New York Journal of Mathematics
New York J. Math.27(2021) 1096–1114.
Reconstruction of holomoprhic tangent bundle of complex projective plane via
tropical Lagrangian multi-section
Yat-Hin Suen
Abstract. In this paper, we study the reconstruction problem of the holo- morphic tangent bundle of the complex projective plane. We introduce the notion of tropical Lagrangian multi-section and cook up one tropicalizing the Chern connection associated the Fubini-Study metric. Then we perform the reconstruction of the tangent bundle from this tropical Lagrangian multi- section. Walling-crossing phenomenon will occur in the reconstruction pro- cess.
Contents
1. Introduction 1096
2. SYZ mirror symmetry ofℙ2 1098
3. A tropical Lagrangian multi-section associated to 𝑇ℙ2 and
reconstruction 1101
Appendix A. Local model for caustics 1109
References 1112
1. Introduction
Mirror symmetry is a duality between symplectic geometry and complex ge- ometry. The famous SYZ conjecture [33] allows mathematicians to construct mirror pairs and explain homological mirror symmetry [28] geometrically via a fiberwise Fourier–Mukai-type transform, which we call theSYZ transform.
The SYZ transform has been constructed and applied to understand mirror symmetry in the semi-flat case [6,31, 30,17] and the toric case [1,2,18, 20, 21, 8, 10, 11, 9, 16,13, 14, 19]. But in all of these works the primary focus was on Lagrangian sections and the mirror holomorphic line bundles the SYZ program produces. For higher rank sheaves, Abouzaid [3] applied the idea of family Floer cohomology to construct sheaves on the rigid analytic mirror of
Received April 15, 2020.
2010Mathematics Subject Classification. 14J33, 53D37, 53-11.
Key words and phrases. Algebraic geometry, symplectic geometry, tropical geometry, mirror symmetry.
The work of Y.-H. Suen was supported by IBS-R003-D1.
ISSN 1076-9803/2021
1096
a given compact Lagrangian torus fibration without singular fiber. Recently, applications of the SYZ transform for unbranched Lagrangian multi-sections have been study by Kwokwai Chan and the author in [15].
In this paper, we study the holomorphic tangent bundle𝑇ℙ2ofℙ2in terms of mirror symmetry. Via SYZ construction, the mirror ofℙ2is given by a Landau- Ginzburg model(𝑌, 𝑊)(see Section2for a brief review). It carries a natural fibration𝑝 ∶ 𝑌 → 𝑁ℝ ≅ ℝ2, which is dual (up to a Legendre transform) to the moment map fibration ofℙ2. Since𝑇ℙ2is naturally an object in the derived category𝐷𝑏Coh(ℙ2), it is natural to ask what is its mirror Lagrangian in𝑌. Be- ing a rank 2 bundle, the mirror Lagrangian of 𝑇ℙ2 is expected to be a rank 2 Lagrangian multi-section of𝑝 ∶ 𝑌 → 𝑁ℝwith certain asymptotic conditions.
As the base𝑁ℝof the fibration𝑝is simply connected, every unbranched cov- ering map is trivial. But𝑇ℙ2 is certainly indecomposable. Hence, we are led to consider branched Lagrangian multi-section and the SYZ transform defined in [6,15,31] cannot be applied directly. To overcome this technicality, we intro- duce the notion oftropical Lagrangian multi-sectionand reconstruct𝑇ℙ2 from this tropical object.
Definition 1.1(=Definition3.1).Let𝐵be an𝑛-dimensional integral affine man- ifold without boundary. Arank 𝑟tropical Lagrangian multi-sectionis a triple 𝕃 ∶= (𝐿, 𝜋, 𝜑), where
a) 𝐿is a topological manifold.
b) 𝜋 ∶ 𝐿 → 𝐵a covering map of degree𝑟with branch locus𝑆 ⊂ 𝐵being a union of locally closed submanifolds of codimension at least 2.
c) 𝜑 = {𝜑𝑈} is a multi-valued function on 𝐿such that on any two affine charts𝑈, 𝑉 ⊂ 𝐿∖𝜋−1(𝑆)(with respective to the induced affine structure on𝐿∖𝜋−1(𝑆)via𝜋),
𝜑𝑈− 𝜑𝑉 = ⟨𝑚, 𝑥⟩ + 𝑏, for some𝑚 ∈ ℤ𝑛and𝑏 ∈ ℝ.
LetΣbe the fan corresponds toℙ2and𝑣0, 𝑣1, 𝑣2be its primitive generators.
The tropical Lagrangian multi-section is obtained by “tropicalizing" the Chern connection associated to the Fubini-Study metric. We will do this in Section 3.1by considering a family of Kähler metrics onℙ2, which gives rise to a family of Chern connections, parameterized by a small real numberℏ > 0. We com- pute the limitℏ → 0of the connections to obtain the “tropical connection", which can be regarded as a singular connection on the total space𝑇𝑁ℝ. From the tropical connection, we can cook up six linear functions𝜑±𝑘,𝑘 = 0, 1, 2, for which𝜑±𝑘 are defined on𝜎𝑘, the cone generated by the rays𝑣𝑖, 𝑣𝑗,𝑖, 𝑗 ≠ 𝑘. Let 𝜎𝑘±be two copies of𝜎𝑘and we consider𝜑+𝑘 (resp. 𝜑−𝑘) as a function defined on 𝜎𝑘+(resp. 𝜎𝑘−). A piecewise linear function𝜑and an integral affine manifold𝐿 can be obtained by gluing𝜑𝑘± ∶ 𝜎𝑘± → ℝtogether in a continuous manner. By projecting𝜎±𝑘 to𝜎𝑘, we obtain a map𝜋 ∶ 𝐿 → 𝑁ℝ. The triple𝕃 ∶= (𝐿, 𝜋, 𝜑)
forms a rank 2 tropical Lagrangian multi-section. See Section3.1for the de- tailed construction.
