Abstract.
We show that the complex pair of pants P∘⊂(ℂ*)nsuperscriptݑƒsuperscriptsuperscriptℂݑ›P^\circ\subset(\mathbbC^{*})^nitalic_P start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ⊂ ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is (ambient) isotopic to a natural polyhedral subcomplex of the product of the two skeleta S×Σ⊂ݒœ×ݒžݑ†ΣݒœݒžS\times\Sigma\subset\mathcalA\times\mathcalCitalic_S × roman_Σ ⊂ caligraphic_A × caligraphic_C of the amoeba ݒœݒœ\mathcalAcaligraphic_A and the coamoeba ݒžݒž\mathcalCcaligraphic_C of P∘superscriptݑƒP^\circitalic_P start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT. We thereby provide the groundwork to be able to isotope the discriminant into codimension 2 in topological SYZ torus fibrations.

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††The research of I.Z. NSF FRG grant DMS-1265228. Simons Collaboration grant A20-0125-001. H.R. was supported by DFG grant RU 1629/4-1. The Department of Mathematics at Universität Hamburg. R. was supported by DFG grant RU 1629/4-1. The Department of Mathematics at Universität Hamburg. was supported by DFG grant RU 1629/4-1 and the Department of Mathematics at Universität Hamburg.

The (n-1)ݑ›1(n-1)( italic_n - 1 )-dimensional pair-of-pants P∘superscriptݑƒP^\circitalic_P start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT is the main building block for many problems in complex and symplectic geometries. Its projection ݒœ⊂ℝnݒœsuperscriptℝݑ›\mathcalA\subset\mathbbR^ncaligraphic_A ⊂ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT under the log|z|ݑ§\log|z|roman_log | italic_z | map is called the amoeba and its projection ݒž⊂ݕ‹nݒžsuperscriptݕ‹ݑ›\mathcalC\subset\mathbbT^ncaligraphic_C ⊂ blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT via the argument map is the coamoeba. Both ݒœݒœ\mathcalAcaligraphic_A and ݒžݒž\mathcalCcaligraphic_C have well known skeleta: the spine S∘⊂ݒœsuperscriptݑ†ݒœS^\circ\subset\mathcalAitalic_S start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ⊂ caligraphic_A, also known as the tropical hyperplane, and Σ⊂ݒžΣݒž\Sigma\subset\mathcalCroman_Σ ⊂ caligraphic_C, whose cover is the boundary of the Ansubscriptݐ´ݑ›A_nitalic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-permutahedron. It is convenient to compactify (ℂ*)nsuperscriptsuperscriptℂݑ›(\mathbbC^{*})^n( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT (and subspaces in it) to the product Δ×ݕ‹nΔsuperscriptݕ‹ݑ›\Delta\times\mathbbT^nroman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, where ΔΔ\Deltaroman_Δ is the standard nݑ›nitalic_n-simplex. We introduce a natural polyhedral complex ݒ«⊂Δ×ݕ‹nݒ«Δsuperscriptݕ‹ݑ›\mathcalP\subset\Delta\times\mathbbT^ncaligraphic_P ⊂ roman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, the ober-tropical pair-of-pants, which is a subcomplex of S×Σݑ†ΣS\times\Sigmaitalic_S × roman_Σ. Its projection to Sݑ†Sitalic_S has equidimensional fibers with the center fiber being ΣΣ\Sigmaroman_Σ. The main result of the paper Theorem 6 states that ݒ«ݒ«\mathcalPcaligraphic_P is (ambient) isotopic to PݑƒPitalic_P (in the PL category).

One can easily visualize an isotopy for the one-dimensional pair of pants. But the simpleness of the n=2ݑ›2n=2italic_n = 2 case is deceptive. To get a hint of how complicated the isotopy problem becomes in higher dimensions, consider a long pentagon (a point in the 3-dimensional pair of pants, n=4ݑ›4n=4italic_n = 4).

