must be nonzero, if , then and must be nonzero. These are the only 3 fixed points of . Shrinking a real torus in these 3 points, we get , , and intersecting sequentially, which is nothing but the toric varieties in the extended core.

Recall that in the category of fibration of toric varieties, the fibers degenerate at the boundary of the Delzant polytopes of the toric varieties(cf. [Bou07]). Our theorem can be viewed as the hyperkähler analog. Using moment map to construct special Lagrangian variety had already been studied by Joyce intensively in [Joy02].

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Craig van Coevering: craigvan@ustc.edu.cn

Wei Zhang: zhangw81@ustc.edu.cn

School of Mathematics

University of Science and Technology of China

Hefei, 230026, P. R.China

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