Section3.2will be devoted to reconstructing𝑇ℙ2from the tropical Lagrangian multi-section𝕃. Let𝑈𝑘be the affine chart corresponds to the cone𝜎𝑘. For each 𝜎𝑘±, we associate a trivial line bundleℒ±𝑘 = 𝒪𝑈
𝑘and so, we have a trivial rank 2bundle on 𝑈𝑘 by taking direct sum. As the summands are indexed by the maximal cones of𝐿, the gluing of𝕃tell us how to glue these trivial rank 2 bun- dles together on𝑈𝑖𝑗 ∶= 𝑈𝑖∩ 𝑈𝑗. Motivated by [15], we can write down three naive transition functions𝜏𝑠𝑓10, 𝜏𝑠𝑓21, 𝜏𝑠𝑓02by considering the slope differences of𝜑′ on different maximal cones. However, this naive gluing is inconsistent due to the affine. In order to obtain a consistent gluing, we modify the naive transi- tion functions by three invertible factorsΘ10, Θ21, Θ02. Put𝜏𝑗′ ∶= 𝜏𝑖𝑗𝑠𝑓Θ𝑖𝑗. These new transition functions satisfy the cocycle condition and hence define a rank 2 holomorphic vector bundle defined onℙ2, which we call it the instanton- corrected mirrorof𝕃. Moreover, we have
Theorem 1.2(=Theorem 3.8). The instanton-corrected mirror of the tropical Lagrangian multi-section𝕃is isomorphic to the holomorphic tangent bundle𝑇ℙ2 ofℙ2.
In the last section, Section3.3, we will discuss the relationship between the factors{Θ𝑖𝑗}and the family Floer theory of the (conjecturally exists) mirror La- grangian of𝑇ℙ2. The Fourier modes𝑚𝑖𝑗∈ 𝑀ofΘ𝑖𝑗, determine three cotangent directions of𝑁ℝthe origin. As the Lagrangian𝕃is tropical, we cannot expect one can determine which fibers of𝑝will bound holomorphic disk with the hon- est Lagrangian. Nevertheless,𝑚𝑖𝑗should be regarded as the normal directions ofwalls1emitting from the branched point0 ∈ 𝑁ℝ.
In Appendix A, we will give a detailed review of Fukaya’s local model on caustic points and his construction of mirror bundle via deformation theory [22], Section 6.4. This gives a symplecto-geometric explanation for the walling- crossing phenomenon in our reconstruction process.
Acknowledgment. The author is grateful to Byung-Hee An, Kwokwai Chan, Ziming Nikolas Ma and Yong-Geun Oh for useful discussions. A special thanks goes to Katherine Lo for her encouragement during this work.
2. SYZ mirror symmetry ofℙ𝟐
We begin with reviewing some elementary facts about the complex projective planeℙ2.
Let𝑁 ≅ ℤ2be a lattice of rank2and set
𝑁ℝ ∶= 𝑁 ⊗ℤℝ, 𝑀 ∶= 𝐻𝑜𝑚ℤ(𝑁, ℤ), 𝑀ℝ∶= 𝑀 ⊗ℤℝ.
1A wall is a codimension 1 submanifold𝑊 ⊂ 𝑁ℝfor which there is a non-trivial holomorphic disk bounded by the Lagrangian and those fibers over𝑊.
LetΣbe the fan with primitive generators
𝑣0∶= (1, 1) 𝑣1 ∶= (−1, 0), 𝑣2 ∶= (0, −1),
The associate toric variety𝑋Σ is the complex projective planeℙ2. The dense torus ofℙ2can be identified with𝑇𝑁ℝ∕𝑁. Let𝑝 ∶ 𝑇𝑁̌ ℝ∕𝑁 → 𝑁ℝbe the natu- ral projection. Denote the coordinates on𝑁ℝby𝜉𝑖and the fiber coordinates of
̌
𝑝by ̌𝑦𝑖. Complex coordinates on𝑇𝑁ℝ∕𝑁are given by 𝑤𝑖 ∶= 𝑒𝑧𝑖 ∶= 𝑒𝜉𝑖+
√
−1 ̌𝑦𝑖.
There is an 1-1 correspondence2between supporting functions on|Σ| = 𝑁ℝ and(ℂ×)2-equivariant line bundles onℙ2. Explicitly, the equivariant line bun- dle𝒪(𝑎0𝐷0+𝑎1𝐷1+𝑎2𝐷2)corresponds to the supporting function𝜑 ∶ 𝑁ℝ→ ℝ, defined by setting𝜑(𝑣𝑖) ∶= 𝑎𝑖.
Let
𝜎0∶= ℝ≥0⟨𝑣1, 𝑣2⟩ 𝜎1∶= ℝ≥0⟨𝑣0, 𝑣2⟩, 𝜎2∶= ℝ≥0⟨𝑣0, 𝑣1⟩.
and𝑈𝑘 ≅ ℂ2be the affine chart corresponds to the cone𝜎𝑘, for𝑘 = 0, 1, 2. We can trivialize𝑇ℙ2 on𝑈𝑘 by
𝜏𝑘 ∶ 𝑇ℙ2|𝑈
𝑘 ∋ ([𝜁0∶ 𝜁1∶ 𝜁2], 𝑣) ↦ (𝑤𝑖𝑘, 𝑤𝑘𝑗, 𝑣𝑖𝑘, 𝑣𝑘𝑗) ∈ ℂ2× ℂ2, for𝑖, 𝑗, 𝑘 = 0, 1, 2distinct and𝑖 < 𝑗. Here,
𝑤𝑘𝑖 ∶= 𝜁𝑖
𝜁𝑘 and𝑣 =∶ 𝑣𝑘𝑖 𝜕
𝜕𝑤𝑘𝑖 + 𝑣𝑗𝑘 𝜕
𝜕𝑤𝑘𝑗 . The transition functions𝜏𝑖𝑗 ∶= 𝜏𝑖◦𝜏𝑗−1are given by
𝜏10=
⎛
⎜
⎜
⎝
− 1
(𝑤10)2 0
− 𝑤
2 0
(𝑤10)2 1 𝑤01
⎞
⎟
⎟
⎠ , 𝜏21=
⎛
⎜
⎜
⎝
1 𝑤21 − 𝑤
0 1
(𝑤12)2
0 − 1
(𝑤12)2
⎞
⎟
⎟
⎠ , 𝜏02=
⎛
⎜
⎜
⎝
− 𝑤
1 2
(𝑤20)2 1 𝑤20
− 1
(𝑤20)2 0
⎞
⎟
⎟
⎠ .
Remark 2.1. We can relate the coordinates𝑤𝑖 on𝑇𝑁ℝ∕𝑁 and the inhomoge- neous coordinates𝑤𝑖0by setting𝑤𝑖 = 𝑤0𝑖|(ℂ×)2.
It is well-known thatℙ2carries a Kähler-Einstein metric, called Fubini-Study metric. It is the Hermitian metric associated to the(1, 1)-form
𝜔𝐹𝑆 ∶= 2
√
−1𝜕 ̄𝜕𝜙(𝜉), where
𝜙(𝜉) ∶= 1
2log(1 + 𝑒2𝜉1+ 𝑒2𝜉2).