This pentagon maps to a point yݑ¦yitalic_y in the face S02:|z0|=|z2|≫|zi|,i≠0,2fragmentssubscriptݑ†02:|subscriptݑ§0||subscriptݑ§2|much-greater-than|subscriptݑ§ݑ–|,i0,2S_02:|z_0|=|z_2|\gg|z_i|,\ i\neq 0,2italic_S start_POSTSUBSCRIPT 02 end_POSTSUBSCRIPT : | italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | = | italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ≫ | italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | , italic_i ≠0 , 2 in the spine Sݑ†Sitalic_S of the amoeba. On the other hand it maps to a point sݑ sitalic_s in the face Σ⟨1,2340⟩subscriptΣ12340\Sigma_\langle 1,2340\rangleroman_Σ start_POSTSUBSCRIPT ⟨ 1 , 2340 ⟩ end_POSTSUBSCRIPT of the skeleton of the coamoeba (the average of the four arguments of z0,z2,z3,z4subscriptݑ§0subscriptݑ§2subscriptݑ§3subscriptݑ§4z_0,z_2,z_3,z_4italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is opposite to the argument of z1subscriptݑ§1z_1italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT). The problem is that, roughly speaking, the two acutest angles do not separate the two longest sides of the pentagon, so that y∈S02ݑ¦subscriptݑ†02y\in S_02italic_y ∈ italic_S start_POSTSUBSCRIPT 02 end_POSTSUBSCRIPT is “very far” from any legitimate strata in the fiber of s∈Σ⟨1,2340⟩ݑ subscriptΣ12340s\in\Sigma_\langle 1,2340\rangleitalic_s ∈ roman_Σ start_POSTSUBSCRIPT ⟨ 1 , 2340 ⟩ end_POSTSUBSCRIPT. (This never happens for triangles because of the larger side lying against larger angle property). So the isotopy cannot be a “small deformation”.

Instead of trying to build an isotopy explicitly (which is an interesting but, perhaps, a rather difficult problem) we build regular cell decompositions of both pairs and show that they are homeomorphic. The cell structures respect the natural stratification of Δ×ݕ‹nΔsuperscriptݕ‹ݑ›\Delta\times\mathbbT^nroman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, so the homeomorphisms will glue well at the boundary. Thus with a little bit of effort the isotopy can be extended to any general affine hypersurface.

