NON-K ¨AHLER SYZ MIRROR SYMMETRY SIU-CHEONG LAU, LI-SHENG TSENG, AND SHING-TUNG YAU Abstract.

SIU-CHEONG LAU, LI-SHENG TSENG, AND SHING-TUNG YAU

Abstract. We study SYZ mirror symmetry in the context of non-K¨ahler Calabi-Yau manifolds. In particular, we study the six-dimensional Type II supersymmetricSU(3) systems with Ramond-Ramond fluxes, and generalize them to higher dimensions. We show that Fourier-Mukai transform provides the mirror map between these Type IIA and Type IIB supersymmetric systems in the semi-flat setting. This is concretely exhibited by nilmanifolds.

1. Introduction

The Strominger-Yau-Zaslow (SYZ) approach [SYZ96] to mirror symmetry has brought many fruitful results in the past two decades. For a mirror pair of Calabi- Yau manifolds (X,X), it asserts that there exist special Lagrangian fibrationsˇ µ: X → B and ˇµ : ˇX → B such that µ⁻¹({b}) and ˇµ⁻¹({b}) are dual tori. As a consequence, Lagrangian sections of µ should correspond to holomorphic line bundles on ˇX, which explains the interchange between symplectic geometry ofX and complex geometry of ˇX. Gross [Gro01] gave a verification of the above picture in the topological level. The Gross-Siebert program [GS11] gave a sophisticated algebraic formulation of the SYZ construction. Moreover, the SYZ program was successfully carried out in various situations for K¨ahler manifolds [LYZ00, Leu05, Aur07, CO06, CL10, FOOO10, CLL12, AAK].

This paper explores the SYZ approach for non-K¨ahler Calabi-Yau manifolds, i.e.

non-K¨ahler almost Hermitian manifolds withc1= 0. We shall consider two classes of non-K¨ahler Calabi-Yau geometries which we will call Type IIA and Type IIB supersymmetric systems. In six dimensions, the systems arise directly from the imposition of supersymmetry in Type IIA/IIB string theory [GMPT05, Tom08, TY]. The six-dimensional Type IIA system is a symplectic manifold (X, ω) with an almost complex structure induced from a complex decomposible three-form Ω with d Re Ω = 0 (instead of d Ω = 0). Type IIB system is a complex threefold ˇX with a holomorphic volume form ˇΩ and a balanced Hermitian metric, whose associated (1,1) form ˇωsatisfies d (ˇω²) = 0 (instead of d ˇω= 0). We also define and study their higher dimensional analogs. Here, we preserve on the IIB side the balanced metric condition which has a number of desirable properties [Mic82, AB93, AB95, AB04].

The mirror dual condition on the symplectic IIA side will be shown to require the introduction of a special real polarization.

Though there is now a fairly decent understanding of mirror symmetry in the K¨ahler situation, the mirror phenomenon for non-K¨ahler Calabi-Yau geometries has not been studied as much and is not well-understood. This motivates us to study mirror symmetry in the presence of the so-called Ramond-Ramond (RR) flux using SYZ transformation. The aim is to construct a duality between Type IIA and Type

1

IIB supersymmetric systems (and their higher dimensional analogs) by using SYZ and Fourier-Mukai transform.

While SYZ fibrations in general may not exist in the presence of flux, the SYZ transformation (when fibations exist) does provide important clues on how mirror symmetry should behave in the non-K¨ahler setting. We expect mirror symmetry between type IIA and IIB systems still to occur in a similar way even in cases when SYZ fibrations may not exist. Namely, the flux source terms (denoted asρ_Aandρ_B later on in this paper) are mirror to each other in the sense of homological mirror symmetry.

Construction of non-K¨ahler Calabi-Yau manifolds has been studied in several recent works [Cal58, STY02, GP04, GGP08, Wu06, FP10, FLY12, FP13]. The survey by Fu [Fu10] gave an excellent introduction to this topic. In this paper we focus on the semi-flat setting [LYZ00, Leu05], namely we consider Lagrangian fibrations away from singular fibers. We will see that interesting correspondences between the Type IIA and IIB systems and their RR fluxes already appear in such a simple setting. Iwasawa manifold and its higher-dimensional analogs serves as important examples. The main theorem is:

Theorem 1.1 (See Theorem 5.1). Under SYZ and Fourier-Mukai transform, a semi-flat supersymmetric Type IIA SU(n) structure is transformed to a semi-flat supersymmetric Type IIBSU(n)structure, and their fluxes also correspond to each other by Fourier-Mukai transform.

The study of non-K¨ahler Calabi-Yau geometries was largely motivated by string theory (see, for example [Str86, BBDG03, BBD⁺04]). In physical terms, the pres- ence of fluxes in supersymmetric systems results in geometries that are generically non-K¨ahler. In Type IIA and IIB string theory, there are mainly two types of fluxes, namely Ramond-Ramond (RR) flux and H-flux (which is also called NS-NS flux).

The settings of Type IIA and IIB supersymmetric systems in this paper follows the form expressed in Tseng-Yau [TY], in which an RR flux is present. These are special cases of the more general systems as given in Grana-Minasian-Petrini-Tomasiello [GMPT05] and Tomasiello [Tom08] which include the H-flux and are expressed in terms of generalized complex geometry [Hit03, Gua11] In this paper, we place our focus on mirrors of balanced metrics and generalize the definition of Type IIA sys- tems to higher dimensions, which does not involve the use of generalized geometry.

T-duality and SYZ in the setting of generalized Calabi-Yau manifolds were stud- ied in [BEM04, BHM04, FMT05, GSN07, CG10] where H-flux played the key role, and also from the bispinor perspective which includes RR fluxes in [GMPT07, GMPW09]. The emphasis of this paper is different: we study SYZ in the presence of RR-flux and in particular focusing on the balanced Hermitian metric condition on non-K¨ahler manifolds, which also appears as part of the Strominger system [Str86].

Acknowledgement

We thank A. Tomasiello and C.-J. Tsai for discussions. We are also grateful to referees for useful comments. This work is supported in part by NSF grants DMS-1159412, 1308244 and PHY-1306313.

2. The supersymmetric systems and their higher-dimensional analogs In this section we introduce Type IIA and IIB supersymmetricSU(n) systems, which are the main objects of study in this paper.

