As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch’s Landau-Ginzburg mirror

TORUS FIBRATIONS ON CERTAIN FLAG VARIETIES

KWOKWAI CHAN, NAICHUNG CONAN LEUNG, AND CHANGZHENG LI

Abstract. We construct pseudotoric structures (`a la Tyurin [26]) on the two- step flag varietyF `1,n−1;n, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry [25]. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch’s Landau-Ginzburg mirror [24, 17, 22, 23].

Contents

1. Introduction 1

Acknowledgment 3

2. Pseudotoric structures (after N. A. Tyurin) 3

2.1. The two-step flag varietyF `_1,n−1;n 6

2.2. The quadrics 7

3. Special Lagrangian torus fibrations 8

3.1. The two-step flag varietyF `_1,n−1;n 9

3.2. The quadrics 13

4. Towards SYZ mirror symmetry for flag varieties 17

4.1. The two-step flag varietyF `_1,n−1;n 18

4.2. The quadrics 19

References 20

1. Introduction

A pseudotoric structure on a K¨ahler manifold X of complex dimension n consists of a HamiltonianT^k-action on (X, ω_X), whereω_X is the K¨ahler structure onX and 1 ≤k ≤n, together with a meromorphic map F :X 99KY from X to a smooth projective toric variety Y of complex dimensionn−ksatisfying certain compatibility conditions. This notion, due to Tyurin [26], is a generalization of toric varieties and was motivated by Auroux’s pioneering work [2, 3] on SYZ mirror symmetry [25].

In his original paper [26], Tyurin explicitly constructed pseudotoric structures on a smooth quadric hypersurface Q in Pⁿ. In this note, we will do the same for the two-step flag variety

F `_1,n−1;n ={W₁6W_n−16Cⁿ|dimW₁= 1,dimW_n−1=n−1}, which is a degree (1,1) hypersurface in

P ∧¹Cⁿ

×P ∧ⁿ⁻¹Cⁿ∼=Pⁿ⁻¹×Pⁿ⁻¹;

1

see Theorem 2.9.

Pseudotoric structures are closely related to SYZ mirror symmetry because, as will be proved in this note, when there is a meromorphic top form ΩX onX with no zeros and only simple poles along an anticanonical divisorD⊂X satisfying the condition

(1.1) ιV₁· · ·ιV_kΩX=dlogF^∗f1∧ · · · ∧dlogF^∗f_n−k,

where{V1, . . . , Vk}is a basis of Hamiltonian vector fields generating theT^k-action andf1, . . . , f_n−k are torus invariant rational functions onY whose norms are Pois- son commuting with respect to the Poisson bracket induced by the toric symplectic sturcture onY, then

ρ:= (µ_Tk;F^∗|f1|, . . . , F^∗|fn−k|) :X\D→B⊂R^d

defines a special Lagrangian torus fibration onX \D, and the SYZ construction [25] may be applied to it to construct a mirror.

We show that meromorphic top forms satisfying (1.1) can be found on both

• smooth quadric hypersurfacesQin Pⁿ (in Section 3.2), and

• the two-step flag varietyF `_1,n−1;n (in Section 3.1),

with suitable choices of the anticanonical divisorD, giving rise to various special Lagrangian torus fibrations (see Theorems 3.5, 3.11 and 3.11). The simplest ex- amples in these two classes are the GrassmannianGr(2,4) and the full flag variety F `₃ respectively.

In [24], Rietsch proposed a Lie-theoretic construction of a Landau-Ginzburg (abbrev. as LG) mirror W^Rie for any flag variety X = G/P with the following properties:

• The number of terms inW^Rie is less than that of Givental’s mirrorW^Giv.

• The superpotentialW^Rie is defined on a space bigger than (C^∗)^dimX and has the correct number of critical points, whileW^Givis defined on (C^∗)^dimX and may not have enough critical points (e.g. in the case ofGr(2,4)).

A natural question is:

How to understand Rietsch’s mirror using SYZ?

We first notice that for both the quadricQand the two-step flag varietyF `_1,n−1;n, there are smooth families of integrable systems connecting the special Lagrangian torus fibration ρ : X\D → B constructed above with the Gelfand-Cetlin fibra- tion (restricted to the preimage of the interior of the Gelfand-Cetlin polytope). We expect that: W^Rieis obtained fromW^Givby “merging” some of the terms which ge- ometrically corresponds togluing of holomorphic disksand a coordinate change coming from the relevant wall-crossing formula.

Evidence for this claim will be provided for bothQandF `_1,n−1;n. In the discus- sion in Section 4, we speculate that, along the smooth family of integrable systems, facets of the Gelfand-Cetlin polytope are being “pushed into” the interior to be- come walls in the baseB. Geometrically, this corresponds to deforming (or partially smoothing of) the anticanonical divisor D⊂X; and accordingly, terms in the su- perpotentialW^Giv are being merged together (as, conjecturally, holomorphic disks are being glued together) to give terms in the new superpotential. For the eventu- al superpotential, the number of terms (which should be less than that ofW^Giv) should coincide with that ofW^Rie.

As an example, let us considerX =F `1,n−1;n. There are 2nterms in Givental’s superpotentialW^Givcorresponding to the 2nfacets in the Gelfand-Cetlin polytope (see e.g. [18]), 4 of which have preimages which arenotalgebraic subvarieties inX. When we move from the Gelfand-Cetlin system to our fibration along the smooth family of integrable systems, 2 of the facets are being “pushed in” to form a wall;

accordingly, 4 terms inW^Giv are merged to give 2 new terms. For the remaining 2n−4 facets, 2n−6 of them come in pairs and each pair is being “pushed in”

to form a wall; accordingly, 2(n−3) terms inW^Giv are being merged in pairs to producen−3 new terms. So the eventual LG superpotential should have

2 + (n−3) + 2 =n+ 1

terms, which is expected to be the number of terms in Rietsch’s superpotential W^Rie. As we will see in Remark 4.3, such expectation holds for n= 3,4.