The isotopy we provide may be applied to several questions in mirror symmetry. One application we have in mind is the following. Given an integral affine manifold with simple singularities [GS06] we want to build a topological SYZ fibration [SYZ96] with discriminant in codimension 2 (rather than codimension one). So far the only examples are the K3 (with discriminant consisting of 24 points in S2superscriptݑ†2S^2italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT) and the quintic 3-fold [G01], cf. [CBM09, EM19]. In a general case, the idea is roughly to modify the local models of the fibration w=f⊂XΣ×(ℂ*)n-kݑ¤ݑ“subscriptݑ‹Σsuperscriptsuperscriptℂݑ›ݑ˜\w=f\\subset X_\Sigma\times(\mathbbC^{*})^n-k italic_w = italic_f ⊂ italic_X start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT × ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT. Here XΣsubscriptݑ‹ΣX_\Sigmaitalic_X start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT is an affine toric variety with ΣΣ\Sigmaroman_Σ a cone over some kݑ˜kitalic_k-dimensional simplex Δ1subscriptΔ1\Delta_1roman_Δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, w:XΣ→ℂ:ݑ¤→subscriptݑ‹Σℂw:X_\Sigma\to\mathbbCitalic_w : italic_X start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT → blackboard_C is a natural toric map, and f:(ℂ*)n-k→ℂ:ݑ“→superscriptsuperscriptℂݑ›ݑ˜ℂf:(\mathbbC^{*})^n-k\to\mathbbCitalic_f : ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT → blackboard_C is a Laurent polynomial with a prescribed Newton polytope, a simplex Δ2subscriptΔ2\Delta_2roman_Δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The local model has the structure of a Tksuperscriptݑ‡ݑ˜T^kitalic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT-bundle over ℝk×(ℂ*)n-ksuperscriptℝݑ˜superscriptsuperscriptℂݑ›ݑ˜\mathbbR^k\times(\mathbbC^{*})^n-kblackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT × ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT with fibers collapsing over ℋ1×H2subscriptℋ1subscriptݐ»2\mathcalH_1\times H_2caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT where ℋ1subscriptℋ1\mathcalH_1caligraphic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the codimension one skeleton of the normal fan to Δ1subscriptΔ1\Delta_1roman_Δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and H2=f=0subscriptݐ»2ݑ“0H_2=\f=0\italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_f = 0 is a hypersurface in (ℂ*)n-ksuperscriptsuperscriptℂݑ›ݑ˜(\mathbbC^{*})^n-k( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT. One then further projects to the ℝn-ksuperscriptℝݑ›ݑ˜\mathbbR^n-kblackboard_R start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT-factor by the LogLog\operatorpage_seo_titleLogroman_Log map in (ℂ*)n-ksuperscriptsuperscriptℂݑ›ݑ˜(\mathbbC^{*})^n-k( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT. To have the Tnsuperscriptݑ‡ݑ›T^nitalic_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT-fibration w=f→ℝn→ݑ¤ݑ“superscriptℝݑ›\w=f\\to\mathbbR^n italic_w = italic_f → blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT with discriminant in codimension 2, we can use our isotopy to replace the pair H2⊂(ℂ*)n-ksubscriptݐ»2superscriptsuperscriptℂݑ›ݑ˜H_2\subset(\mathbbC^{*})^n-kitalic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊂ ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT with a pair ℋ2⊂(ℂ*)n-ksubscriptℋ2superscriptsuperscriptℂݑ›ݑ˜\mathcalH_2\subset(\mathbbC^{*})^n-kcaligraphic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊂ ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n - italic_k end_POSTSUPERSCRIPT such that now ℋ2subscriptℋ2\mathcalH_2caligraphic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is mapped to the spine of the amoeba of H2subscriptݐ»2H_2italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The details are in our forthcoming paper [RZ20].

Thanks

The authors would like to thank the organizers of the Tropical workshop at Oberwolfach in May 2019 where the second author came up with the new object ݒ«ݒ«\mathcalPcaligraphic_P for the first time (hence the page_seo_title). Our gratitude for hospitality additionally goes to the Mittag-Leffler institute, Institute of the Mathematical Sciences of the Americas and MATRIX in Creswick.

2. Geometry of the complex pair of pants

2.1. Notations

We set n^:=0,…,nassign^ݑ›0…ݑ›\hatn:=\0,\dots,n\over^ start_ARG italic_n end_ARG := 0 , … , italic_n . We will think of (ℂ*)n≅(ℂ*)n+1/ℂ*superscriptsuperscriptℂݑ›superscriptsuperscriptℂݑ›1superscriptℂ(\mathbbC^{*})^n\cong(\mathbbC^{*})^n+1/\mathbbC^{*}( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ≅ ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT / blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT as the product Δ∘×ݕ‹nsuperscriptΔsuperscriptݕ‹ݑ›\Delta^\circ\times\mathbbT^nroman_Δ start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT where Δ∘superscriptΔ\Delta^\circroman_Δ start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT is the interior of the nݑ›nitalic_n-simplex