2We use the convention that if a piecewise linear function𝑓is given by𝑓(𝑣𝑖) = 𝑎𝑖, then the corresponding line bundle is given by𝒪(∑2
𝑖=0𝑎𝑖𝐷𝑖
) .
Let ∇𝐹𝑆 be the Chern connection associated to the the Fubini-Study metric.
With respective to the holomorphic frame{ 𝜕
𝜕𝑤1, 𝜕
𝜕𝑤2}, it can be written as
∇𝐹𝑆|𝑈0 = 𝑑 − 1
1 + |𝑤|2 [(2|𝑤1|2 𝑤1𝑤̄2
0 |𝑤1|2) 𝑑𝑧1+ (|𝑤2|2 0
̄
𝑤1𝑤2 2|𝑤2|2) 𝑑𝑧2] , where|𝑤|2∶= |𝑤1|2+|𝑤2|2. For the purpose of this paper, we need the formula of∇𝐹𝑆|𝑈0in terms of the frame{ 𝜕
𝜕𝑧1, 𝜕
𝜕𝑧2}. We have
∇𝐹𝑆( 𝜕
𝜕𝑧𝑖) = ∇𝐹𝑆(𝑤𝑖 𝜕
𝜕𝑤𝑖) = 𝑑𝑧𝑖⊗ 𝜕
𝜕𝑧𝑖 + 𝑤𝑖∇𝐹𝑆( 𝜕
𝜕𝑤𝑖) . Hence,∇𝐹𝑆|𝑈0 is equal to𝑑minus
1
1 + |𝑤|2[(|𝑤1|2− |𝑤2|2− 1 |𝑤2|2
0 |𝑤1|2) 𝑑𝑧1+ (|𝑤2|2 0
|𝑤1|2 |𝑤2|2− |𝑤1|2− 1) 𝑑𝑧2] . 2.1. SYZ mirror symmetry ofℙ𝟐. Now, we jump to SYZ mirror symmetry of ℙ2. The mirror ofℙ2is given by the Landau-Ginzburg model(𝑌, 𝑊), where
𝑌 ∶=𝑇∗𝑁ℝ∕𝑀 𝑊(𝑧1, 𝑧2) ∶=𝑧1+ 𝑧2+ 𝑞
𝑧1𝑧2,
and𝑞 > 0is a positive constant. The complex coordinates𝑧𝑗are given by 𝑧𝑖 ∶= 𝑒𝑥𝑖+
√
−1𝑦𝑖,
where𝑥𝑖is the affine coordinates on𝑃̊and𝑦𝑖are the fiber coordinates. One can equip𝑌with the standard symplectic structure
𝜔𝑌 ∶= 𝑑𝜉1∧ 𝑑𝑦1+ 𝑑𝜉2∧ 𝑑𝑦2, and the holomorphic volume form
Ω𝑌 ∶= 𝑑𝑧1 𝑧1 ∧ 𝑑𝑧2
𝑧2 .
Let 𝑝 ∶ 𝑌 → 𝑁ℝ be the natural projection, which is clearly dual to𝑝 ∶̌ 𝑇𝑁ℝ∕𝑁 → 𝑁ℝ. The homological mirror symmetry conjecture predicts that Lagrangian branes in𝑌should be mirror to coherent shaves inℙ2. In [8], Chan consider Lagrangian sections of𝑝 ∶ 𝑌 → 𝑁ℝwith certain decay conditions at infinity and define its SYZ mirror line bundle onℙ2. Roughly speaking, given such a Lagrangian section𝐿, one can associate a line bundle ̌𝐿 → ℙ2together with a connection∇̌𝐿, so that with respective to a local unitary framě1, one has
∇̌𝐿 ∶= 𝑑 −
√
−1(
𝑓1(𝜉)𝑑 ̌𝑦1+ 𝑓2(𝜉)𝑑 ̌𝑦2) .
where(𝑓1, 𝑓2)are local defining equations of𝐿. In fact, every such connec- tion is compatible with the metric𝑒−2𝐹, where𝐹 is a local potential function of the Lagrangian section𝐿. In terms of the local holomorphic frame𝑒−𝐹̌1, the connection becomes
𝑑 −(
𝑓1(𝜉)𝑑𝑧1+ 𝑓2(𝜉)𝑑𝑧2) .
As an example, by using the potential function𝜙of𝜔𝐹𝑆, for each𝑘 ∈ ℤ, we can define the Lagrangian section
𝐿𝑘 ∶= {(𝜉, 𝑘 ⋅ 𝑑𝜙(𝜉)) ∈ 𝑌 ∶ 𝜉 ∈ 𝑁ℝ}.
The mirror bundle of𝐿𝑘is the line bundle𝒪(𝑘)together with the connection
∇𝑘 = 𝑑 − ( 𝑘|𝑤1|2
1 + |𝑤|2𝑑𝑧1+ 𝑘|𝑤2|2 1 + |𝑤|2𝑑𝑧2) .
Note that𝑘⋅𝜙is a smoothing of the supporting function corresponds to𝒪(𝑘𝐷0) ≅ 𝒪ℙ2(𝑘)as
𝑡→∞lim 𝑘
2log𝑡(1 + 𝑡2𝜉1 + 𝑡2𝜉2) = max{0, 𝑘𝜉1, 𝑘𝜉2}.
Thus the differential of supporting functions should be regarded as singular Lagrangian sections of𝑝 ∶ 𝑌 → 𝑁ℝ.
3. A tropical Lagrangian multi-section associated to𝑻ℙ𝟐 and reconstruction
In this section, we introduce the notion of tropical Lagrangian multi-section and construct one by tropicalizing the Chern connection associated to the Fubini- Study metric. We then perform the reconstruction of𝑇ℙ2 from the tropical La- grangian multi-section. The wall-crossing phenomenon will be discussed in the last subsection.
We now introduce the following
Definition 3.1. Let𝐵be an𝑛-dimensional integral affine manifold without bound- ary. Arank𝑟tropical Lagrangian multi-sectionis a triple𝕃 ∶= (𝐿, 𝜋, 𝜑), where
a) 𝐿is a topological manifold.
b) 𝜋 ∶ 𝐿 → 𝐵a covering map of degree𝑟with branch locus𝑆 ⊂ 𝐵and ramification locus𝑆′⊂ 𝐿being a union of locally closed submanifolds of codimension at least 2.
c) 𝜑 = {𝜑𝑈}is a collection of local continuous functions on𝐿such that on any two affine charts𝑈, 𝑉 ⊂ 𝐿∖𝑆′(with respective to the induced affine structure on𝐿∖𝑆′via𝜋),
𝜑𝑈− 𝜑𝑉 = ⟨𝑚, 𝑥⟩ + 𝑏, for some𝑚 ∈ ℤ𝑛and𝑏 ∈ ℝ.