Recently, the above speculations have been realized by the work of Hong, Kim and Lau [12], at least for the case of Gr(2, n) (and also for OG(1,5)). The idea is that, by deforming the Gelfand-Cetlin fibration to the special Lagrangian torus fibrationρ,¹the non-torus fibers of the Gelfand-Cetlin fibration, which are known to be non-trivial objects in the Fukaya category [20], are deformed to certainimmersed Lagrangians.

In the case ofGr(2, n), it has already been shown by Nohara and Ueda [21] that potential functions of the Lagrangian torus fibers of differentGelfand-Cetlin–type fibrations (which correspond to triangulations of the regular n-gon) can be glued (via cluster transformations) to give an open dense subset of Marsh-Rietsch’s mirror [17]. But this is still insufficient as the open dense subset misses some of the critical points of Marsh-Rietsch’s mirrors.

What Hong, Kim and Lau showed in [12] was that deformations of the immersed Lagrangian above (a local model of which was given in [6]) produces the final missing chart in Marsh-Rietsch’s mirror ofGr(2, n). This completes the SYZ construction, namely, by applying SYZ (or family Floer theory) to regular as well as singular fibers of the special Lagrangian torus fibrationρ, one can recover Marsh-Rietsch’s mirror. Similar constructions should be applicable to the two-step flag varieties F `_1,n−1;n [13].

Acknowledgment. We thank Yoosik Kim and Siu-Cheong Lau for detailed ex- planations of their work and very useful discussions. We are also indebted to the anonymous referee for insightful comments and suggestions which helped to improve the exposition of this article.

K. Chan was supported by Hong Kong RGC grant CUHK14300314 and direc- t grants from CUHK. N. C. Leung was supported by Hong Kong RGC grants CUHK14302215 & CUHK14303516 and direct grants from CUHK. C. Li was sup- ported by NSFC grants 11822113, 11831017 and 11521101.

2. Pseudotoric structures (after N. A. Tyurin)

LetX be a K¨ahler manifold of complex dimensionnwith K¨ahler structureω_X.

1In fact, Hong, Kim and Lau applied the technique in Abouzaid-Auroux-Katzarkov’s work [1], instead of Tyurin’s pseudotoric structures, to construct the Lagrangian torus fibrations, but their fibrations have essentially the same structures as the ones we constructed here.

Definition 2.1(cf. Definition 1 in [26]). Apseudotoric structureonX consists of the following data:

(1) a HamiltonianT^k-action on (X, ωX), where1≤k≤n, and

(2) a meromorphic map F :X 99KY fromX to a toric K¨ahler manifoldY of complex dimensionn−kequipped with a toric K¨ahler structure ωY, satisfying the following conditions:

(i) the base locus of F is a T^k-invariant subvariety B of complex dimension k−1 in X,

(ii) the holomorphic map F:X\B →Y isT^k-invariant, and (iii) for any smooth functionhonY, we have the relation

(2.1) VF^∗h=f· ∇FVh,

whereV_F^∗_h andV_h are the Hamiltonian vector fields of the functions F^∗h and h on X \B and Y with respect to ω_X and ω_Y respectively, f is a real function on X \B, ∇_F is the symplectic connection induced by the symplectic fibration F, and ∇FVh denotes the horizontal lift of the vector fieldXh toX\B.

The complex dimension of Y is called therankof the pseudotoric structure.

Remark 2.2. Our definition is slightly different from that of Tyurin[26, Definition 1], which only assumes thatX is a symplectic manifold.

By fixing a basis {v1, . . . , v_k} of the Lie algebra Lie(T^k) (which gives rise to a basis {V1, . . . , V_k} of Hamiltonian vector fields), we can write the moment map associated to the HamiltonianT^k-action on (X, ω_X) as

µ= (µ₁, . . . , µ_k) :X →R^k∼= Lie(T^k)^∗, i.e. ι_Viω_X=dµ_i fori= 1, . . . , k.

Lemma 2.3. For1≤i≤kand for any smooth functionhon Y, we have {µ_i, F^∗h}=ω_X(V_i, V_F^∗_h) = 0,

where{·,·}denotes the Poisson bracket onX induced by ω_X.

Proof. SinceF :X\B→Y is aT^k-invariant symplectic fibration,Vi is tangent to the fibers of F for eachi. On the other hand, for any vector field W lying in the vertical subbundle Ver := ker(dF :T(X\B)→T Y), we have

ωX(VF^∗h, W) =d(F^∗h)(W) = (F^∗(dh))(W) =dh(dF(W)) = 0.

The result follows.

Let ν₁, . . . , ν_n−k be functions on the image U ⊂ Y of the holomorphic map F :X\B→Y which arePoisson commutingwith respect to the Poisson bracket induced by the toric symplectic structure ω_Y on Y. For example, such functions are given by the components of the toric moment map

ν= (ν₁, . . . , ν_n−k) :Y →R^n−k∼= Lie(T^n−k)^∗ associated to the toric manifoldY.

Lemma 2.4. For1≤j, `≤n−k, we have

{F^∗ν_j, F^∗ν_{`</sub>}=ω<sub>X</sub>(V<sub>F</sub><sup>∗</sup><sub>νj</sub>, V<sub>F</sub><sup>∗</sup><sub>ν`}) = 0.

Proof. By the condition (2.1), we have

ω_X(V_F^∗_νj, V_F^∗_{ν`</sub>) =d(F<sup>∗</sup>ν<sub>j</sub>)(f∇FV<sub>ν`}) =f F^∗(dν_j(V_{ν`</sub>)) =f F<sup>∗</sup>{νj, ν<sub>`}}= 0.