Δ:=(x0,…,xn)∈ℝn+1:xi≥0,∑xi=1assignΔconditional-setsubscriptݑ¥0…subscriptݑ¥ݑ›superscriptℝݑ›1formulae-sequencesubscriptݑ¥ݑ–0subscriptݑ¥ݑ–1\Delta:=\left\(x_0,\dots,x_n)\in\mathbbR^n+1\ :\ x_i\geq 0,\ \sum x% _i=1\right\roman_Δ := ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT : italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0 , ∑ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1
and ݕ‹n:=(ℝ/2πℤ)n+1/(ℝ/2πℤ)assignsuperscriptݕ‹ݑ›superscriptℝ2ݜ‹ℤݑ›1ℝ2ݜ‹ℤ\mathbbT^n:=(\mathbbR/2\pi\mathbbZ)^n+1/(\mathbbR/2\pi\mathbbZ)blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT := ( blackboard_R / 2 italic_π blackboard_Z ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT / ( blackboard_R / 2 italic_π blackboard_Z ) is the nݑ›nitalic_n-torus with homogeneous coordinates [θ0,…,θn]subscriptݜƒ0…subscriptݜƒݑ›[\theta_0,\dots,\theta_n][ italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ]. It is more natural to work with closed spaces, especially for the gluing purposes, so we will compactly (ℂ*)nsuperscriptsuperscriptℂݑ›(\mathbbC^{*})^n( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT to Δ×ݕ‹nΔsuperscriptݕ‹ݑ›\Delta\times\mathbbT^nroman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT and all subspaces in (ℂ*)nsuperscriptsuperscriptℂݑ›(\mathbbC^{*})^n( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT by taking their closures in Δ×ݕ‹nΔsuperscriptݕ‹ݑ›\Delta\times\mathbbT^nroman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. We will denote by bsdΔbsdΔ\operatorpage_seo_titlebsd\Deltaroman_bsd roman_Δ the first barycentric subdivision of ΔΔ\Deltaroman_Δ. We will also consider the dualizing subdivision dsdΔdsdΔ\operatorpage_seo_titledsd\Deltaroman_dsd roman_Δ which is a coarsening of bsdΔbsdΔ\operatorpage_seo_titlebsd\Deltaroman_bsd roman_Δ by combining all simplices in bsdΔbsdΔ\operatorpage_seo_titlebsd\Deltaroman_bsd roman_Δ from a single interval [I,J]ݐ¼ݐ½[I,J][ italic_I , italic_J ] together. That is, a cell ΔIJsubscriptΔݐ¼ݐ½\Delta_IJroman_Δ start_POSTSUBSCRIPT italic_I italic_J end_POSTSUBSCRIPT in dsdΔdsdΔ\operatorpage_seo_titledsd\Deltaroman_dsd roman_Δ is ΔIJ:=ConvΔ^K:I⊆K⊆J,assignsubscriptΔݐ¼ݐ½Conv:subscript^Δݐ¾ݐ¼ݐ¾ݐ½\Delta_IJ:=\operatorpage_seo_titleConv\\hat\Delta_K\ :\ I\subseteq K\subseteq J\,roman_Δ start_POSTSUBSCRIPT italic_I italic_J end_POSTSUBSCRIPT := roman_Conv over^ start_ARG roman_Δ end_ARG start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT : italic_I ⊆ italic_K ⊆ italic_J , where Δ^Ksubscript^Δݐ¾\hat\Delta_Kover^ start_ARG roman_Δ end_ARG start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT stands for the barycenter of ΔKsubscriptΔݐ¾\Delta_Kroman_Δ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT.

The dualizing subdivision can be applied to any polytope Qݑ„Qitalic_Q where it interpolates between the cone structure of the normal fan in the interior of Qݑ„Qitalic_Q and the original face stratification at the boundary of Qݑ„Qitalic_Q. For a general polytope Qݑ„Qitalic_Q the cells in dsdQdsdݑ„\operatorpage_seo_titledsdQroman_dsd italic_Q need not be polytopes anymore, but it is still a regular CW-complex.