Definition 2.1. Let X be a real manifold of dimension 2n. An SU(n) structure onX is a pair (ω,Ω)of differential forms satisfying the following conditions:

(1) Ω is a nowhere-vanishing decomposible complex-valued n-form on X such that by defining

T^(0,1)X:={v∈T X⊗C:ιvΩ = 0}

andT^(1,0)X to be the complex conjugate of T^(0,1)X, one has a splitting T X⊗C=T^(1,0)X⊕T^(0,1)X

which induces an almost complex structure J on X. Then Ω is an (n,0) form with respect to this almost complex structureJ. (Note that we do not required Ω = 0here.)

(2) ωis a non-degenerate real¹(1,1)-form with respect to this complex structure J such thatω(·, J·)defines a Hermitian metric onX. (Note that we do not requiredω= 0here.) This in particular implies that Ω∧ω= 0.

Since both(Ω∧Ω)/i¯ ⁿ andωⁿ are real nowhere-vanishing top-forms, we have Ω∧Ω =¯ iⁿF·ωⁿ

n!

for some nowhere-vanishing functionF onX. F is called to be the conformal factor of theSU(n)structure.

In other words anSU(n) structure is a pointwise Calabi-Yau structure without imposing any integrability condition. Supersymmetry imposes integrability condi- tions onSU(n) structures. In string theory one mostly focuses onn= 3.

Definition 2.2. AnSU(3)structure(X, ω,Ω)is said to be supersymmetric of type IIB ifΩdefines an honest complex structure and ω defines a balanced metric, that is, d Ω = 0andd (ω²) = 0.

Let F be the conformal factor of theSU(3) structure. Define ρB = 2i∂∂ F¯ ⁻¹· ω

which is an exact(2,2) form.

In conclusion, we have the following system of equations for a supersymmetric SU(3) structure of type IIB:

d Ω = 0, (2.1)

d (ω²) = 0, (2.2)

Ω∧Ω =¯ −iF·ω³ 6 , (2.3)

2i∂∂ F¯ ⁻¹·ω

=ρ_B. (2.4)

1More generally we should includeB-field here, andωcan be complex-valued. Such a com- plexification is necessary for making the mirror map to be an isomorphism. This paper focuses on RR flux and we omit it for simplicity.

Whenω is symplectic (that is dω= 0) andF is constant, (X, ω,Ω) is Calabi-Yau, and the RR flux source currentρB is zero. In such case the corresponding metric hasSU(3) holonomy.

Definition 2.3. AnSU(3)structure(X, ω,Ω)is said to be supersymmetric of Type IIA ifdω= 0 andd Re Ω = 0.

Let F be the conformal factor of theSU(3) structure. We define ρA= d d^Λ(F·Im Ω)

which is an exact three-form. Here

d^Λ= d Λ−Λd

is the symplectic adjoint operator, where Λ is the adjoint of the Lefschetz operator L=ω∧(·).

Thus the Type IIA supersymmetricSU(3) structure satisfies the following system of equations:

dω= 0, (2.5)

d (Re Ω) = 0, (2.6)

Ω∧Ω =¯ −iF·ω³ 6 , (2.7)

d d^Λ(F·Im Ω) =ρA. (2.8)

When d Ω = 0 andF is constant, Ω defines an honest complex structure andX is Calabi-Yau; the RR flux source currentρA is zero.

Type IIB supersymmetric structures have a natural generalization to all dimen- sions:

Definition 2.4. AnSU(n)structure(X, ω,Ω)is said to be supersymmetric of type IIB ifΩdefines an honest complex structure and ω defines a balanced metric, that is, d Ω = 0andd (ωⁿ⁻¹) = 0.

Remark 2.5. It is well known that supersymmetric SU(3) system satisfies the equation of motion. See, for example [GMPT07].

Nilmanifolds introduced in Section 7 serve as examples of such systems.

However, generalizing Type IIA supersymmetricSU(n) structure to all dimen- sions is not as obvious as it appears. The naive definition, which is to require d ReΩ = 0 as in dimension three, does not match with SYZ transformation. It turns out that choosing a special real polarization is important in the definition of Type IIA supersymmetric structure in higher dimensions.

A complex structure automatically comes with a complex polarization, which is the splitting

T X⊗C=T^(1,0)X⊕T^(0,1)X.

This gives rise to the notion of (p, q) forms. On the other hand, given a symplectic structure, it does not automatically come with a real polarization, and we have to fix one in order to define Type IIA supersymmetric structure in higher dimensions.

Definition 2.6. Let X be a real2n-fold andω a non-degenerate two-form. A real polarization with respect toω is an integrable Lagrangian distribution {Λx⊂TxX : x∈U} over an open dense subset U ⊂X. (Lagrangian means thatω|Λx = 0 for allx∈U.)

Let (X, ω,Ω) be an SU(n) structure. A real polarization (U,Λ) is said to be special with respect toΩ of phaseθ∈R/2πZif Ω|Λ_x∈R>0·e^iθ for allx∈U.

Note that the symbol Λ appearing in d^Λ refers to the adjoint of the Lefzchetz operator. We will also use Λ to denote a real polarization. It should be clear from the context which meaning we refer to.

Given a real polarization on (X, ω), the metric onX induces a splittingT U = Λ⊕Λ^⊥. We have

Ωⁿ_U = M

p+q=n

Ω^(p,q)_U ^Λ,

where forms in Ω^(p,q)_U ^Λ are defined overU and consist ofpΛ-directions andq Λ^⊥- directions, and the corresponding projection operators π_Λ^p,q : Ωⁿ_U → Ω^(p,q)_U ^Λ. Let Ω^(p,q)Λ be the space of differential forms on X whose restriction toU belongs to Ω^(p,q)_U ^Λ.

Then define a generalization of Definition 2.3 as follows:

Definition 2.7. Let(X, ω,Ω)be anSU(n)structure with a special real polarization (U,Λ). (X, ω,Ω)is said to be supersymmetric of Type IIA if

(1) dω= 0.

(2) d (π_Λ^n,0·Ω|U) = 0.

(3) d (π_Λ^1,n−1·Ω|U) = 0.

Definition 2.8. LetF be the conformal factor of anSU(n)structure(X, ω,Ω). If (X, ω,Ω)is supersymmetric of Type IIA (with respect to a special real polarization (U,Λ)), define

ρA=−id d^Λ(F·(π^n−1,1_Λ ·Ω|U+π^0,n_Λ ·Ω|U)).

If (X, ω,Ω)is supersymmetric of Type IIB, define ρB= 2i∂∂ F¯ ⁻¹·ω

. Definition 2.7 reduces to Definition 2.3 whenn= 3:

Proposition 2.9. Let (X, ω,Ω)be anSU(3)structure with a special real polariza- tion (U,Λ) of phase zero or π. Thend (Re Ω) = 0 if and only if d (π_Λ^3,0·Ω|U) = d (π_Λ^1,2·Ω|_U) = 0.