These lemmas prove the following

Proposition 2.5. The functions

µ1, . . . , µk, F^∗ν1, . . . , F^∗ν_n−k

form a set of Poisson commuting algebraically independent functions over X\B.

Therefore the map

ρ:= (µ₁, . . . , µ_k, F^∗ν₁, . . . , F^∗ν_n−k) :X\B→Rⁿ defines a completely integrable system onX\B.

Pseudotoric structures are difficult to find in general. But in [26], Tyurin gave a (rather restrictive) sufficient condition, which we reformulate as follows.

For any smooth functionhonY, the Hamiltonian vector fieldsV_h andV_F^∗_hare respectively defined by

ωY(Vh,·) =dh(·) andωX(VF^∗h,·) = (d(F^∗h))(·).

Applying the pullbackF^∗ to the former equation gives ω_Y(V_h, dF(·)) = (d(F^∗h))(·)

as 1-forms on X. On the other hand, by the definition of the horizontal lift, we havedF(∇_FV_h) =V_h and hence

(F^∗ωY)(∇FVh,·) =ωY(Vh, dF(·)).

So altogether we have

ω_X(V_F∗h,·) = (F^∗ω_Y)(∇FV_h,·).

Hence, a sufficient condition to guarantee that condition (2.1) holds is the following:

The 1-forms ωX(∇FVh,·)and(F^∗ωY)(∇FVh,·)are parallel to each other overX\B.

Notice that bothωX(∇FVh,·) and (F^∗ωY)(∇FVh,·) annihilate the vertical subbun- dle Ver⊂T X, so they are determined by their values on the horizontal subbundle Hor.

Proposition 2.6. IfY is of complex dimension one, then condition (2.1)is auto- matically satisfied.

Proof. When dim_CY = 1, the horizontal subbundle Hor is of rank 2, so V2

Hor^∗ is of rank 1. Hence ωX(·,·) and (F^∗ωY)(·,·) only differ by a real function over

Hor.

This is why the condition (2.1) did not appear in the cohomogeneity one examples in [2, 3, 5].

Example 2.7. Suppose that bothX andY are submanifolds of a complex projective spaceP^N and their symplectic structuresω_X andω_Y are restrictions of the Fubini- Study formω_FSonP^N. Also suppose that the mapF :X→Y is the restriction of a projection map F˜ :P^N →P^N onto a linear subspace L⊂P^N. Then, by a linear change of coordinates if necessary, we may assume that Lis given by the common

zero set of a subset of the coordinates. In this case, a straightforward computation shows thatωX(∇FVh,·)is parallel to(F^∗ωY)(∇FVh,·)over the horizontal subbun- dle Hor (i.e., differ by a constant at every point x∈ X), and so condition (2.1) holds.

Lemma 2.8. Condition (2.1) remains valid after restriction to a sub-fibration.

More precisely, if F˜ :M → N is a symplectic fibration for which condition (2.1) holds, and ifF:X→Y is a symplectic fibration such thatX andY are symplectic submanifolds ofM andN respectively, andF is the restriction ofF˜ toX, i.e., the following diagram commutes

X

F

 ^ιX //M

F˜

Y  ^ιY //N

Then condition (2.1)also holds for F:X→Y for any function onY coming from the restriction of a function ofN.

Proof. The result follows from the following two observations:

(1) For a symplectic submanifold ι_X : X ,→ M and a smooth function f : M →R, the Hamiltonian vector field V_ι^∗

Xf of the restriction ι^∗_Xf is given by restricting the Hamiltonian vector field V_f to X and projecting to the subbundleT X ⊂T M.

(2) On the other hand, sinceιX(F⁻¹(y)) =ιX(X)∩( ˜F)⁻¹(ιY(y)), if the decom- position ofT Minto the direct sum of the vertical and horizontal subbundles with respect to the fibration ˜F :M →N is given by

T M = Ver⊕Hor,

then the corresponding decomposition ofT X with respect toF :X →Y is precisely

T X= (Ver∩T X)⊕(Hor∩T X).

Also, ˜F◦ιX=ι_Y◦F, so the horizontal lift∇FV_ι^∗

Yhis given by restricting the horizontal lift∇F˜V_htoXand projecting to the subbundle Hor∩T X ⊂Hor.

2.1. The two-step flag varietyF `_1,n−1;n. The two-step flag variety

F `1,n−1;n={W16Wn−16Cⁿ|dimW1= 1,dimWn−1=n−1}

is a homogeneous variety of the form SL(n,C)/P. Denote by [x₁ : · · · : x_n] the homogenous coordinates of P(V1

Cⁿ), and by [xˆ1 : · · · : xˆn] the homogeneous coordinates ofP(Vn−1

Cⁿ); here we are using the notational convention x_ˆj:=x12···(j−1)(j+1)···n.

Then the well-knownPl¨ucker embeddingofX =F `1,n−1;ninP(V1

Cⁿ)×P(Vn−1

Cⁿ)∼= Pⁿ⁻¹×Pⁿ⁻¹is given by

x₂xˆ2=x₁xˆ1+

n

X

j=3

(−1)^j−1x_jxˆj.