The hypersimplex Δn(2)⊂ΔnsuperscriptΔݑ›2superscriptΔݑ›\Delta^n(2)\subset\Delta^nroman_Δ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 2 ) ⊂ roman_Δ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is obtained from the ordinary simplex by cutting the corners half-way. That is,

Δn(2):=(x0,…,xn)∈ℝn+1:∑xi=2π and 0≤xi≤π.assignsuperscriptΔݑ›2conditional-setsubscriptݑ¥0…subscriptݑ¥ݑ›superscriptℝݑ›1subscriptݑ¥ݑ–2ݜ‹ and 0subscriptݑ¥ݑ–ݜ‹\Delta^n(2):=\(x_0,\dots,x_n)\in\mathbbR^n+1\ :\ \sum x_i=2\pi% \text and 0\leq x_i\leq\pi\.roman_Δ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 2 ) := ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT : ∑ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2 italic_π and 0 ≤ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_π .
We will use 2π=12ݜ‹12\pi=12 italic_π = 1 for the amoeba and 2π=6.28…2ݜ‹6.28…2\pi=6.28\dots2 italic_π = 6.28 … for the coamoeba.

The cyclic polytope Cdsubscriptݐ¶ݑ‘ݑŸC_ditalic_C start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_r ) is the convex hull of the points x(t1),…,x(tr)ݑ¥subscriptݑ¡1…ݑ¥subscriptݑ¡ݑŸx(t_1),\dots,x(t_r)italic_x ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_x ( italic_t start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) in ℝdsuperscriptℝݑ‘\mathbbR^dblackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, where x(t)=(t,t2,…,td)ݑ¥ݑ¡ݑ¡superscriptݑ¡2…superscriptݑ¡ݑ‘x(t)=(t,t^2,\dots,t^d)italic_x ( italic_t ) = ( italic_t , italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , … , italic_t start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) and t1<⋯
2.2. A cell decomposition of the complex pair of pants

The (n-1)ݑ›1(n-1)( italic_n - 1 )-dimensional pair-of-pants P∘superscriptݑƒP^\circitalic_P start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT is the complement of n+1ݑ›1n+1italic_n + 1 generic hyperplanes in ℂℙn-1ℂsuperscriptℙݑ›1\mathbbC\mathbbP^n-1blackboard_C blackboard_P start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT. By an appropriate choice of coordinates we can identify P∘superscriptݑƒP^\circitalic_P start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT with the affine hypersurface in (ℂ*)n+1/ℂ*superscriptsuperscriptℂݑ›1superscriptℂ(\mathbbC^{*})^n+1/\mathbbC^{*}( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT / blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT given in homogeneous coordinates by

z0+z1+⋯+zn=0.subscriptݑ§0subscriptݑ§1⋯subscriptݑ§ݑ›0z_0+z_1+\dots+z_n=0.italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ⋯ + italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 .
We define the compactified pair-of-pants PݑƒPitalic_P to be the closure of P∘superscriptݑƒP^\circitalic_P start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT in Δ×ݕ‹nΔsuperscriptݕ‹ݑ›\Delta\times\mathbbT^nroman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT via the map