Definition3.1is a straightforward generalization of the notion of polariza- tion in the famous Gross-Siebert program [23,25,26]. We also remark that the domain𝐿can be disconnected in general. But in this paper,𝐿is connected and the multi-valued function𝜑is a single-valued continuous function.
3.1. Construction of the tropical Lagrangian multi-section. In order to obtain a tropical Lagrangian for𝑇ℙ2, we need to “tropicalize" the Chern con- nection associated to the Fubini-Study metric.
To do this, we construct a family{(𝑋ℏ, 𝜔ℏ)}ℏ>0of Kähler manifolds. Let 𝑃 ∶= {(𝑥1, 𝑥2) ∈ 𝑀ℝ∶ 𝑥1, 𝑥2 ≥ 0, 𝑥1+ 𝑥2≤ 1}
be a moment polytope ofℙ2and𝑃̊ be its interior. Let 𝑔ℏ∶= 𝑔𝑃+ ℏ−1𝜓.
where
𝑔𝑃(𝑥) ∶=1
2(𝑥1log(𝑥1) + 𝑥2log(𝑥2) + (1 − 𝑥1− 𝑥2) log(1 − 𝑥1− 𝑥2)) , 𝜓(𝑥) ∶=𝑥21+ 𝑥22+ 𝑥1𝑥2− 𝑥1− 𝑥2.
The Legendre dual coordinates are denoted by 𝜉ℏ𝑖 ∶= 𝜕𝑔ℏ
𝜕𝑥𝑖. They are related to the original coordinates via
𝜉ℏ𝑖 = 𝜉𝑖+ ℏ−1𝜕𝜓
𝜕𝑥𝑖. Put
𝑤𝑖ℏ∶= 𝑒𝑧𝑖ℏ ∶= 𝑒𝜉ℏ𝑖+
√
−1 ̌𝑦𝑖, 𝑖 = 1, 2.
A straightforward calculation shows that 𝐻𝑒𝑠𝑠(𝑔𝑃+ ℏ−1𝜓) >0,
det(𝐻𝑒𝑠𝑠(𝑔𝑃+ ℏ−1𝜓)) = 1
𝛼ℏ(𝑥)𝑥1𝑥2(1 − 𝑥1− 𝑥2),
for some smooth function𝛼ℏ∶ 𝑃 → ℝso that if we chooseℏ > 0small enough, 𝛼ℏ(𝑥) > 0, for all𝑥 ∈ ̊𝑃. These are precisely the compatibility conditions stated in [4,5], which guarantee the complex coordinates𝑤ℏ𝑖 can be extended to𝑈0 ⊂ ℙ2. Thus, we get a family of complex manifolds{𝑋ℏ}ℏ>0. Each member of this family can be identified withℙ2via
𝜄ℏ∶ (𝑤1ℏ, 𝑤2ℏ) ↦ [1 ∶ 𝑤ℏ1∶ 𝑤ℏ2].
We define
𝜔ℏ∶= 𝜄∗ℏ𝜔𝐹𝑆
and the associated connection by∇ℏ, which is given by
∇ℏ= 𝑑 − 1
1 + |𝑤ℏ|2[(|𝑤ℏ1|2− |𝑤ℏ2|2− 1 |𝑤2ℏ|2 0 |𝑤1ℏ|2) 𝑑𝑧1ℏ + (|𝑤ℏ2|2 0
|𝑤ℏ1|2 |𝑤2ℏ|2− |𝑤1ℏ|2− 1) 𝑑𝑧2ℏ] ,
under the frame{ 𝜕
𝜕𝑧𝑖
ℏ
}and the coordinates{𝑧𝑖ℏ}. We want to compute limℏ→0∇ℏ𝜕
𝜕𝑧𝑖ℏ
𝜕
𝜕𝑧𝑗ℏ
= lim
ℏ→0
∑2 𝑘=1
Γ𝑘𝑖𝑗(ℏ) 𝜕
𝜕𝑧𝑘ℏ. First of all, it is clear that
𝜕
𝜕𝑧ℏ𝑖 = 1 2
⎛
⎜
⎝
∑2 𝑗=1
𝐻𝑒𝑠𝑠(ℏ−1𝜓 + 𝑔𝑃)𝑖𝑗 𝜕
𝜕𝑥𝑗 −
√
−1 𝜕
𝜕 ̌𝑦𝑗
⎞
⎟
⎠
→ −
√
−1 2
𝜕
𝜕 ̌𝑦𝑖. at every point asℏ → 0. Put𝑔𝑖 ∶= 𝜕𝑔𝑃
𝜕𝑥𝑖 and𝜓𝑖 ∶= 𝜕𝜓
𝜕𝑥𝑖. Note that 𝜓1= 2𝑥1+ 𝑥2− 1,
𝜓2= 𝑥1+ 2𝑥2− 1 and𝜓1− 𝜓2= 𝑥1− 𝑥2. To computelim
ℏ→0Γ𝑘𝑖𝑗(ℏ), we decompose𝑃into three pieces 𝑃0∶= 𝑃 ∩ {𝜓1≤ 0, 𝜓2≤ 0},
𝑃1∶= 𝑃 ∩ {𝜓1≥ 0, 𝜓1≥ 𝜓2}, 𝑃2∶= 𝑃 ∩ {𝜓2≥ 0, 𝜓2≥ 𝜓1}.