In particular, it is of complex dimension 2n−3. In terms of the inhomogeneous coordinates

z_j= x_j x1

, z_ˆj= xˆj

xnˆ

in the open subsetx₁6= 0, x_nˆ6= 0, it is given by z₂z_ˆ2=z_ˆ1+ (−1)ⁿ⁻¹z_n+

n−1

X

j=3

(−1)^j−1z_jz_ˆj. The maximal complex torus

T_C={t= (t1, . . . , tn)∈(C^∗)ⁿ|t1t2· · ·tn = 1},

as a subgroup ofSL(n,C) and of dimensionn−1, and hence induces an action on P(V1

Cⁿ)×P(Vn−1

Cⁿ) given by

(2.2) t·([x₁:· · ·:x_n],[x_ˆ1:· · ·:x_nˆ]) = ([t₁x₁:· · ·:t_nx_n],[t_ˆ1x_ˆ1:· · ·:t_nˆx_nˆ]) where we denote tˆj := t1t2· · ·t_j−1tj+1· · ·tn. Clearly, the T_C-action preserves X. Moreover, the induced action of the maximal compact torusT ⊂T_C onX, where t= (exp

√−1θ₁,· · · ,exp

√−1θ_n)∈T, is Hamiltonian with moment map µ= (µ₁, . . . , µ_n−1) :X −→Rⁿ⁻¹.

We define a rational map

F :F `1,n−1;n99KPⁿ⁻² by setting

y0=x1xˆ1, y1=x3xˆ3, y2=x4xˆ4, . . . , yn−2=xnxˆn.

The base locus ofFis given by the Schubert subvarietyBdefined byx1xˆ1=x3xˆ3=

· · ·=xnxˆn= 0, which is of complex dimensionn−2 inF `_1,n−1;n.

Theorem 2.9. This defines a pseudotoric structure on the two-step flag variety F `_1,n−1;n.

Proof. To see that this produces a pesudotoric structure, we embed Pⁿ⁻¹×Pⁿ⁻¹ and henceF `_1,n−1;n intoP^N, whereN =n²−1, by the Segre embedding. We also embed Pⁿ⁻² into the sameP^N in a natural way so that F is the restriction of a projection map ˜F : P^N → P^N onto a linear subspace (image of Pⁿ⁻²). Then the

result follows from Example 2.7 and Lemma 2.8.

2.2. The quadrics. This example was treated in Tyurin [26, Theorem 2], where he showed that an arbitrary smooth quadric hypersurface inPⁿ admits a pseudotoric structure.

Forn= 2m, consider the quadric

Q={x²₀+x₁x₂+· · ·+x_2m−1x_2m= 0}.

This quadric is a homogeneous variety G/P forG=SO(2m+ 1,C). A maximal torusT_C^mofSO(2m+ 1,C) acts on the quadric by

(2.3) (t1, . . . , tm)·[x0:x1:· · ·:x2m] = [x0, t1x1, t⁻¹₁ x2, . . . , tmx2m−1, t⁻¹_mx2m].

Define the rational map

F :P^2m99KP^m by setting

y₀=x²₀, y₁=x₁x₂, . . . , y_m=x_2m−1x_2m.

This restricts toQto give aT^m-invariant rational map F :Q99KP^m−1,

whereT^mdenotes the standard maximal compact subtorus ofT_C^m, andP^m−1⊂P^m is defined by

y0+y1+· · ·+ym= 0.

The base locus ofF is given by the union of a set of linear subspaces of dimension min P^2m, so the base locusB is of complex dimension m−1 inQ.

Similarly, whenn= 2m+ 1, consider the quadric

Q={x0x1+x2x3+· · ·+x2mx2m+1= 0}.

This quadric is a homogeneous variety G/P forG=SO(2m+ 2,C). A maximal complex torusT_C^m+1 ofSO(2m+ 2,C) acts on the quadric by

(2.4)

(t0, . . . , tm)·[x0:x1:· · ·:x2m+1] = [t0x0, t⁻¹₀ x1, t1x2, t⁻¹₁ x3, . . . , tmx2m, t⁻¹_mx2m+1].

Define the rational map

F:P^2m+199KP^m by setting

y0=x0x1, y1=x2x3, . . . , ym=x2mx2m+1. This restricts toQto give aT^m+1-invariant rational map

F :Q99KP^m−1,

whereT^m+1denotes the standard maximal compact subtorus ofT_C^m+1,P^m−1⊂P^m is defined by

y₀+y₁+· · ·+y_m= 0.

The base locus ofF is given by the union of a set of linear subspaces of dimension min P^2m+1, so the base locusB is of complex dimensionm−1 inQ.

Theorem 2.10 (Theorem 2 in [26]). This defines a pseudotoric structure on the smooth quadric.

Proof. The proof is very similar to the previous example, but this time we embed P^2minto P^N, whereN = ^2m+2₂

−1, by the Veronese embedding, and also embed P^m into the same P^N in a natural way. Then we see that F comes from the restriction of a projection map ˜F :P^N →P^N onto a linear subspace (image ofP^m).

So the result again follows from Example 2.7 and Lemma 2.8.

3. Special Lagrangian torus fibrations

In this section, we explain how pseudotoric structures are related to special La- grangian fibrations, and hence SYZ mirror symmetry [25]. Recall that a submani- fold L⊂X is calledspecial Lagrangianif it is Lagrangian and Ime

√−1θΩX|L= 0 for someθ∈R. We have the following proposition:

Proposition 3.1. Let (X, ωX)be a K¨ahler manifold equipped with a pseudotoric structure, i.e. with a Hamiltonian T^k-action on (X, ωX)for some 1≤k≤nand a meromorphic mapF :X 99KY from X to a toric K¨ahler manifoldY of complex dimensionn−kequipped with a toric K¨ahler structureωY satisfying the conditions in Definition 2.1. Suppose that there is an anticanonical divisor D∈ | −KX|and a meromorphic n-form Ω_X on X which has no zeros and only simple poles along the divisorD, satisfying the following conditions:

(i) D isT^k-invariant.

(ii) The holomorphic mapF :X\B→Y restricts to a mapF :X\D→Y\E, whereE is the toric anticanonical divisor in Y.