μ1×μ2:(ℂ*)n+1/ℂ*→Δ×ݕ‹n,[z0,…,zn]↦(|z0|∑|zi|,…,|zn|∑|zi|;[argz0,…,argzn]):subscriptݜ‡1subscriptݜ‡2formulae-sequence→superscriptsuperscriptℂݑ›1superscriptℂΔsuperscriptݕ‹ݑ›maps-tosubscriptݑ§0…subscriptݑ§ݑ›subscriptݑ§0subscriptݑ§ݑ–…subscriptݑ§ݑ›subscriptݑ§ݑ–subscriptݑ§0…subscriptݑ§ݑ›\mu_1\times\mu_2:(\mathbbC^{*})^n+1/\mathbbC^{*}\to\Delta\times% \mathbbT^n,\quad[z_0,\dots,z_n]\mapsto\left(\fracz_i,\dots,\fracz_n;[\arg z_0,\dots,\arg z_n]\right)italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ( blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT / blackboard_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT → roman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , [ italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] ↦ ( divide start_ARG | italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | end_ARG start_ARG ∑ | italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | end_ARG , … , divide start_ARG | italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | end_ARG start_ARG ∑ | italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | end_ARG ; [ roman_arg italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , roman_arg italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] )
where we abused notation when writing the quotient of ݕ‹n+1superscriptݕ‹ݑ›1\mathbbT^n+1blackboard_T start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT modulo the diagonal ݕ‹1superscriptݕ‹1\mathbbT^1blackboard_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT simply by ݕ‹nsuperscriptݕ‹ݑ›\mathbbT^nblackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT (with coordinates θ1-θ0,…,θn-θ0subscriptݜƒ1subscriptݜƒ0…subscriptݜƒݑ›subscriptݜƒ0\theta_1-\theta_0,...,\theta_n-\theta_0italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT). The closure PݑƒPitalic_P is a manifold with boundary, and it can be thought of as a real oriented blow-up of ℂℙn-1ℂsuperscriptℙݑ›1\mathbbC\mathbbP^n-1blackboard_C blackboard_P start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT along its intersections with the coordinate hyperplanes in ℂℙnℂsuperscriptℙݑ›\mathbbC\mathbbP^nblackboard_C blackboard_P start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT.

Next we review a natural stratification Pσ,JsubscriptݑƒݜŽݐ½\P_\sigma,J\ italic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POSTSUBSCRIPT of PݑƒPitalic_P from [KZ18] induced from a product stratification of Δ×ݕ‹nΔsuperscriptݕ‹ݑ›\Delta\times\mathbbT^nroman_Δ × blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. The stratification ΔJsubscriptΔݐ½\\Delta_J\ roman_Δ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT of the simplex ΔΔ\Deltaroman_Δ is given by its face lattice, page_seo_titlely by the non-empty subsets J⊆n^ݐ½^ݑ›J\subseteq\hatnitalic_J ⊆ over^ start_ARG italic_n end_ARG. The set of the hyperplanes θi=θj,i,j∈n^formulae-sequencesubscriptݜƒݑ–subscriptݜƒݑ—ݑ–ݑ—^ݑ›\theta_i=\theta_j,\ i,j\in\hatnitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_i , italic_j ∈ over^ start_ARG italic_n end_ARG, stratifies the torus ݕ‹nsuperscriptݕ‹ݑ›\mathbbT^nblackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT by cyclic orderings of the points θ0,…,θnsubscriptݜƒ0…subscriptݜƒݑ›\theta_0,\dots,\theta_nitalic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT on the circle. The strata ݕ‹σnsubscriptsuperscriptݕ‹ݑ›ݜŽ\mathbbT^n_\sigmablackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT are labeled by cyclic partitions σ=⟨I1,…,Ik⟩ݜŽsubscriptݐ¼1…subscriptݐ¼ݑ˜\sigma=\langle I_1,\dots,I_k\rangleitalic_σ = ⟨ italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ of the set n^^ݑ›\hatnover^ start_ARG italic_n end_ARG, that is, n^=I1⊔⋯⊔Ik^ݑ›square-unionsubscriptݐ¼1⋯subscriptݐ¼ݑ˜\hatn=I_1\sqcup\dots\sqcup I_kover^ start_ARG italic_n end_ARG = italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊔ ⋯ ⊔ italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and the sets I1,…,Iksubscriptݐ¼1…subscriptݐ¼ݑ˜I_1,\dots,I_kitalic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT are cyclically ordered. The elements within each Iisubscriptݐ¼ݑ–I_iitalic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are not ordered. We call the Iisubscriptݐ¼ݑ–I_iitalic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT the parts of σݜŽ\sigmaitalic_σ and we set |σ|:=kassignݜŽݑ˜|\sigma|:=k| italic_σ | := italic_k. If all parts are 1-element sets then we will write σ=⟨i0,…,in⟩ݜŽsubscriptݑ–0…subscriptݑ–ݑ›\sigma=\langle i_0,\dots,i_n\rangleitalic_σ = ⟨ italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩.