For𝑥 ∈ ̊𝑃0, we have
ℏ→0lim 1
1 + |𝑤ℏ|2 = lim
ℏ→0
1
1 + 𝑒2𝑔1𝑒2ℏ−1𝜓1+ 𝑒2𝑔2𝑒2ℏ−1𝜓2 = 1
ℏ→0lim
|𝑤1ℏ|2
1 + |𝑤ℏ|2 = lim
ℏ→0
𝑒2𝑔1𝑒2ℏ−1𝜓1
1 + 𝑒2𝑔1𝑒2ℏ−1𝜓1+ 𝑒2𝑔2𝑒2ℏ−1𝜓2 = 0,
ℏ→0lim
|𝑤2ℏ|2
1 + |𝑤ℏ|2 = lim
ℏ→0
𝑒2𝑔2𝑒2ℏ−1𝜓2
1 + 𝑒2𝑔1𝑒2ℏ−1𝜓1+ 𝑒2𝑔2𝑒2ℏ−1𝜓2 = 0, as𝜓1, 𝜓2< 0. For𝑥 ∈ ̊𝑃1, we have
limℏ→0
1
1 + |𝑤ℏ|2 = lim
ℏ→0
𝑒−2ℏ−1𝜓1
𝑒−2ℏ−1𝜓1+ 𝑒2𝑔1 + 𝑒2𝑔2𝑒2ℏ−1(𝜓2−𝜓1) = 0, limℏ→0
|𝑤ℏ1|2
1 + |𝑤ℏ|2 = lim
ℏ→0
𝑒2𝑔1
𝑒−2ℏ−1𝜓1+ 𝑒2𝑔1 + 𝑒2𝑔2𝑒2ℏ−1(𝜓2−𝜓1) = 1, limℏ→0
|𝑤ℏ2|2
1 + |𝑤ℏ|2 = lim
ℏ→0
𝑒2𝑔2
𝑒−2ℏ−1𝜓2+ 𝑒2𝑔1𝑒2ℏ−1(𝜓1−𝜓2)+ 𝑒2𝑔2 = 0.
Similarly, we have, for𝑥 ∈ ̊𝑃2, limℏ→0
1
1 + |𝑤ℏ|2 =0 limℏ→0
|𝑤1ℏ|2 1 + |𝑤ℏ|2 =0,
limℏ→0
|𝑤2ℏ|2 1 + |𝑤ℏ|2 =1.
The potential function 𝜙of the Fubini-Study metric defines an isomorphism 𝑑𝜙 ∶ 𝑁ℝ→ ̊𝑃, which is known as theLegendre transform. Since𝑑𝜙maps ̊𝜎𝑖 to 𝑃̊𝑖, we have a singular connection∇𝑡𝑟𝑜𝑝𝐹𝑆 on the total space𝑇𝑁ℝ, define by
∇𝑡𝑟𝑜𝑝𝐹𝑆 ∶=
⎧
⎪⎪
⎨⎪
⎪
⎩ 𝑑 −√
−1 (−1 0
0 0) 𝑑 ̌𝑦1−√
−1 (0 0
0 −1) 𝑑 ̌𝑦2 if𝜉 ∈ ̊𝜎0, 𝑑 −√
−1 (1 0
0 1) 𝑑 ̌𝑦1−√
−1 (0 0
1 −1) 𝑑 ̌𝑦2 if𝜉 ∈ ̊𝜎1, 𝑑 −√
−1 (−1 1
0 0) 𝑑 ̌𝑦1−√
−1 (1 0
0 1) 𝑑 ̌𝑦2 if𝜉 ∈ ̊𝜎2, with respective to the frame{√
−1 𝜕
𝜕 ̌𝑦𝑖}. we can diagonalize the two non-diagonal matrices by
( 1 0
−1 1) (0 0
1 −1) (1 0
1 1) = (0 0 0 −1) , (1 −1
0 1 ) (−1 1 0 0) (1 1
0 1) = (−1 0 0 0) .
These amount to a gauge transform of∇𝑡𝑟𝑜𝑝𝐹𝑆 . Hence with respective to the new frame,
∇𝑡𝑟𝑜𝑝𝐹𝑆 =
⎧
⎪⎪
⎨⎪
⎪
⎩ 𝑑 −
√
−1 (−1 0
0 0) 𝑑 ̌𝑦1−
√
−1 (0 0
0 −1) 𝑑 ̌𝑦2 if𝜉 ∈ ̊𝜎0, 𝑑 −√
−1 (1 0
0 1) 𝑑 ̌𝑦1−√
−1 (0 0
0 −1) 𝑑 ̌𝑦2 if𝜉 ∈ ̊𝜎1, 𝑑 −√
−1 (−1 0
0 0) 𝑑 ̌𝑦1−√
−1 (1 0
0 1) 𝑑 ̌𝑦2 if𝜉 ∈ ̊𝜎2, (1)
Remark 3.2. It is straightforward to show that the above gauge change is the ℏ → 0limit of the (non-holomorphic) change of frame
𝐻ℏ∶
⎧
⎨
⎩
𝜕
𝜕𝑧1ℏ ↦ 𝜕
𝜕𝑧1ℏ − |𝑤
1 ℏ|2 1+|𝑤ℏ|2
𝜕
𝜕𝑧ℏ2,
𝜕
𝜕𝑧2ℏ ↦ 𝜕
𝜕𝑧2ℏ − |𝑤
2 ℏ|2 1+|𝑤ℏ|2
𝜕
𝜕𝑧ℏ1.
One can easily check that𝑑𝐻ℏ⋅ 𝐻ℏ−1→ 0asℏ → 0and obtain (1).
Now we construct a tropical Lagrangian multi-section from∇𝑡𝑟𝑜𝑝𝐹𝑆 . For each cone𝜎𝑘, we let𝜎±𝑘 be two copies of𝜎𝑘. Define six linear functions𝜑±𝑘 ∶ 𝜎𝑘±→ ℝ by
𝜑−0 ∶ (𝜉1, 𝜉2) ↦ −𝜉1, 𝜑+1 ∶ (𝜉1, 𝜉2) ↦ 𝜉1,
𝜑−2 ∶ (𝜉1, 𝜉2) ↦ 𝜉2. 𝜑+0 ∶ (𝜉1, 𝜉2) ↦ −𝜉2, 𝜑−1 ∶ (𝜉1, 𝜉2) ↦ 𝜉1− 𝜉2, 𝜑+2 ∶ (𝜉1, 𝜉2) ↦ 𝜉2− 𝜉1
We obtain a topological space𝐿by gluing𝜎±0 with𝜎∓1 and𝜎2∓along𝑣1and𝑣2, respectively, and glue𝜎±1 with𝜎∓2 along𝑣0. The topological space𝐿is homeo- morphic to𝑁ℝand by choosing a branch cut, say along𝑣0, it is easy to see that the projection map𝜋 ∶ 𝐿 → |Σ| ≅ 𝑁ℝgiven by mapping𝜎±𝑘 → 𝜎𝑘can be iden- tified with the square map𝑧 ↦ 𝑧2onℂ. Moreover,{𝜑±𝑘}glue to a continuous piecewise linear function𝜑on𝐿. See Figure1. If we take the “trace" of𝜑, we obtain a piecewise linear function that defines the line bundle𝒪(3), which is of course isomorphic todet(𝑇ℙ2)as a holomorphic line bundle.