Proof. SinceU is an open dense subset ofX, d (Re Ω) = 0 if and only if d (Re Ω|U) = 0. Forx0∈U, let (θ1, θ2, θ3, r1, r2, r3) be local coordinates aroundxsuch that each leaf of Λ is given by (r1, r2, r3) = (c1, c2, c3) for some constants ci’s. Then Ω can be written as

Ω =f(θ, r)·(dθ₁+iψ₁)∧(dθ₂+iψ₂)∧(dθ₃+iψ₃)

where ψi = J ·dθi are one-forms and f(θ, r) is complex-valued. Since (U,Λ) is special of phase zero orπ, Ω|Λ_x =f(θ, r)·dθ1∧dθ2∧dθ3is real-valued, and hence f(θ, r) is real-valued. Then

Re Ω =f(θ, r)·(dθ₁∧dθ₂∧dθ₃−dθ₁∧ψ₂∧ψ₃−ψ₁∧dθ₂∧ψ₃−ψ₁∧ψ₂∧dθ₃)

= (π_Λ^3,0⊕π^1,2_Λ )·Ω.

It follows that d (Re Ω) = 0 if and only if d (π^3,0_Λ ·Ω|U) = d (π_Λ^1,2·Ω|U) = 0.

Note also whenn= 3, Im Ω =−i(π^2,1_Λ ·Ω +π_Λ^0,3·Ω). Therefore, to conclude, a Type IIA supersymmetricSU(n) system satisfies

dω= 0, (2.9)

d ((π_Λ^n,0⊕π_Λ^1,n−1)·Ω) = 0, (2.10)

Ω∧Ω =¯ −iⁿF· ωⁿ n!, (2.11)

−id d^Λ(F·(π^n−1,1_Λ ·Ω +π_Λ^0,n·Ω)) =ρ_A. (2.12)

A Type IIB supersymmetricSU(n) system satisfies d Ω = 0, (2.13)

d (ωⁿ⁻¹) = 0, (2.14)

Ω∧Ω =¯ −iF·ωⁿ n!, (2.15)

2i∂∂ F¯ ⁻¹·ω

=ρB. (2.16)

Remark 2.10. The general SU(n) systems we introduced here have similarities with the higher dimensional supersymmetric compactifications in string theory. For instance, for n=4, they are very similar to special cases of the supersymmetric equations in [PT13, Ros14]. It is interesting to expand upon this issue in future works.

3. SYZ transformation

The SYZ transform we use in this paper follows from [LYZ00, Leu05]. In this section we briefly review the setting and definitions which will be used in the rest of the paper.

Let (X, ω) be a symplectic manifold andπ:X→Ba Lagrangian torus bundle.

By Arnold-Liouville’s action-angle coordinates [Arn91, Section 50], for eachp∈B there exists open subsetU ⊂Bcontainingp, action coordinatesr₁, . . . , r_nofU and symplectomorphism (π⁻¹(U), ω) ∼= (T^∗U/Λ^∗, ω_can), where Λ^∗ is a lattice bundle generated by dr1, . . . ,drn. The corresponding fiber coordinates ofT^∗Uare denoted asθi’s, and soω=Pn

i=1dθi∧dri.

The dual torus bundle is defined as the set of all fiberwise flatU(1)-connections:

Xˇ :={(r,∇) :r∈B,∇ is a flatU(1) connection overFr}

whereFrdenotes the fiber ofπatr. The torus bundle map ˇπ: ˇX →B is given by forgetting the fiberwise flatU(1) connections∇.

Xˇ is endowed with a canonical complex structure. Locally for eachp∈B, there exists open subset U ⊂ B containing pand a biholomorphism ˇπ⁻¹(U) ∼=T U/Λ, where Λ ⊂ T U is the lattice bundle generated by _∂r^∂

1, . . . ,_∂r^∂

n, and r_i’s are the action coordinates mentioned before. The corresponding fiber coordinates of T U are denoted as ˇθi’s, and so complex coordinates of ˇX can be taken to be ζi :=

exp(ˇθi+iri) orzi:= ˇθi+iri fori= 1, . . . , n.

The transition between any two local action-coordinate systems ofBbelongs to GL(n,Z)nRⁿ, and soBis endowed with a tropical affine structure. We assume that the action coordinate systems can be chosen such that all the transitions belong to

SL(n,Z)n Rⁿ. In such a situation ˇX has a holomorphic volume form ˇΩ, which is locally written as

Ω = dˇ z₁∧. . .∧dz_n= (d ˇθ₁+idr₁)∧. . .∧(d ˇθ_n+idr_n).

(X, ω) and ( ˇX,Ω) are said to form a semi-flat mirror pair. The semi-flatˇ A-branes onX are mirror to semi-flat B-branes on ˇX under Fourier-Mukai transform. The readers are referred to [LYZ00] for details.

In this paper, we mostly focus on Fourier-Mukai transform of differential forms, which will be discussed in the next section. We will consider a semi-flat real (1,1) form ˇω on ˇX, which is given by

ˇ ω= i

2 X

i,j

µ_ij(r) dz_i∧dz_j whereP

i,jµij(r)dridrj defines a Riemannian metric onB(that is,µij is symmet- ric oni, jand positive definite). We will impose the balanced condition d ˇωⁿ⁻¹= 0 instead of the K¨ahler condition d ˇω = 0, and consider its mirror structure by ap- plying the Fourier-Mukai transform.

4. Fourier-Mukai transform of differential forms

Let (X, ω) and ( ˇX,Ω) be a semi-flat mirror pair of real dimension 2nˇ as in Section 3. In this section we introduce Fourier-Mukai transform for differential forms on X and ˇX. We will restrict the construction toT-invariant differential forms, while keeping in mind that Fourier-Mukai transform can also be done for general differ- ential forms, which is an important tool for quantum corrections [CL10, CLL12].

Fourier-Mukai transform of differential forms was used to study SYZ mirror symme- try without flux in [Leu05, CL10, CLL12, CLM11]. There is an important subtlety in the treatment here, namely we take a switch from the complex to the real po- larization (Definition 4.2) along with the transform. A new result in this section is a direct relation between the Doubealt operators ( ¯∂, ∂) on ˇX and the differentials (d,d^Λ) onX.