We can view points in Pσ,JsubscriptݑƒݜŽݐ½P_\sigma,Jitalic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POSTSUBSCRIPT as (degenerate) convex polygons in the plane defined up to rigid motions and scaling. The edges represent the complex numbers z0,…,znsubscriptݑ§0…subscriptݑ§ݑ›z_0,\dots,z_nitalic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ordered counterclockwise according to σݜŽ\sigmaitalic_σ. The edges within each Issubscriptݐ¼ݑ I_sitalic_I start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT are parallel and not ordered. The edges of the polygon which are not in Jݐ½Jitalic_J have zero length but their directions are still recorded.

Each ݕ‹σnsubscriptsuperscriptݕ‹ݑ›ݜŽ\mathbbT^n_\sigmablackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT can be thought of as the interior of the simplex

(1) Δσ:=(α1,…,αk)∈ℝk:αi≥0,∑αi=2πassignsubscriptΔݜŽconditional-setsubscriptݛ¼1…subscriptݛ¼ݑ˜superscriptℝݑ˜formulae-sequencesubscriptݛ¼ݑ–0subscriptݛ¼ݑ–2ݜ‹\Delta_\sigma:=\left\(\alpha_1,\dots,\alpha_k)\in\mathbbR^k\ :\ % \alpha_i\geq 0,\ \sum\alpha_i=2\pi\right\roman_Δ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT := ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT : italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0 , ∑ italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2 italic_π
The coordinates αisubscriptݛ¼ݑ–\alpha_iitalic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT play the rôle of exterior angles of the convex polygons. The precise relation between αݛ¼\alphaitalic_α’s and the original arguments θݜƒ\thetaitalic_θ’s is as follows: θi=θjsubscriptݜƒݑ–subscriptݜƒݑ—\theta_i=\theta_jitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT for i,j∈Isݑ–ݑ—subscriptݐ¼ݑ i,j\in I_sitalic_i , italic_j ∈ italic_I start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT and θi+αs=θjsubscriptݜƒݑ–subscriptݛ¼ݑ subscriptݜƒݑ—\theta_i+\alpha_s=\theta_jitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT for i∈Is,j∈Is+1formulae-sequenceݑ–subscriptݐ¼ݑ ݑ—subscriptݐ¼ݑ 1i\in I_s,j\in I_s+1italic_i ∈ italic_I start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , italic_j ∈ italic_I start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT. Here we assume the periodic indexing Is+k=Issubscriptݐ¼ݑ ݑ˜subscriptݐ¼ݑ I_s+k=I_sitalic_I start_POSTSUBSCRIPT italic_s + italic_k end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT.

The inclusion of closed strata Pσ′,J′⊆Pσ,JsubscriptݑƒsuperscriptݜŽ′superscriptݐ½′subscriptݑƒݜŽݐ½P_\sigma^\prime,J^\prime\subseteq P_\sigma,Jitalic_P start_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_J start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⊆ italic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POSTSUBSCRIPT gives a partial order among the pairs: (σ′,J′)⪯(σ,J)precedes-or-equalssuperscriptݜŽ′superscriptݐ½′ݜŽݐ½(\sigma^\prime,J^\prime)\preceq(\sigma,J)( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_J start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⪯ ( italic_σ , italic_J ) if σݜŽ\sigmaitalic_σ is a refinement of σ′superscriptݜŽ′\sigma^\primeitalic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT (we write σ′⪯σprecedes-or-equalssuperscriptݜŽ′ݜŽ\sigma^\prime\preceq\sigmaitalic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ) and J′⊆Jsuperscriptݐ½′ݐ½J^\prime\subseteq Jitalic_J start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_J. To simplify notations we will drop the index Jݐ½Jitalic_J from the subscript if J=n^ݐ½^ݑ›J=\hatnitalic_J = over^ start_ARG italic_n end_ARG.