(iii) Given a basis {V1, . . . , V_k} of Hamiltonian vector fields generating theT^k- action on (X, ω_X), we have

(3.1) ι_V1· · ·ι_VkΩ_X=dlogF^∗f₁∧ · · · ∧dlogF^∗f_n−k

for someT^k-invariant meromorphic functions f₁, . . . , f_n−k onY.²

(iv) The functions ν_j := |f_j|, j = 1, . . . , n−k on the open dense orbit Y₀ :=

Y\E∼= (C^∗)^n−k are Poisson commuting with respect to the Poisson bracket induced by the toric symplectic structure ωY onY.

Then the map

ρ:= (µ1, . . . , µk, F^∗ν1, . . . , F^∗νn−k) :X\D→Rⁿ

defines a special Lagrangian fibration onX\D, where the special condition is with respect to the holomorphic volume formΩX onX\D.

Proof. Proposition 2.5 already shows thatρis a Lagrangian torus fibration. So we only need to prove that the fibers are special with respect to Ω_X.

The following argument is along the lines of [2, Proof of Proposition 5.2]. First note that the torus T^k acts on X by symplectomorphisms and the holomorphic map F : X \B → Y is T^k-invariant. Hence the parallel transport induced by the symplectic connection ∇F is T^k-equivariant. In particular, any Lagrangian fiberLofρis invariant under the parallel transport along the corresponding fiber of the map ν = (ν1, . . . , ν_n−k) : Y0 → R^n−k. Thus the horizontal lifts ∇FV_|fj| (j= 1, . . . , n−k) are tangent to L. But condition (2.1) says that each∇_FV_|fj| is parallel toV_F∗|f_j|. Therefore, the tangent space to any point onLis spanned by the vector fieldsV1, . . . , Vk (which generate theT^k-action) andV_F∗|f1|, . . . , V_F∗|fn−k|.

Now at every point on the Lagrangian fiber L, we have, by condition (iii) or more precisely equation (3.1), that

Ω_X|_L(V₁, . . . , V_k, V_F∗|f₁|, . . . , V_F∗|fn−k|)

=(ι_V1· · ·ι_VkΩ_X)|L(V_F∗|f1|, . . . , V_F∗|fn−k|)

=(dlogF^∗f1∧ · · · ∧dlogF^∗f_n−k)|L(V_F^∗_|f1|, . . . , V_F^∗_|fn−k|).

Since VF^∗|fj| is tangent to the level sets of F^∗|fj|, we have either Im ΩX|L = 0 (whenn−kis even) or Re ΩX|L = 0 (whenn−kis odd).

3.1. The two-step flag variety F `<sub>1,n−1;n</sub>. In this subsection, we show that the condition (3.1) is satisfied for the pseudotoric structure on the two-step flag variety F `_1,n−1;n we constructed in Theorem 2.9.

Take any anti-canonical divisor D ∈ | −K_X| of X, we obtain an open Calabi- Yau manifoldX\D. In this section, we will specify several choices ofD, and then construct special Lagrangian fibrations on X\D with respect to a meromorphic volume form ΩX on X that has no zeros and only simple poles along D. One of such choices will be given by Schubert divisors; we refer to [8, Lemma 3.5] for the description of the anti-canonical divisor class [−KX] by Schubert divisor classes

2Note that the LHS of (3.1) differs only by a scalar multiple for different choices of the basis {V1, . . . , Vk}, so this condition is independent of such a choice.

for a general homogeneous variety G/P. We notice that the two-step flag variety F `1,n−1;n and the quadrics are all special cases of such homogeneous varieties.

Recall that the natural action ofT ={diag(t1, . . . , tn)∈(S¹)ⁿ |t1t2· · ·tn = 1}

(as a maximal compact torus of the maximal complex torus T_C of SL(n,C)) on X was given in (2.2). We shall specify a basisV₁, . . . , V_n−1 of Hamiltonian vector fields generated by the torus action of T. As we will see later, our choices of Ω_X are very nice in the sense that the condition

ιV₁· · ·ιVn−1ΩX=±dlogf1∧ · · ·dlogfn−2

is satisfied, and from this we can construct a special Lagrangian torus fibration (with singular fibers)

Φ := (µ1, . . . , µn−1,|f1|, . . . ,|fn−2|) :X\D−→R²ⁿ⁻³,

where (µ₁, . . . , µ_n−1) :X−→Rⁿ⁻¹ is the moment map of the torus action ofT. 3.1.1. Contraction by torus-invariant holomorphic vector fields. On the open subset x₁6= 0, xnˆ6= 0, we have local coordinates (z2, z₃, . . . , z_n, z_ˆ2, . . . , z_[n−1) ofX.

Letθk denote the weight oftk. Then for 1≤j ≤n−1, the weight of the action ofT on the coordinatezj+1 is given by

ψ_j :=θ_j+1−θ₁.

For 2≤k≤n−1, the weight of the action ofT on the coordinatezˆk is given by θ_n−θ_k=ψ_n−1−ψ_k−1.

LetVi be the Hamiltonian vector field corresponding to the weight ψi. Denote bycy the coefficient ofψi in the weight for a coordinate. Then we have

ιVi =X

y

cy·y·ι∂y,

where ι_∂y denotes the contraction with the holomorphic vector field _∂y^∂ (on the left). We simply denote ι_∂y as ι_y. We also adapt the notation convention that z₁= 1, z_ˆn= 1, and denote

Ω :=dz2dzˆ2· · ·dzn−1dz_n−1[dzn, Aj :=zjzˆj, j= 1, . . . , n.

Lemma 3.2.