Let Ω^k_B(X,C) denote the space of complex-valuedk-forms on X which depend only on base, that is,φ∈Ω^k_B(X,C) is locally written as

φ= X

I=(i₁,...,i_p) J=(j₁,...,j_q)

p+q=k

a_IJ(r) dθ_i1∧. . .∧dθ_ip∧dr_j1∧. . .∧dr_jq

where (r1, . . . , rn, θ1, . . . , θn) is an action-angle coordinate system ofX, andaIJ(r)’s are complex-valued functions onB. Such aφis also said to beT-invariant, where T denotes a torus fiber ofX →B.

Definition 4.1. A semi-flat (supersymmetric)SU(n)structure is a (supersymmet- ric)SU(n)structure(X, ω,Ω)such thatX has a Lagrangian torus bundle structure X →B andω,Ω∈Ω^∗_B(X,C).

Similarly, let Ω^p,q_B ( ˇX) denote the space of (p, q)-forms on ˇX which depend only on base. ˇφ∈Ω^p,q_B ( ˇX) is locally of the form

φˇ= X

I=(i1,...,ip) J=(j₁,...,j_q)

aIJ(r) dzi₁∧. . .∧dzi_p∧dzj₁∧. . .∧dzj_q.

The above expression of ˇφ is written in terms of complex polarization. We can switch to real polarization by the following definition:

Definition 4.2(Polarization switch operator). The polarization switch operatorP onΩ^∗_B( ˇX)is defined by sending

φˇ= X

I=(i₁,...,i_p) J=(j₁,...,j_q)

a_IJ(r) dz_i1∧. . .∧dz_ip∧dz_j1∧. . .∧dz_jq

to

P ·φˇ= X

I=(i₁,...,i_p) J=(j₁,...,j_q)

a_IJ(r) d ˇθ_i1∧. . .∧d ˇθ_ip∧dr_j1∧. . .∧dr_jq.

Now we define Fourier-Mukai transform of differential forms onX and ˇX, which gives an isomorphism

Ω^k_B(X,C)∼=

k

M

p=0

Ω^p,n−k+p_B ( ˇX) .

Consider the following commutative diagram:

X×_BXˇ

X Xˇ

B

@

@

@ R

@

@

@@R

π πˇ

whereπ:X →Band ˇπ: ˇX →B are dual torus bundle maps given in the semi-flat setting, and the fiber product X ×BXˇ is defined using these bundle maps. We have the universal connection onX×BX, which is d +ˇ iθˇ_idθ_i−iθ_id ˇθ_i in terms of semi-flat local coordinates. The curvature of the universal connection is

F= 2i

n

X

i=1

d ˇθi∧dθi.

Definition 4.3(Fourier-Mukai transform). Letφˇbe aT-invariant differential form onXˇ. The Fourier transform ofφˇis defined as

FT·φˇ:=π_∗

(ˇπ^∗(P ·φ))ˇ ∧exp F 2i

where the push-forward π∗ induced by π : X → B is defined by integration along torus fibers.

Similarly for a T-invariant differential form φ on X, its Fourier transform is defined as

FT·φ:=P⁻¹·

ˇ π_∗

(π^∗φ)∧exp−F 2i

.

By carefully keeping track of the sign in the transform, we see that applying Fourier-Mukai transform two times is simply identity up to a sign only dependent on the dimension.

Proposition 4.4.

FT◦FT = (−1)^n(n−1)2 wheren= dimB.

Proof. It suffices to consider the basic elements dz_I ∧dz_J = dz_i1 ∧. . .∧dz_ip∧ dz_j1∧. . .∧dz_jq. Switching to real polarization gives d ˇθ_I ∧dr_J = d ˇθ_i1∧. . .∧ d ˇθ_ip∧dr_j1∧. . .∧dr_jq. Multiplication by

exp X

i

d ˇθi∧dθi

!

=

n

X

p=0

(−1)^p(p−1)2 X

K={i1,...,ip}

d ˇθK∧dθK

gives

(−1)(n−p)(n−p−1)

2 d ˇθ_Ic∧dθ_Ic∧d ˇθ_I∧dr_J = (−1)(n−p)(n−p−1)

2 (−1)^p(n−p)d ˇθ_Ic∧d ˇθ_I∧dθ_Ic∧dr_J whereI^c:={1, . . . , n} −I andp=|I|. Then integrating along the fiber directions θˇi’s gives

(−1)(n−p)(n−p−1)

2 (−1)^p(n−p)sign(I^c, I)dθI^c∧drJ = (−1)(n−p)(n−p−1)

2 sign(I, I^c)dθI^c∧drJ. Thus we have

(4.1) FT·(dzI∧dzJ) = (−1)(n−p)(n−p−1)

2 sign(I, I^c)dθI^c∧drJ. Now we take Fourier transform again. Multiplying dθI^c∧drJby exp −P

id ˇθi∧dθi gives gives

(−1)^p(−1)^p(p−1)2 d ˇθI∧dθI∧dθI^c∧drJ = (−1)^p(−1)^p(p−1)2 (−1)^npdθI∧dθI^c∧d ˇθI∧drJ. Integrating along the fiber directionsθi’s gives

(−1)^p(−1)^p(p−1)2 (−1)^npsign(I, I^c)d ˇθI∧drJ. Switching the real polarization back to complex polarization, we have

FT(dθI^c∧drJ) = (−1)^p(−1)^p(p−1)2 (−1)^npsign(I, I^c)dzI∧dzJ. Hence FT◦FT(dzI∧dzJ) equals to

(−1)(n−p)(n−p−1)

2 (−1)^p(−1)^p(p−1)2 (−1)^np·dzI ∧dzJ= (−1)^n(n−1)2 dzI∧dzJ. In the theorem below we see that d on Ω^∗_B(X,C) is transformed to ¯∂on Ω^∗_B( ˇX,C), and d^Λ on Ω^∗_B(X,C) is transformed to ∂ on Ω^∗_B( ˇX,C). We recall that d^Λ :=

d ◦Λ−Λ◦d which decreases the degree by one, where Λ is the dual Lefschetz operator. To our knowledge the theorem has not appeared in previous literature.

Theorem 4.5. Fourier-Mukai transform gives an isomorphism

Ω^∗_B(X,C),(−1)ⁿi

2 d,(−1)ⁿi 2 d^Λ

∼= Ω^∗_B( ˇX,C),∂, ∂¯ as double complexes.