We have a natural surjection Δσ→ݕ‹σn¯→subscriptΔݜŽ¯subscriptsuperscriptݕ‹ݑ›ݜŽ\Delta_\sigma\rightarrow\overline\mathbbT^n_\sigmaroman_Δ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT → over¯ start_ARG blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_ARG which is bijective away from the vertices (all the vertices map to 0∈ݕ‹n0superscriptݕ‹ݑ›0\in\mathbbT^n0 ∈ blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT). Since Pσ,JsubscriptݑƒݜŽݐ½P_\sigma,Jitalic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POSTSUBSCRIPT does not have any points lying over the vertices of ΔσsubscriptΔݜŽ\Delta_\sigmaroman_Δ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT, we may view Pσ,JsubscriptݑƒݜŽݐ½P_\sigma,Jitalic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POSTSUBSCRIPT to be sitting as Pσ,J⊆ΔJ×Δσ.subscriptݑƒݜŽݐ½subscriptΔݐ½subscriptΔݜŽP_\sigma,J\subseteq\Delta_J\times\Delta_\sigma.italic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POSTSUBSCRIPT ⊆ roman_Δ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT × roman_Δ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT . We set σ0=⟨0,1,…,n⟩subscriptݜŽ001…ݑ›\sigma_0=\langle 0,1,\dots,n\rangleitalic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ⟨ 0 , 1 , … , italic_n ⟩ and Δ0:=Δσ0assignsubscriptΔ0subscriptΔsubscriptݜŽ0\Delta_0:=\Delta_\sigma_0roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := roman_Δ start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT. For J=n^ݐ½^ݑ›J=\hatnitalic_J = over^ start_ARG italic_n end_ARG and σ=σ0ݜŽsubscriptݜŽ0\sigma=\sigma_0italic_σ = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT we denote the corresponding maximal stratum of PݑƒPitalic_P by P0⊆Δ×Δ0subscriptݑƒ0ΔsubscriptΔ0P_0\subseteq\Delta\times\Delta_0italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊆ roman_Δ × roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. All maximal strata are isomorphic to P0subscriptݑƒ0P_0italic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. We say that σݜŽ\sigmaitalic_σ divides Jݐ½Jitalic_J, and write σ|JconditionalݜŽݐ½\sigma|Jitalic_σ | italic_J, if

Jݐ½Jitalic_J contains elements in at least two of the subsets I1,…,Iksubscriptݐ¼1…subscriptݐ¼ݑ˜I_1,\dots,I_kitalic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT of σ=⟨I1,…,Ik⟩ݜŽsubscriptݐ¼1…subscriptݐ¼ݑ˜\sigma=\langle I_1,\dots,I_k\rangleitalic_σ = ⟨ italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩. The face poset of a stratum Pσ,JsubscriptݑƒݜŽݐ½P_\sigma,Jitalic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POSTSUBSCRIPT consists of pairs (σ′,J′)superscriptݜŽ′superscriptݐ½′(\sigma^\prime,J^\prime)( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_J start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) such that σ′⪯σprecedes-or-equalssuperscriptݜŽ′ݜŽ\sigma^\prime\preceq\sigmaitalic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⪯ italic_σ, J′⊆Jsuperscriptݐ½′ݐ½J^\prime\subseteq Jitalic_J start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_J and σ′superscriptݜŽ′\sigma^\primeitalic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT divides J′superscriptݐ½′J^\primeitalic_J start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. This, in particular, means that |σ′|≥2superscriptݜŽ′2|\sigma^\prime|\geq 2| italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≥ 2 and |J′|≥2superscriptݐ½′2|J^\prime|\geq 2| italic_J start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≥ 2. If σݜŽ\sigmaitalic_σ does not divide Jݐ½Jitalic_J the stratum Pσ,JsubscriptݑƒݜŽݐ½P_\sigma,Jitalic_P start_POSTSUBSCRIPT italic_σ , italic_J end_POS

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