ιV₁· · ·ιVn−1Ω = (−1)ⁿA1dA3· · ·dAn+

n

X

j=3

(−1)^n+jAjdA1dA3· · ·dAdj· · ·dAn. Proof. We notice thatιVi−1 =ziιz_i−zˆiιz_ˆi fori = 2, . . . , n−1, and thatιVn−1 = z_nι_z5+Pn−1

j=2 z_ˆjι_zˆ

j. Thus for 2≤i≤n−1, we have ι_Vi−1dz_idz_ˆi=z_idz_i+z_ˆidz_ˆi= d(z_izˆi) =dA_i. SinceA₂=A₁+Pn

j=1(−1)^j−1A_j, we have ιV₁· · ·ιVn−1Ω =AndA2dA3· · ·dAn−1+

n−1

X

j=2

ιV₁· · ·ιVn−2zˆjιz_ˆjdz1dzˆ1· · ·dzn−1dz_n−1[dAn

=

n

X

j=2

(−1)^n−jA_jdA₂· · ·dAd_j· · ·dA_n

= (−1)ⁿ⁻²A₂dA₃· · ·dA_n+

n

X

j=3

(−1)^n−jA_j dA₁+ (−1)^j−1dA_j)dA₃· · ·dAd_j· · ·dA_n

=

n

X

k=2

(−1)^n−kA_kdA₃· · ·dA_n+

n

X

j=3

(−1)^n−jA_jdA₁dA₃· · ·dAd_j· · ·dA_n

= (−1)ⁿA₁dA₃· · ·dA_n+

n

X

j=3

(−1)^n+jA_jdA₁dA₃· · ·dAd_j· · ·dA_n. 3.1.2. Anti-canonical divisor by Schubert divisors. There is an anti-canonical divi- sor ofX defined as follows, which consists of Schubert divisors ofX up to transla- tions by the Weyl group ofSL(n,C).

D^Sch:={x1xˆ1x3xˆ3· · ·xnxˆn= 0} ∩X,

where we treat the intersection as that for subvarieties inPⁿ⁻¹×Pⁿ⁻¹. Take the nowhere vanishing holomorphic volume form Ω^Sch_X onX\D^Sch given by

Ω^Sch_X := Ω A1A3A4· · ·An

on the open subsetx16= 0, xnˆ6= 0. It is meromorhphic onX, and has simple poles along the divisorD^Sch.

Proposition 3.3. We have

ιV₁· · ·ιV_n−1Ω^Sch_X = (−1)ⁿ⁻¹dlogf1∧dlogf2· · · ∧dlogf_n−2, wheref1= ^A_A¹

3 =^x_x^1xˆ1

3x_ˆ3, andfj= ^A_A^j+2

3 =^xj+2_x ^x[j+2

3x_ˆ3 forj= 2, . . . , n−2.

Proof. Denote by RHS by the expression on the right hand side of the expected equality. By direct calculations, we have

RHS = (−1)ⁿ⁻¹(dA1

A₁ −dA3

A₃ )(dA4

A₄ −dA3

A₃ )· · ·(dAn

A_n −dA3

A₃ )

= (−1)ⁿ⁻¹A3dA1−A1dA3

A1A3

·A3dA4−A4dA3

A3A4

· · ·A3dAn−AndA3

A3An

= (−1)ⁿ⁻¹−A1dA3dA4· · ·dAn+Pn

j=3(−1)^j+1AjdA1dA3· · ·dAdj· · ·dAn

A1A3A4· · ·An

= ιV1· · ·ιVn−1Ω A1A3A4· · ·An

.

The last equality follows from Lemma 3.2.

Let us consider some other choices of meromorphic volume forms onX. Denote B_j :=

n

X

k=j

(−1)^k−jx_kxˆk, j = 3, . . . , n.

By abuse of notations, we also denote by Bj the function on the open subset x16= 0, xnˆ6= 0. Notice that

An=Bn, Aj =Bj+Bj+1, j= 3, . . . , n−1.

For any 3≤j ≤n, we consider a divisor ofX defined by D^(j):={x1x_ˆ1x₃x_ˆ3· · ·x_j−1x_[

j−1B_j· · ·B_n= 0}.

Notice that whenj=n, we haveD⁽ⁿ⁾=D^Sch. We also remark thatD^Rie:=D⁽³⁾ is the anticanonical divisor which corresponds to Rietsch’s mirror [24].

Proposition 3.4. For any 3≤j≤n, we have ιV₁· · ·ιVn−1Ω^(j)_X = (−1)^n−jdlogx1xˆ1

B_j ∧dlogx3xˆ3

B_j ∧· · ·∧dlogxj−1x_[j−1

B_j ∧dlogBj+1

B_j ∧· · ·∧dlogBn

B_j, where

Ω^(j)_X := Ω

A₁A₃· · ·A_j−1B_j· · ·B_n.

Proof. Denote by RHS^(j)the right hand side of the expected equality. SinceD⁽ⁿ⁾= D^Sch, by Proposition 3.3 we have

ιV1· · ·ιVn−1Ω^Sch_X = (−1)ⁿ⁻¹dlogf1∧dlogf2· · · ∧dlogfn−2

= (−1)ⁿ⁻¹(dlogf1−dlogfn−2)∧ · · · ∧(dlogfn−3−dlogfn−2)∧dlogfn−2

= (−1)ⁿ⁻¹dlog x₁x_ˆ1 xnxnˆ

∧dlog x₄x_ˆ4 xnxnˆ

∧ · · · ∧dlogx_n−1x_n−1[ xnxnˆ

∧dlogx_nx_nˆ x3xˆ3

=dlogx1xˆ1

B_n ∧dlogx3xˆ3

B_n ∧ · · · ∧dlogxn−1x_[n−1 B_n . Namely ifj=n, then the statement holds.