Proof. Letφ=φI,JdzI∧dzJ where I andJ are multi-indices. By assumingφI,J

is skew-symmetric onI andJ, the expression is independent of the orderings ofI andJ. We need to prove that

FT◦∂¯·φ=(−1)ⁿi

2 ·d ◦FT·φ and

FT◦∂·φ= (−1)ⁿi

2 ·d^Λ◦FT·φ.

Consider the first equation. By Equation (4.1), Fourier transform ofφis (4.2) FT·φ= (−1)(n−p)(n−p−1)

2 sign(I, I^c)φ_I,Jdθ_Ic∧dr_J

where I^c ={1, . . . , n} −I. Note that the above expression is independent of the orderings ofI^c andJ.

Letp:=|I|. We have FT◦∂¯·φ= FT·

i

2(∂_iφ_I,J)dz_i∧dz_I∧dz_J

=i

2(−1)(n−p)(n−p−1)

2 sign(I, I^c)(∂iφI,J)(−1)^pdθI^c∧dri∧drJ

=i

2(−1)(n−p)(n−p−1)

2 sign(I, I^c)(∂iφI,J)(−1)ⁿdri∧dθI^c∧drJ. Comparing with Equation 4.2, we see that FT◦∂¯·φ= (−1)ⁿⁱ₂·d ◦FT·φ.

To prove the second equation, let’s recall that d^Λ= d ◦Λ−Λ◦d , where Λφ=

n

X

i=1

ιriιθiφ

forω=P

idθi∧dri. We remark that more generally forω=P

i,jωijdxi∧dxj, Λφ=1

2 X

i,j

(ω⁻¹)^ijιx_iιx_jφ.

WritingK=I^c− {i}, we have FT◦∂·φ= FT· −i

2 X

i∈I^c

∂_iφ_I,Jdz_i∧dz_I∧dz_J

!

= −i

2 (−1)(n−p−1)(n−p−2)

2 X

i∈I^c

sign(i, I, K)(∂iφI,J) dθK∧drJ.

On the other hand, let’s compute d ◦Λ◦FT·φand Λ◦d ◦FT·φ.

d ◦FT·φ= (−1)(n−p)(n−p−1)

2 sign(I, I^c)X

i6∈J

(∂iφI,J) dri∧dθI^c∧drJ.

Again the above expression is independent of orderings of I^c and J. Since Λ = P

jιr_jιθ_j, takingI^c= (j, K) andJ = (j, L), we have Λ◦d ◦FT·φ=(−1)(n−p)(n−p−1)

2 sign(I, I^c) X

j=i6∈J j∈I^c

(∂iφI,J) (−dθK∧drJ)

+ (−1)(n−p)(n−p−1)

2 sign(I, I^c) X

i6∈J j6=i j∈I^c∩J

(∂iφI,J) dri∧((−1)^n−p−1dθK∧drL).

To compute d ◦Λ◦FT·φ(again takingI^c= (j, K) andJ = (j, L)), we have Λ◦FT·φ= (−1)(n−p)(n−p−1)

2 sign(I, I^c) X

j∈I^c∩J

φI,J(−1)^n−p−1dθK∧drL

and hence

d ◦Λ◦FT·φ=(−1)(n−p−2)(n−p−1)

2 sign(I, I^c) X

i=j∈I^c∩J

∂jφI,Jdrj∧dθK∧drL

+ (−1)(n−p)(n−p−1)

2 sign(I, I^c) X

i6∈J i6=j j∈I^c∩J

∂_iφ_I,Jdr_i∧(−1)^n−p−1dθ_K∧dr_L.

We see that the second terms of Λ◦d ◦FT·φand d ◦Λ◦FT·φare equal. Thus d^Λ◦FT·φ=(−1)(n−p−2)(n−p−1)

2 sign(I, I^c) X

j∈I^c∩J

∂_jφ_I,Jdr_j∧dθ_K∧dr_L

−(−1)(n−p)(n−p−1)

2 sign(I, I^c)X

i6∈J j∈I^c

(∂iφI,J) (−dθK∧drJ)

=(−1)(n−p)(n−p−1)

2 X

i∈I^c

sign(I, I^c)(∂iφI,J) dθK∧drJ

=(−1)ⁿ⁻¹(−1)(n−p−2)(n−p−1)

2 X

i∈I^c

sign(i, I, K)(∂iφI,J) dθK∧drJ

=(−1)ⁿ i

2 −1

FT·(∂φ).

5. SYZ between type-A and type-B supersymmetric systems Let (X, ω) and ( ˇX,Ω) be a semi-flat mirror pair as described in Section 3. (X, ω)ˇ has the required real polarization Λ given by the given Lagrangian torus bundle (where Λx is the tangent space of the torus fiber atx∈X). The purpose of this section is to prove Theorem 1.1, namely Fourier-Mukai transform introduced in Section 4 gives a symmetry between Type-A and Type-B supersymmetric SU(n) structures.

Theorem 5.1 (Detailed version of Theorem 1.1). Let ωˇ be a torus-invariant real (1,1)-form on Xˇ andΩthe Fourier-Mukai transform of e^{2 ˇω}.

(1) ( ˇX,ω,ˇ Ω)ˇ forms anSU(n)structure if and only if(X, ω,Ω)forms anSU(n) structure. For such a pair the conformal factorF of(X, ω,Ω)is related to the conformal factorFˇ of( ˇX,ω,ˇ Ω)ˇ byFˇ= 2²ⁿF⁻¹.

(2) (X, ω,Ω)is supersymmetric of Type-A if and only if( ˇX,ω,ˇ Ω)ˇ is supersym- metric of Type-B.

(3) Letρ_Abe the Type-A RR flux source current of(X, ω,Ω). Then its Fourier- Mukai transform equals to the Type-B RR flux source currentρˇBof( ˇX,ω,ˇ Ω)ˇ up to a constant multiple.

Proof. Let (θ₁, . . . , θ_n, r₁, . . . , r_n) be local semi-flat coordinates. The (1,1) form ˇω can be written as

ˇ ω=i

2 X

i,j

µ_ij(r)dz_i∧dz_j

=X

i,j

µij(r)d ˇθi∧drj+i 2



 X

i<j

(µij−µji)(d ˇθi∧d ˇθj+ dri∧drj)



.

Since ˇω is real,µ_ij =µ_jiare real-valued. Defining the one-forms µi:=X

j

µijdrj

onB, ˇω can also be written as ˇ ω=X

i

d ˇθi∧µi. Changing to real polarization,

P ·exp(2ˇω) = exp



iX

i,j

µ_ij(r)d ˇθ_i∧dr_j



.