Figure 1. The tropical Lagrangian𝕃. In summary, we obtain
Proposition 3.3. The data𝕃 ∶= (𝐿, 𝜋, 𝜑)defines a tropical Lagrangian multi- section.
Remark 3.4. In [32], instead of Hermitian structure, Payne used equivariant structure of𝑇ℙ2 to construct the same tropical Lagrangian multi-section.
Remark 3.5. One should think of the topological space𝐿and the differential of the function𝜑as the tropical limit of certain rank 2 Lagrangian multi-section of 𝑝 ∶ 𝑌 → 𝑁ℝ that is mirror to𝑇ℙ2. If we formally apply the SYZ transform to the Lagrangian multi-section𝑑𝜑 ∶ 𝐿 → 𝑌, we obtain (1). See[15]for precise definition of SYZ transform of unbranched Lagrangian multi-sections.
3.2. Reconstructing𝑻ℙ𝟐. As in Remark3.5, the function𝜑should be thought of as the potential function of a Lagrangian multi-section and thus “𝑑𝜑" is the Lagrangian itself. For each𝜎±𝑘, letℒ±𝑘 be the trivial line bundle𝒪𝑈𝑘. Put
ℰ0∶= ℒ−0 ⊕ ℒ+0,
ℰ1∶= ℒ+1 ⊕ ℒ−1, ℰ2∶= ℒ−2 ⊕ ℒ+2.
Let𝑒±𝑘 be a global frame ofℒ±𝑘. They give a natural ordered frame forℰ𝑘. Ac- cording to the gluing of(𝐿, 𝜑), we should glue these ordered frame as follows
(𝑒−0, 𝑒+0) ↔ (𝑒+1, 𝑒−1), (𝑒+1, 𝑒−1) ↔ (𝑒−2, 𝑒+1), (𝑒−2, 𝑒+2) ↔ (𝑒+0, 𝑒−0).
The composition of these maps is the monodromy𝑒±0 ↦ 𝑒∓0 of the unbranced covering map𝜋|𝐿∖𝜋−1(0) ∶ 𝐿∖𝜋−1(0) → 𝑁ℝ∖{0}. In view of the SYZ transform defined in [15], we need to weight each gluing by the exponential of the differ- ence of local potential functions, which is an affine function. Motivated by this, we should look at the monomial associated to the minus of the slope difference in our case. This gives the following set of naive transition functions
𝜏𝑠𝑓10 ∶=⎛
⎜
⎝
𝑎0 (𝑤01)2 0
0 𝑏0
𝑤01
⎞
⎟
⎠
, 𝜏𝑠𝑓21 ∶=⎛
⎜
⎝
𝑏1 𝑤12 0
0 𝑎1
(𝑤12)2
⎞
⎟
⎠
, 𝜏𝑠𝑓02 ∶=⎛
⎜
⎝
0 𝑏2
𝑤02 𝑎2 (𝑤02)2 0
⎞
⎟
⎠ ,
for some constants𝑎𝑖, 𝑏𝑖 ∈ ℂ×. Since𝑡𝑟(𝜑)is a piecewise linear function defin- ing𝒪(3), we should impose the condition
∏2 𝑖=0
𝑎𝑖𝑏𝑖 = −1.
This condition can be regarded as a rank 1 system on 𝐿∖𝜋−1(0) with mon- odromy−1as follows. For each maximal cone𝜎±𝑖 ⊂ 𝐿, choose a neighborhood 𝑉𝑖± ⊂ 𝐿∖𝜋−1(0)of 𝜎±𝑖 ∖𝜋−1(0)such that𝑉𝑖± ∩ 𝑉𝑗± is non-empty if and only if 𝜎𝑖±∩ 𝜎𝑗±≠ 𝜋−1(0). Hence, by our convention, only𝑉𝑖+∩ 𝑉𝑗−are non-empty, for 𝑖, 𝑗distinct. Define a rank 1 local systemℒon𝐿∖𝜋−1(0)by setting its transition functions to be
𝑎0on𝑉−0 ∩ 𝑉1+, 𝑎1on𝑉+1 ∩ 𝑉2−, 𝑎2on𝑉−2 ∩ 𝑉0+, 𝑏0on𝑉+0 ∩ 𝑉1−, 𝑏1on𝑉−1 ∩ 𝑉2+, 𝑏2on𝑉+2 ∩ 𝑉0−. The condition∏2
𝑖=0𝑎𝑖𝑏𝑖 = −1simply says thatℒhas monodromy−1. So the constants in the naive transition functions are coupled with the local system data. However, it is clear that{𝜏𝑠𝑓𝑖𝑗}doesn’t satisfy the cocycle condition. This
is due to the non-trivial affine monodromy of the branched covering map𝜋 ∶ 𝐿 → 𝑁ℝ.
Remark 3.6. Although{𝜏𝑖𝑗𝑠𝑓}does not form a vector vector bundle onℙ2, they do form a rank 2 bundleℰ𝑠𝑓on the singular space𝐷0∪ 𝐷1∪ 𝐷2because each divisor 𝐷𝑘 is covered by the two charts𝑈𝑖∩ 𝐷𝑘, 𝑈𝑗∩ 𝐷𝑘for𝑖, 𝑗, 𝑘 = 0, 1, 2begin distinct and there are no triple intersections, so the cocycle condition is vacuous. Since the torus(ℂ×)2⊂ ℙ2corresponds to the point0 ∈ 𝑁ℝ, which is exactly the branched point of𝜋 ∶ 𝐿 → 𝑁ℝ, from the SYZ transform perspective,ℰ𝑠𝑓should be regarded as the mirror bundle of𝕃0 ∶= (𝐿∖𝜋−1(0), 𝜋|𝐿∖𝜋−1(0), 𝜑|𝐿∖𝜋−1(0))and thus deserve the namesemi-flat mirror bundle of𝕃.