Now we consider 3≤j < nand recallAj=Bj+Bj+1. Therefore

j+1

X

k=j

(−1)^k+nBkdA1dA3· · ·dA_j−1dAjdBj+1· · ·dBdk· · ·dBn

=(−1)^j+n(Aj−Bj+1)dA1dA3· · ·dAj−1dBj+1dBj+2· · ·dBn

+ (−1)^j+1+nBj+1dA1dA3· · ·dA_j−1dBjdBj+2· · ·dBn

=(−1)^j+nAjdA1dA3· · ·dA_j−1dBj+1dBj+2· · ·dBn

+ (−1)^j+1+nB_j+1dA₁dA₃· · ·dA_j−1dA_jdB_j+2· · ·dB_n

Thus by expanding RHS^(j), we have

D^(j)·RHS^(j)= (−1)ⁿA1dA3· · ·dAj−1dBj· · ·dBn

+

j−1

X

i=3

(−1)ⁱ⁺ⁿAidA1dA3· · ·dAdi· · ·dAj−1dBj· · ·dBn

+

n

X

k=j

(−1)^k+nB_kdA₁dA₃· · ·dA_j−1dB_j· · ·dBd_k· · ·dB_n

= (−1)ⁿA1dA3· · ·dA_j−1dAjdBj+1· · ·dBn

+

j

X

i=3

(−1)ⁱ⁺ⁿAidA1dA3· · ·dAdi· · ·dA_j−1dAjdBj+1· · ·dBn

+

n

X

k=j+1

(−1)^k+nBkdA1dA3· · ·dAj−1dAjdBj+1· · ·dBdk· · ·dBn

=D^(j+1)·RHS^(j+1).

By induction onj, we conclude that, for 3≤j≤n,

D^(j)·RHS^(j)=D⁽ⁿ⁾·RHS⁽ⁿ⁾=ι_V1· · ·ι_Vn−1Ω.

To apply Proposition 3.1, we need to check that the functions |f1|, . . . ,|f_n−2| are Poisson commuting with respect to the Poisson bracket induced by the Fubini- Study K¨ahler form ωF S on Pⁿ⁻². To see this, first note that the norms of the inhomogeneous coordinates w1 := ^y_y⁰

1, w2 := ^y_y²

1, . . . , w_n−2 := ^yn−2_y

1 are Poisson commuting since they define the log map

Log:Pⁿ⁻²\E∼= (C^∗)ⁿ⁻²→Rⁿ⁻²,(w1, . . . , wn−2)7→(log|w1|, . . . ,log|wn−2|), which is a Lagrangian torus fibration; hereEis the union of coordinate hyperplanes defined by y₀y₁· · ·y_n−2 = 0. But the functions f₁, . . . , f_n−2 are pullbacks of the inhomogeneous coordinatesw1, . . . , w_n−2 by linear isomorphismsσ:Pⁿ⁻²→Pⁿ⁻² (given by lower triangular matrices) which obviously preserve ωF S and hence the Poisson structure. So|f1|, . . . ,|f_n−2| are Poisson commuting as well. Proposition 3.4 together with Proposition 3.1 then give the following theorem.

Theorem 3.5. For each D=D^(j), the map

ρ:= (µ1, . . . , µk, F^∗ν1, . . . , F^∗νn−k) :X\D→Rⁿ

defines a special Lagrangian fibration onX\D, where the special condition is with respect to the holomorphic volume formΩ^(j)_X on X\D.

Observe that the functions µ₁, . . . , µ_k, F^∗ν₁, . . . , F^∗ν_n−k are defined onX \B, so we obtain completely integrable systems on these varieties in the sense of [11, Definition 2.1]. But the fibers over the image ofD\B are collapsed tori.

3.2. The quadrics. In this subsection, we show that the condition (3.1) is satisfied for the pseudotoric structure on the quadric hypersurfaces in Pⁿ constructed by Tyurin (as in Theorem 2.10).

3.2.1. Even-dimensional quadric. Here we consider the even-dimensional quadric Q2m={x0x1+x2x3+· · ·+x2mx2m+1= 0} ⊂P^2m+1.

Recall that the natural action of a maximal compact torus T^m+1 ⊂ T^m+1

C of SO(2m+ 2,C) on the quadric was given in (2.4). On the affine plane x0 6= 0, the quadric is defined by z1+z2z3+· · ·+z2mz2m+1 = 0, where zi = _x^xi

0 are the inhomogeneous coordinates.

Letθk denote the weight of tk. Then for 1≤j≤m, the weight of the action of T^m+1 on the inhomogeneous coordinatez_2j (resp. z_2j+1) is given byθ_j−θ₀(resp.

−θj−θ₀). Denote

ψm+1=−θm−θ0; ψj=θj−θ0, j= 1, . . . , m.

Then the weight onz2j+1 is given byψm+ψm+1−ψj for 1≤j≤m.

Let {V1, . . . , Vm+1} denote the basis of Hamiltonian vector fields generated by the torus action ofT^m+1 and corresponding to the weights ψ_j. Then we have



















ι_Vj =z_2jι_z2j −z_2j+1ι_z2j+1, j= 1, . . . , m−1;

ι_Vm =z_2mι_z2m+

m−1

X

i=1

z_2i+1ι_z2i+1;

ι_Vm+1=z_2m+1ι_z2m+1+

m−1

X

i=1

z_2i+1ι_z2i+1.

Similar to the case of two-step flag varieties, we take an anti-canonical divisor D^Sch ofX=Q2mdefined by Schubert divisors, namely, given by

D^Sch:={x0x1· · ·x_2m−3x2mx2m+1= 0}, and consider the meromorphic volume form

Ω^Sch_X := dz2dz3· · ·dz2mdz2m+1

z1· · ·z_2m−3z2mz2m+1

. We denote

Ω :=dz₂dz₃· · ·dz_2mdz_2m+1 andA_j:=z_2jz_2j+1 forj= 1, . . . , m, so that we can also write

Ω^Sch_X = Ω

z1A1A2· · ·A_m−2z2mz2m+1

. Lemma 3.6. ForΩ =dz2dz3· · ·dz2mdz2m+1, we have

ι_Vm+1ι_Vmι_V1· · ·ι_Vm−1Ω =

m

X

j=1

(−1)^m+jA_jdA₁· · ·dAd_j· · ·dA_m. Proof. For 1≤j≤m−1, we have

z_2j+1ι_z2j+1dA_j=A_j, ι_Vjdz_2jd_z2j+1 =d(z_2jz_2j+1) =dA_j. Moreover, ιV_m+1ιV_m =Amιz_2m+1ιz_2m +Pm−1

j=1 z2j+1ιz_2j+1(z2mιz_2m −z2m+1ιz_2m+1).