Then (5.1)

Ω = FT·exp(2ˇω) =π_∗·exp −X

i

d ˇθ_i∧dθ_i

!

∧exp



iX

i,j

µ_ij(r)d ˇθ_i∧dr_j





=π∗·exp −X

i

d ˇθi∧(dθi+iµi)

!

= (−1)^n(n−1)2 (dθ1+iµ1)∧. . .∧(dθn+iµn).

Ω defines a complex structureJ onX by

J·dθ_i=−µ_i, J·µ_i= dθ_i

if and only if the one-forms µ_i for i= 1, . . . , n are linearly independent, which is equivalent to the condition thatω is non-degenerate.

Moreover ω=X

ij

dθ_i∧dr_i=X

ij

ηi+ηi

2 ∧µ⁻¹_ji (η_j−η_j)

2i = i

2 X

ij

µ⁻¹_ji η_i∧η_j

has no (2,0) and (0,2) components becauseµ⁻¹_ji is symmetric in i, j, where µ⁻¹ is the inverse matrix of µij and ηi := dθi+iµi are (1,0)-forms with respect to Ω.

Thusω is a real (1,1) form with respect toJ. Finally

(ω, J) is Hermitian⇔µij is positive definite

⇔(µ⁻¹)ij is positive definite⇔(ˇω,J) is Hermitian.ˇ

This proves ( ˇX,ω,ˇ Ω) forms anˇ SU(n) structure if and only if (X, ω,Ω) forms an SU(n) structure.

Recall that the conformal factorF for (X, ω,Ω) is defined by Ω∧Ω =¯ iⁿ·F·ωⁿ

n!

and similarly for the conformal factor ˇF of ( ˇX,ω,ˇ Ω). By direct computations, weˇ have

Fˇ= 2²ⁿ

F = 2ⁿ(−1)^n(n+1)2 det(µij) .

Now we consider the supersymmetry conditions. d ˇω= 0 and d (π_Λ^n,0·Ω|U) = 0 are automatic. It suffices to prove that d (π^1,n−1_Λ ·Ω|U) = 0 if and only if d ˇωⁿ⁻¹= 0.

Thek-th summand of

P ·exp 2ˇω=

n

X

k=0

(2P ·ω)ˇ ^k k!

hask torus-fiber directions (d ˇθ) and k base directions (dr). After Fourier-Mukai transform it hasn−ktorus-fiber directions (dθ) andkbase directions (dr). Thus π^n−k,k_Λ ·Ω is the Fourier-Mukai transform of (2ˇω)^k/(k)! (up to a constant multiple).

By Theorem 4.5, d is transformed to ¯∂(up to a constant multiple). Thus d (π_Λ^1,n−1· Ω) = 0 if and only if ¯∂(ˇωⁿ⁻¹) = 0. Since ˇω is real, ∂(ˇωⁿ⁻¹) = ¯∂(ˇωⁿ⁻¹). Thus d (π_Λ^1,n−1·Ω) = 0 if and only if d ˇωⁿ⁻¹= 0.

Now consider the fluxes of the supersymmetric systems, which are defined by ρ_A=−id d^Λ(F·(π_Λ^n−1,1·Ω +π_Λ^0,n·Ω)),

ˇ

ρ_B= 2i∂∂¯ Fˇ⁻¹·ωˇ .

From aboveπ^n−1,1_Λ ·Ω is the Fourier-Mukai transform of 2ˇω, and ˇF = 2²ⁿF⁻¹. The componentπ^0,n_Λ ·Ω is the transform of ˇωⁿ, which contributes zero to ˇρB. More- over by Theorem 4.5, ⁽⁻¹⁾₂ⁿⁱ ·d is transformed to ¯∂ and ⁽⁻¹⁾₂ⁿⁱ ·d^Λ is transformed to∂. It follows that the Fourier transform ofρA is 2²ⁿ⁺²ρˇB.

6. Deformation theory

Since Type IIA and IIB supersymmetricSU(n) systems correspond to each other under the SYZ mirror transform, their deformations should also correspond to each other. The following cohomologies are keys to study their deformations:

Definition 6.1 ([BC65, Aep65]). Let Xˇ be a complex manifold. The Bott-Chern cohomology of Xˇ is defined as

H_B.C.^p,q ( ˇX) := Ker(d )∩Ω^p,q( ˇX) Im(∂∂)¯ ∩Ω^p,q( ˇX)

Definition 6.2 ([TY12a, TY12b]). Let X be a symplectic manifold. The Tseng- Yau symplectic cohomology ofX is defined as

H_{d +d}^k Λ(X) := Ker(d + d^Λ)∩Ω^k(X) Im(d d^Λ)∩Ω^k(X) .

Indeed we have a more refined version for a given real polarization (U,Λ) onX. Definition 6.3. Let X be a symplectic manifold with a real polarization (U,Λ).

Define

H_{d +d}^(p,q)Λ_Λ(X) := Ker(d + d^Λ)∩Ω^(p,q)Λ(X) Im(d d^Λ)∩Ω^(p,q)Λ(X)

where we recall thatΩ^(p,q)Λ(X)consists of forms whose restrictions toU consist of pΛ-directions andq Λ^⊥-directions.

In the semi-flat setting given in Section 4, we also have the torus-invariant coun- terparts

(6.1) H_B,B.-C.^p,q ( ˇX) := Ker(d )∩Ω^p,q_B ( ˇX) Im(∂∂)¯ ∩Ω^p,q_B ( ˇX) and

(6.2) H_{B,d +d}^(p,q)Λ Λ(X) := Ker(d + d^Λ)∩Ω^(p,q)_B ^Λ(X) Im(d d^Λ)∩Ω^(p,q)_B ^Λ(X)

.

For a Type IIB supersymmetric SU(3) system ( ˇX, ω,Ω), suppose that ωt for t∈(−, ) is a deformation of the balanced metricω(meaningωt=0=ω) such that ( ˇX, ω_t,Ω) is still a Type IIB supersymmetricSU(3) system, with a fixed conformal factorF and fixed type-B flux ρB. As argued in [TY, Section 2.1], the first order deformation δ(ω²) = _∂t^∂|_t=0ω²_t belongs to H_B.C.^2,2 ( ˇX)∩ L · P^1,1( ˇX), where P^1,1( ˇX) denotes the space of primitive (1,1) classes andLdenotes the Lefschetz actionω∧·.