In order to obtain a consistent gluing, we have to modify each𝜏𝑠𝑓𝑖𝑗 by an in- vertible factor. We choose the correction factors to be
Θ10 ∶=𝐼 +
⎛
⎜
⎝
0 0
−𝑎0𝑏1𝑎2𝑤
2 0
𝑤10 0
⎞
⎟
⎠
∈ 𝐴𝑢𝑡( ℰ0|𝑈10)
,
Θ21 ∶=𝐼 +⎛
⎜
⎝
0 −𝑎0𝑎1𝑏2𝑤
0 1
𝑤21
0 0
⎞
⎟
⎠
∈ 𝐴𝑢𝑡( ℰ1|𝑈
21
),
Θ02 ∶=𝐼 +⎛
⎜
⎝
0 0
−𝑏0𝑎1𝑎2𝑤
1 2
𝑤02 0
⎞
⎟
⎠
∈ 𝐴𝑢𝑡( ℰ2|𝑈02)
,
For each𝑗 = 0, 1, 2, the factorΘ𝑖𝑗is written in terms of the frame on𝑈𝑗and is only defined on𝑈𝑖𝑗. Define𝜏𝑖𝑗′ ∶= 𝜏𝑖𝑗𝑠𝑓Θ𝑖𝑗. A straightforward calculation shows that
Proposition 3.7. With the condition∏
𝑖𝑎𝑖𝑏𝑖 = −1, we have
𝜏02′ 𝜏′21𝜏′10= 𝐼. (2) Thus we obtain a rank 2 holomorphic vector bundleℰonℙ2, which we called theinstanton-corrected mirrorof the tropical Lagrangian multi-section𝕃. Fur- thermore, we have
Theorem 3.8. The instanton-corrected mirror of the tropical Lagrangian multi- section𝕃is isomorphic to the holomorphic tangent bundle𝑇ℙ2ofℙ2.
Proof. We define𝑓 ∶ 𝑇ℙ2 → ℰby
𝑓|𝑈0 ∶= 𝑓0∶= (1 0 0 𝑎0𝑏1𝑎2) , 𝑓|𝑈1 ∶= 𝑓1∶= (−𝑎0 0
0 −𝑎1−1𝑏−12 ) , 𝑓|𝑈
2 ∶= 𝑓2∶= (−𝑎0𝑏1 0
0 𝑏2−1) .
Using∏
𝑖𝑎𝑖𝑏𝑖 = −1, one can check that
𝜏′02𝑓2= 𝑓0𝜏02, 𝜏′21𝑓1= 𝑓2𝜏21, 𝜏′10𝑓0= 𝑓1𝜏10.
Hence,𝑓defines an isomorphism.
Remark 3.9. As pointed out by Fukaya [22], Section 6.4, the local system ℒ (which he called an orientation twist) is related to the orientation of certain mod- uli space of holomorphic disks.
3.3. The wall-crossing factors. In Section3.2, we have introduced three in- vertible factors
Θ10 ∶=𝐼 +⎛
⎜
⎝
0 0
−𝑎0𝑏1𝑎2𝑤
2 0
𝑤10 0
⎞
⎟
⎠
∈ 𝐴𝑢𝑡( ℰ0|𝑈
10
),
Θ21 ∶=𝐼 +⎛
⎜
⎝
0 −𝑎0𝑎1𝑏2𝑤
0 1
𝑤21
0 0
⎞
⎟
⎠
∈ 𝐴𝑢𝑡( ℰ1|𝑈21)
,
Θ02 ∶=𝐼 +
⎛
⎜
⎝
0 0
−𝑏0𝑎1𝑎2𝑤
1 2
𝑤02 0
⎞
⎟
⎠
∈ 𝐴𝑢𝑡( ℰ2|𝑈02)
,
to modify the naive transition functions𝜏𝑖𝑗𝑠𝑓. In this section, we give a heuris- tic explanation about how Θ𝑖𝑗 are related to holomorphic disks bounded by the (conjecturally exists) mirror Lagrangian of𝑇ℙ2. In terms of the coordinates 𝑤𝑖 = 𝑤𝑖0, each factorΘ𝑖𝑗determines aFourier mode3𝑚𝑖𝑗∈ 𝑀, where
𝑚10∶= (1, −1), 𝑚21 ∶= (0, 1), 𝑚02∶= (−1, 0).
Let𝑛𝑖𝑗 ∈ 𝑁be the primitive integral tangent vector so that if we identify𝑁ℝ, 𝑀ℝ withℝ2and the natural pairing⟨−, −⟩with the standard inner product onℝ2, {𝑛𝑖𝑗, 𝑚𝑖𝑗}forms an orientable orthonormal basis with respective to the standard volume form𝑑𝑥 ∧ 𝑑𝑦onℝ2. Then we have
𝑛10∶= (−1, −1) 𝑛21 ∶= (1, 0), 𝑛02= (0, 1), . See Figure2.
3By our convention, the Fourier mode of𝑒(𝑚,𝑧)is−𝑚 ∈ 𝑀
Figure 2
Hence, the cocycle condition (2) can be understood as the wall-crossing dia- gram as shown in Figure2. Furthermore, in view of [27], the walls of the mir- ror Lagrangian should concentrate on a small neighborhood of⋃
𝑖≠𝑗ℝ≥0⟨𝑛𝑖𝑗⟩ asℏ → 0.
The closed string wall-crossing phenomenon has been studied in [29,24,12].
The wall-crossing factors are elements of the so called tropical vertex group. They are responsible for correcting the semi-flat complex structure by Maslov index 0 disks bounded by those SYZ fibers over a wall. In our case, which is an open theory, the factors{Θ𝑖𝑗}are responsible for correcting the “semi-flat bun- dle"{𝜏𝑖𝑗𝑠𝑓}by non-trivial holomorphic disks bounded by SYZ fibers over walls and the mirror Lagrangian of𝑇ℙ2.
We end this section by stating the following
Conjecture 3.10. There exists a connected rank 2 Lagrangian multi-section𝕃of the Lagrangian torus fibration𝑝 ∶ 𝑇∗𝑁ℝ∕𝑁 → 𝑁ℝso that for any𝜖 > 0, there exists𝛿 > 0such that ifℏ ∈ (0, 𝛿), the𝜖-tubular neighborhood𝑈𝜖of
⋃
𝑖≠𝑗
ℝ≥0⟨𝑛𝑖𝑗⟩
contains the walls of𝕃. Furthermore, as an object in the Fukaya-Seidel category of𝑇∗𝑁ℝ∕𝑁,𝕃is quasi-isomorphic to a cone between the zero section𝐿0and the direct sum𝐿⊕31 .
As the SYZ transform of𝐿0 and𝐿1 is given by the structural sheaf 𝒪 and the line bundle𝒪(1), respectively, this conjecture is nothing but a symplecto- geometric analog of the Euler sequence forℙ2.