Therefore

ι_Vm+1ι_Vmι_V1· · ·ι_Vm−1Ω =ι_Vm+1ι_VmdA₁· · ·dA_m−1dz_2mdz_2m+1

=A_mdA₁· · ·dA_m−1+

m−1

X

j=1

(−1)^m−1z_jι_z2j+1dA₁· · ·dA_m−1dA_m

=

m

X

j=1

(−1)^m+jA_jdA₁· · ·dAd_j· · ·dA_m.

Proposition 3.7. ForΩ^Sch_X =_z^{dz2dz3···dz2mdz2m+1}

1z₂···z2m−3z_2mz_2m+1, we have

ι_Vm+1ι_Vmι_V1· · ·ι_Vm−1Ω^Sch_X =dlogf₁∧dlogf₂· · · ∧dlogf_m−2∧dlogf_m, wherefj= _x^Aj

0x₁ =^x2j_x^x2j+1

0x₁ forj∈ {1, . . . , m} \ {m−1}.

Proof. We notice thatz1+Pm

j=1Aj = 0. Similar to the proof of Proposition 3.3, we calculate the right hand side RHS of the above expression as follows.

z1A1· · ·Am−2Am·RHS

=(z₁A₁· · ·A_m−2A_m)(dA1

A₁ −dz1

z₁ )· · ·(dA_m−2 A_m−2 −dz1

z₁ )(dAm

A_m −dz1

z₁ )

=z1dA1· · ·dAm−2dAm+ X

1≤j≤m j6=m−1

AjdA1d· · ·dAj−1(−dz1)dAj+1· · ·dAm−2dAm

=z1dA1· · ·dA_m−2dAm+ X

1≤j≤m j6=m−1

AjdA1d· · ·dA_j−1(dAj+dA_m−1)dAj+1· · ·dA_m−2dAm

=(z₁+ X

1≤j≤m j6=m−1

A_j)dA₁· · ·dA_m−2dA_m+ X

1≤j≤m j6=m−1

(−1)^m−jA_jdA₁· · ·dAd_j· · ·dA_m

=z₁A₁· · ·A_m−2A_m·LHS.

The last equality follows from Lemma 3.6.

Following [23], we consider another anti-canonical divisor:

D^Rie:={x0x1B1· · ·B_m−2x2mx2m+1= 0}, where Bj = Pj

i=0x2ix2i+1 for 1 ≤ j ≤m−2; more generally, as in the case of the two step-flag variety F `_1,n−1;n, we may consider a sequence of anticanonical divisors of the form

D^(j):={x0x1A1· · ·Aj−1Bj· · ·Bm−2x2mx2m+1= 0}, 1≤j≤m−1 so thatD^(m−1)=D^Sch andD⁽¹⁾=D^Rie.

Then by similar calculations in the proof of Proposition 3.4, we obtain the fol- lowing.

Proposition 3.8. We have

ιVm+1ιVmιV1· · ·ιVm−1Ω^(j)_X =dlogg1· · · ∧dloggm−2∧dloggm, where

Ω^(j)_X := Ω

x0x1x2x3· · ·x2j−2x2j−1Bj· · ·Bm−2x2mx2m+1

andg_k= _x^Ak

0x1 for1≤k≤j−1,g_k= _x^Bk

0x1 forj≤k≤m−2, andg_m=^x2m_x^x2m+1

0x1 . By similar reasoning as in the paragraph right before Theorem 3.5, we can check that the functions|f1|, . . . ,|fm|as well as|g1|, . . . ,|gm|are Poisson commuting. So Propositions 3.7 and 3.8 together with Proposition 3.1 give the following theorem.

Theorem 3.9. For each D=D^(j), the map

ρ:= (µ1, . . . , µk, F^∗ν1, . . . , F^∗ν_n−k) :X\D→Rⁿ

defines a special Lagrangian fibration onX\D, where the special condition is with respect to the holomorphic volume formΩ^(j)_X on X\D.

3.2.2. Odd-dimensional quadric. Here we consider the odd-dimensional quadric Q_2m−1={x²₀+x₁x₂+· · ·+x_2m−1x_2m= 0} ⊂P^2m.

Recall that the action of a maximal compact torusT^m⊂T_C^mofSO(2m+ 1,C) on the quadric was given in (2.3). On the affine planex2m6= 0, the quadric is defined byz²₀+z1z2+· · ·+z2m−3z2m−2+z2m−1= 0, wherezi=_x^xi

2m are the inhomogeneous coordinates.

Letθ_k denote the weight of t_k. Then for 1≤j≤m, the weight of the action of T^m+1 on the inhomogeneous coordinatez_2j−1(resp. z2j) is given byθj+θm(resp.

−θj+θm) forj= 1, . . . , m−1, and the weight onz0is given byθm.

Let {V1, . . . , Vm} denote a basis of Hamiltonian vector fields generated by the torus action ofT^mand corresponding to the weights θj. Then we have

ιV_m =X^m−1

i=0 ziιz_2i and ιV_j =z_2j−1ιz_2j−1−z2jιz_2j, j= 1, . . . , m−1.

Similar to the case of even-dimensional quadric, we denote A_j = z_2j−1z_2j (or A_j = x_2j−1x_2j when we refer to homogeneous coordinates) for j = 1, . . . , m by