It easily generalizes to all dimensions:

Proposition 6.4. Let( ˇX, ωt,Ω)fort∈(−, )be a one-parameter family of Type IIB supersymmetricSU(n)systems with a fixed conformal factor. Then the infini- tesimal deformation _∂t^∂

_t=0(ω_tⁿ⁻¹)represent a class in H_B.C.^n−1,n−1( ˇX)∩ Lⁿ⁻²· P^1,1( ˇX)B.C.. Proof. The balanced condition d (ω_tⁿ⁻¹) = 0 implies that d _∂t^∂

_t=0(ωⁿ⁻¹_t ) = 0, and hence _∂t^∂

_t=0(ω_tⁿ⁻¹) represents a class in H_B.C.^n−1,n−1( ˇX). Taking _∂t^∂ _t=0 on Ωˇ∧Ω =¯ˇ iⁿFˇ·ωˇⁿ_t

n!

where ˇF is the conformal factor, we haveωⁿ⁻¹∧_∂t^∂

_t=0ω_t= 0, and hence _∂t^∂ _t=0ω_t is a primitive (1,1) form. Thus _∂t^∂

_t=0(ω_tⁿ⁻¹) = (n−1)ωⁿ⁻²∧ _∂t^∂

_t=0(ωt) belongs

toLⁿ⁻²· P^1,1( ˇX)_B.C..

Remark 6.5. Indeed as in[TY]we can say more: if we also fix the Type-B flux of ( ˇX, ω_t,Ω), then the infinitesimal deformation _∂t^∂

_t=0(ωⁿ⁻¹_t ) is actually harmonic with respect to the Bott-Chern cohomology.

Now consider a Type IIA supersymmetric SU(n) system (X, ω,Ω). Its defor- mation theory is more tricky than the n = 3 case because it depends on a real polarization.

Proposition 6.6. Let(X, ω,Ωt)be a one-parameter family of Type IIA supersym- metricSU(n)systems with a fixed special real polarization(U,Λ)onX and a fixed conformal factor. Thenπ^1,n−1_Λ · _∂t^∂

_t=0Ωt

U represents a class in H_{d +d}^(1,n−1)Λ ^Λ(U)∩[π^1,n−1_Λ ·Ω^(n−1,1)(U)]

where Ω^(n−1,1) denotes the space of (n−1,1) forms (with respect to the almost complex structure induced from Ω = Ω_t=0).

Proof. _∂t^∂

_t=0Ω_t only has (n,0) and (n−1,1) components. Taking _∂t^∂ _t=0 on Ω_t∧Ω¯_t=iⁿF·^ω_n!ⁿ whereFis the conformal factor, we have _∂t^∂

t=0Ω_t

∧Ω = 0, and¯ hence _∂t^∂

_t=0Ω_thas no (n,0) component, and hence belongs to Ω^(n−1,1)(X). By the condition d (π_Λ^1,n−1·Ωt|U) = 0, we have d

π_Λ^1,n−1· _∂t^∂

_t=0Ωt

= 0. Moreover Ωt

is a primitive form asω∧Ω_t= 0, and hence _∂t^∂

_t=0Ω_tis also primitive. In particular d^Λ· _∂t^∂

_t=0Ωt= 0. Thusπ^1,n−1_Λ · _∂t^∂

_t=0Ωt|U defines a class inH_{d +d}^(1,n−1)_Λ ^Λ(U).

From Section 5, we see that SYZ and Fourier-Mukai transform gives the mirror map from the Type IIA moduli space to the Type IIB moduli space of the mirror.

Indeed the above cohomologies are also mirror to each other by Fourier-Mukai transform:

Theorem 6.7. Let (X, ω) → B be a Lagrangian torus bundle and ( ˇX,Ω) be its mirror. Fourier-Mukai transform gives an isomorphism

H_{B,d +d}^(n−p,q)_Λ^Λ(X,C)∼=H_B,B.C.^p,q ( ˇX)

whereΛ is the real polarization onX induced from the bundle structure.

Proof. Fourier-Mukai transform maps aT-invariant (p, q) form on ˇXto aT-invariant form on X with (n−p) fiber directions andq base directions. By Theorem 4.5,

∂¯ and ∂ on ˇX correspond to d and d^Λ on X respectively under Fourier-Mukai transform. Hence d,∂∂¯ on ˇX correspond to d + d^Λ and d d^Λ onX respectively.

By Equation (6.1) and (6.2), statement follows.

7. Nilmanifolds

We describe an important example of a mirror pair of non-K¨ahler supersymmet- ric Type-A and Type-B systems constructed from nilmanifolds. We first describe the three-dimensional case, and extend the discussion to all dimensions.

7.1. Three dimensional case. The three-dimensional real Iwasawa manifold B is given by the quotient of the Heisenberg group, which consists of matrices of the form





1 r₁ r₃ 0 1 r₂

0 0 1



, r₁, r₂, r₃∈R,

by the discrete subgroup Γ consisting of matrices of the same form withr₁, r₂, r₃ being integers.

Explicitly Γ acts on the Heisenberg group by affine transformations: for γ=





1 a c 0 1 b 0 0 1



∈Γ we have

γ·



 r1

r2

r3



=





1 0 0 0 1 0 0 a 1







 r1

r2

r3



+



 a b c



. ThusB is an affine manifold.

Now consider X = T^∗B/Λ^∗ and ˇX = T B/Λ. Xˇ has a canonical complex structure, and its complex coordinates are given byζi= ˇθi+iri, i= 1,2,3. It has a holomorphic volume form

Ω = dˇ ζ1∧dζ2∧dζ3. X has a canonical symplectic formω=P3

i=1dθi∧dri. Take the Hermitian two-form

ˇ ω= i

2 dζ1∧d ¯ζ1+ dζ2∧d ¯ζ2+ (dζ3−r1dζ2)∧(d ¯ζ3−r1d ¯ζ2)

= d ˇθ₁∧dr₁+ d ˇθ₂∧dr₂+ (d ˇθ₃−r₁d ˇθ₂)∧(dr₃−r₁dr₂) on ˇX. Notice that d ˇω6= 0 while d ˇω²= 0 by direct computation. Moreover,

ˇ

ω³= d ˇθ₁∧dr₁∧d ˇθ₂∧dr₂∧d ˇθ₃∧dr₃

which is a constant multiple of ˇΩ∧Ω. Thus ( ˇ¯ˇ X,ω,ˇ Ω) forms a type IIB supersym-ˇ metric system with a constant conformal factor.