Appendix A. Local model for caustics
In this appendix, we give a review on Fukaya’s local model on caustic points [22], Section 6.4.
Let𝐵 ∶= ℂand𝑋 ∶= ℂ2. Equip𝑋with the standard symplectic structure 𝜔 = 𝑑𝑥1∧ 𝑑𝑦1+ 𝑑𝑥2∧ 𝑑𝑦2
and holomorphic volume form
Ω = 𝑑𝑧1∧ 𝑑𝑧2, where𝑧𝑖 = 𝑥𝑖 +
√
−1𝑦𝑖 are the standard complex coordinates on𝑋. After a hyperkähler rotation, we have complex coordinates
𝑥 = 𝑥1+
√
−1𝑥2, 𝑦 = 𝑦1−
√
−1𝑦2. Fukaya considered the Lagrangian
𝐿 ∶= {(𝑥2, ̄𝑥)|𝑥 ∈ 𝐵} ⊂ 𝑋
With respective to𝑝 ∶ 𝑋 → 𝐵, the projection onto the first coordinate,𝐿is a special Lagrangian multi-section of rank 2. Parameterizing𝐿in terms of polar coordinates:
𝐿 = {(𝑟𝑒
√
−1𝜃,√ 𝑟𝑒−
√
−1𝜃
2) ∈ ℂ2∶ 𝑟 ≥ 0, 𝜃 ∈ ℝ}.
Let𝑢 ∶ [0, 1] × [−1, 1] → 𝑋be given by 𝑢(𝑠, 𝑡) ∶= (𝑠𝑟𝑒
√
−1𝜃, 𝑡√ 𝑠𝑟𝑒−
√
−1𝜃
2).
Define
𝑓𝐿(𝑥) ∶= ∫
1
−1
∫
1
0
𝑢∗𝜔.
An elementary calculation shows that 𝑓𝐿(𝑥) = 4
3𝑟
3
2 cos (3𝜃 2 ) .
Proposition A.1. There are precisely three gradient flow lines of𝑓𝐿starting from the origin.
Proof. In polar coordinates, we have
∇𝑓𝐿= 2√
𝑟 cos (3𝜃 2 ) 𝜕
𝜕𝑟 −2ℏ−1
√𝑟
sin (3𝜃 2 ) 𝜕
𝜕𝜃. Then the gradient flow equation
( ̇𝑟, ̇𝜃) = ∇𝑓𝐿(𝑟, 𝜃) has solution given by
𝑟
2
3 sin (3𝜃 2 ) = 𝐶,
where𝐶is a real constant. If the gradient flow lines start from the origin, then we have𝐶 = 0. Hence, the gradient flow lines are precisely those straight lines along the directions
𝜃 = 0, 𝜃 = 2𝜋
3 , 𝜃 = 4𝜋 3 .
The three gradient flow lines emitting from the origin of𝐵, namely, (0,1
2) ∋ 𝑡 ↦ 𝑡𝑒
√
−1𝜃, with𝜃 = 0,2𝜋 3 ,4𝜋
3 ,
should correspond to three holomorphic disks bounded by𝐿and those fibers of𝑝 ∶ 𝑋 → 𝐵supported on these rays.
Let𝑝 ∶ ̌̌ 𝑋 → 𝐵be the dual fibration of𝑝 ∶ 𝑋 → 𝐵and define 𝐵>0∶=𝐵∖{𝑥 ∈ ℂ ∶ 𝑥1≤ 0},
𝐵>0∶=𝐵∖{𝑥 ∈ ℂ ∶ 𝑥1≥ 0}, 𝑋̌>0∶= ̌𝑝−1(𝐵>0),
𝑋̌<0∶= ̌𝑝−1(𝐵<0).
Equip𝐿0∶= 𝐿∖{0}with a local systemℒwith holonomy−1. The mirror bundle ℰ0of(𝐿∖{0}, ℒ)has local holomorphic frame ̌𝑒1, ̌𝑒2on𝑋̌>0. More precisely,
̌𝑒𝑗 = 𝑒−
2𝜋
ℏ𝑓𝑖̌1𝑖, 𝑖 = 1, 2,
The monodromy action around the fiber{(0, 0)} × 𝑇2is given by
̌𝑒1↦ ̌𝑒2,
̌𝑒2↦ − ̌𝑒1.
Let ̄𝜕0be the Dolbeault operator ofℰ0. We need to extend this complex struc- ture to the whole space𝑋. Let’s delete a small disk𝐷 around the origin of𝐵. Let𝛿 > 0be small and𝔟𝛿be a 1-from onℝ, supported on[−𝛿, 𝛿]and∫ℝ𝔟𝛿 = 1. Define three elements in𝐴0,1( ̌𝑋>0, 𝐸𝑛𝑑(ℰ0)):
𝐵̌0∶= − 𝐻𝑒𝑣(⟨𝑥, 𝑣0⟩)ℱ(Π∗0𝔟𝛿) ̌𝑒∗1⊗ ̌𝑒2, 𝐵̌1∶=𝐻𝑒𝑣(⟨𝑥, 𝑣1⟩)ℱ(Π∗1𝔟𝛿) ̌𝑒∗2⊗ ̌𝑒1, 𝐵̌2∶=𝐻𝑒𝑣(⟨𝑥, 𝑣1⟩)ℱ(Π∗2𝔟𝛿) ̌𝑒∗2⊗ ̌𝑒1, where𝐻𝑒𝑣 ∶ ℝ → ℝis the Heaviside function:
𝐻𝑒𝑣(𝑥) = { 1 if𝑥 ≥ 0, 0 if𝑥 < 0,
Π𝑗 ∶ ℝ2→ ℝ ⋅ 𝑣𝑗⟂,𝑗 = 0, 1, 2, are the orthogonal projections:
Π𝑗(𝑥) ∶= 𝑥 − ⟨𝑥, 𝑣𝑗⟩𝑣𝑗,
andℱ is the Fourier transform sending𝑑𝑥𝑖 to𝑑 ̄𝑧𝑖. By choosing𝛿 > 0small enough (such a choice of𝛿depends on the radius of the disk𝐷), we may assume 𝐵̌0, ̌𝐵1, ̌𝐵2have disjoint support on𝑝̌−1(𝐵>0∖𝐷). Let
𝐵 ∶= ̌̌ 𝐵0+ ̌𝐵1+ ̌𝐵2 ∈ 𝐴0,1( ̌𝑝−1(𝐵>0∖𝐷),End(ℰ0)).