The type-B flux source is

ρB = 2i∂∂¯ωˇ ∼dr1∧d ˇθ1∧dr2∧d ˇθ2

up to some constant multiple. ρ_B is the Poincar´e dual of the complex manifold defined by r₁ =r₂ = ˇθ₁ = ˇθ₂ = 0, which can also be written as T C/(C∩Λ) for C={r₁=r₂= 0} ⊂B.

Now take the Fourier-Mukai transform ofe^{2 ˇω}. Write ˇ

ω= i

2 dζ1∧d ¯ζ1+ dζ2∧(d ¯ζ2−r1d ¯ζ3+r²₁d ¯ζ2) + dζ3∧(d ¯ζ3−r1d ¯ζ2) . Switching polarization (Equation (4.2)) gives

P ·ωˇ= i

2 d ˇθ₁∧dr₁+ d ˇθ₂∧(dr₂−r₁dr₃+r₁²dr₂) + d ˇθ₃∧(dr₃−r₁dr₂) . Then

e^{2 ˇω}e^{P3i=1d ˇθi∧dθi} = exp(d ˇθ1∧(dθ1+idr1) + d ˇθ2∧(dθ2+i(dr2−r1dr3+r²₁dr2)) + d ˇθ₃∧(dθ₃+i(dr₃−r₁dr₂))).

Thus the Fourier-Mukai transform is

Ω = (dθ1+idr1)∧(dθ2+i(dr2−r1dr3+r²₁dr2))∧(dθ3+i(dr3−r1dr2))

= (dθ1+idr1)∧((dθ2+r1dθ3) +idr2)∧(dθ3+i(dr3−r1dr2))

on X. It is easy to verify that d (Re Ω) = 0 and Ω∧Ω equals to¯ ω³ up to a constant multiple. Thus (X, ω,Ω) is a Type IIA supersymmetric SU(n) system with a constant conformal factor.

The type-A flux source is

ρA= d d^Λ(Im Ω)∼dr1∧dr2∧dθ3

up to a constant multiple. ρA is the Poincar´e dual of the Lagrangian submanifold defined by r1 =r2 =θ3 = 0, which can also be written as the conormal bundle N^∗C/(C∩Λ^∗), whereC ={r1 =r2= 0} ⊂B is as defined above. Note that the type-A flux currentρ_Aof X and type-B flux currentρ_B of the mirror ˇX correspond to each other by Fourier-Mukai transform.

7.2. General dimension. We may extend the above example to general dimen- sions to construct type II-A and type II-B supersymmetric SU(n) systems which are mirror to each other. Consider the vector spaceV of upper unitriangular ma- trices (rij)^K_i,j=1, that is,rij = 0 fori > j and rii = 1 for alli= 1, . . . , K. Then we consider the lattice Γ of upper unitriangular matrices (aij)^K_i,j=1with integer entries, aij = 0 fori > jandaii= 1 for alli= 1, . . . , K. Γ acts onV by left multiplication, and the real Iwasawa manifoldB is the quotient V /Γ. Explicitlya= (aij)i<j∈Γ acts onr= (rij)i<j∈V bya·r=r⁰, where

r_ik⁰ =rik+

k−1

X

j=i+1

aijrjk+aik

for all i < k. In particular when k = i+ 1, r⁰_i,i+1 = r_i,i+1+a_i,i+1 for all i = 1, . . . , K−1.

B is an affine manifold. We have the following global one-forms which will be useful to construct the non-K¨ahler geometries:

Proposition 7.1. Define inductively overk−i∈Nthe following one forms onV:

eik:= drik−

k−1

X

j=i+1

rijejk

ande_i,i+1:= dr_i,i+1 for alli= 1, . . . , K−1. These one forms are invariant under the action of Γ and hence descend to be global one-forms onB=V /Γ.

Proof. We prove by induction overk−i∈N. Fork−i= 1, sincea= (aij)i<j∈Γ acts by mappingri,i+1tori,i+1+ai,i+1, it follows thatei,i+1= dri,i+1 is invariant under Γ. Now consider

eik= drik−

k−1

X

j=i+1

rijejk

which is sent to

(drik+ai,i+1dri+1,k+. . .+a_i,k−1dr_k−1,k)

−

k−1

X

j=i+1

(r_ij+a_i,i+1r_i+1,j+. . .+a_i,j−1r_j−1,j+a_ij)e_jk

=e_ik+



a_i,i+1dr_i+1,k+. . .+a_i,k−1dr_k−1,k−

k−1

X

j=i+1

(a_i,i+1r_i+1,j+. . .+a_i,j−1r_j−1,j +a_ij)e_jk





under the action, because by inductive assumption ejk is invariant for all j = i+ 1, . . . , k−1. Consider the second term in the last expression:

ai,i+1dri+1,k+. . .+ai,k−1drk−1,k−

k−1

X

j=i+1

(ai,i+1ri+1,j+. . .+ai,j−1rj−1,j+aij)ejk

=

k−1

X

j=i+1

aijdrjk− aij+

j−1

X

l=i+1

ailrlj

! ejk

!

=

k−1

X

j=i+1



a_ij

k−1

X

l=j+1

r_jle_lk−

j−1

X

l=i+1

a_ilr_lj

! e_jk





=

k−1

X

j=i+1 k−1

X

l=j+1

a_ijr_jle_lk−

k−1

X

j=i+1 j−1

X

l=i+1

a_ilr_lje_jk

=0

where we have used the equality

drjk−ejk=

k−1

X

l=j+1

rjlelk

by definition ofe_jk. Thuse_ikis invariant under the action.

Now considerX =T B/Λ, which is automatically a complex manifold. Suppose θij are the fiber coordinates of T B corresponding to the coordinates rij onB for i < j. Thenθij transforms in the similar way under the action ofa= (aij)i<j∈Γ:

θ⁰_ik=θ_ik+

k−1

X

j=i+1

a_ijθ_jk.

The global one-forms onB in Proposition 7.1 pulls back to be global one-forms on X. We also have the following global one-forms onX:

fik:= dθik−

k−1

X

j=i+1

rijfjk

defined inductively onk−i∈Nas in the definition ofeik’s.

The holomorphic volume form onX is defined as Ω =^

i<j

dz_ij

wherezij=θij+irij. It automatically satisfies Equation (2.13). (We fix an order, say dictionary order for{(i, j) : i < j}, to define the wedge product in the above expression.) Take

ω:=X

i<j

e_ij∧f_ij

Proposition 7.2. ω is a Hermitian (1,1)-form onX. Moreover,Ω∧Ω¯ equals to ωⁿ up to some constant multiple, that is, the conformal factor is a constant.