LLAGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n PENG ZHOUA
BSTRACT . Let W ( z , · · · , z n ) : ( C ∗ ) n → C be a Laurent polynomial in n variables, and let H be ageneric smooth fiber of W . In [RSTZ] Ruddat-Sibilla-Treumann-Zaslow give a combinatorial recipefor a skeleton for H . In this paper, we show that for a suitable exact symplectic structure on H , theRSTZ-skeleton can be realized as the Liouville Lagrangian skeleton. Let ( M, ω = dλ ) be an exact symplectic manifold, and let X = X λ be the Liouville vector fielddefined by ι X ω = − λ . If ( M, ω, λ, X ) is a Liouville manifold (see [CE, Chapter 11] for definition),then X shrinks M to a compact isotropic (possibly singular) submanifold Λ , called the Liouvilleskeleton . The Liouville skeleton is useful for sympletic topology, since the tubular neighborhoodof the skeleton is symplectomorphic to the original manifold up to rescaling the symplectic form.A large class of Liouville manifolds come from Stein manifolds, e.g. affine hypersurfaces H in ( C ∗ ) n . Given an exhausting psh function ϕ on the Stein manifold, we can define the Liouvillestructure by setting ω = − dd c ϕ and λ = − d c ϕ . In [RSTZ], Ruddat-Sibilla-Treumann-Zaslowgive a combinatorial recipe for a topological skeleton in affine hypersurfaces. The RSTZ-skeletondepends on the Newton polytope Q of the defining polynomial for the hypersurface and a startriangulation of Q .It is conjectured that the RSTZ-skeleton can be realized as a Liouville skeleton for a suitablechoice of Liouville structure on the hypersurface. Here we construct such Liouville structure usingtropicalization. The main idea is contained in the following example:0.1. Example: the pair-of-pants.
Consider the hypersurface H = { x + y = 1 } , x, y ∈ C ∗ . The hypersurface can be identified as C \{ , } , a ’pair-of-pants’. A topological skeleton can beconstructed as following: fix an arbitrarily small positive number (cid:15) , and define the skeleton as Λ = ( {| x | = (cid:15) } ∪ {| y | = (cid:15) } ∪ { x ≥ (cid:15), y ≥ (cid:15) } ) (cid:92) { x + y = 1 } . Thus Λ has the shape of two circles connected by an interval. To realize it as a Lagrangian skeleton,we need to choose an exact symplectic structure. Consider the following function ϕ on ( C ∗ ) andits restriction on the pair-of-pants ϕ ( x, y ) = (log | x | − log (cid:15) ) + (log | y | − log (cid:15) ) . It is easy to check that ϕ is a psh function on ( C ∗ ) , and restricts to be a psh function on anycomplex submanifold of ( C ∗ ) . Geometrically, ϕ is constructed by taking the projection map Log = log | · | : ( C ∗ ) → R Date : October 31, 2018.This work is supported by an IHES Simons Postdoctoral Fellowship as part of the Simons Collaboration on HMS. . a r X i v : . [ m a t h . S G ] O c t PENG ZHOU p (cid:15) F IGURE
1. RSTZ-skeleton (on the left) and its embedding for the pair-of-pants.and then taking Euclidean distance on R to a point ϕ ( z ) = | Log( z ) − p (cid:15) | , p (cid:15) = (log (cid:15), log (cid:15) ) . The hypersurface { x + y = 1 } projects under log | · | to an ’amoeba’ shaped region in R , with threetenacles asymptotic to the three rays (drawn as dashed lines in the figure) { log | x | = 0 , log | y | (cid:28) } , { log | x | (cid:28) , log | y | = 0 } , { log | x | = log | y | (cid:29) } . The Louville flow X λ on H induced by λ = − d c ϕ is the same as the negative gradient flow −∇ ω ( ϕ ) of ϕ with respect to the Kahler metric ω = − dd c ϕ . The critical point of ϕ on H can be identifiedwith the critical points on the amoeba Log( H ) with respect to the distance function to point p (cid:15) .The unstable manifold of −∇ ϕ on H is topologically two circles together with an interval.0.2. Set-up and Summary of results.
To state our main result precisely, we need some notation.Let
M, N be dual lattices of rank n . Let T = R / π Z . For any abelian group G , e.g. G = C ∗ , R , T ,we define M G := M ⊗ Z G and similarly for N G . If we fix a basis of M , then M ∼ = Z n , and M C ∗ ∼ = ( C ∗ ) n , M R ∼ = R n , M T ∼ = T n .Let Q ⊂ N R be a integral convex polytope of full-dimension containing . Let T be a coherentstar triangulation of Q based at , and let ∂ T be the subset of T with simplices not containing asa vertex. Let Σ T be the simplicical fan spanned by the simplices in T . Let A denote the vertices of T , and ∂A that of ∂ T , so that A = ∂A ∪ { } .We fix two functions h : A → R , Θ : A → T, such that h induces the coherent star triangulation of T . Without loss of generality, we let h (0) = 0 , Θ(0) = π. Let (cid:98) h : Q → R denote the convex piecewise linear function on Q extending h .We define conical Lagrangian Λ T , Θ ⊂ M T × N R ∼ = T ∗ M T by Λ T , Θ := (cid:91) τ ∈ ∂ T { θ ∈ M T : (cid:104) α, θ (cid:105) = Θ( α ) for all vertices α ∈ τ } × cone( τ ) (0.1)where we used the pairing (cid:104)− , −(cid:105) : M T × N R → T induced by the canonical pairing between M, N , and cone( τ ) = R ≥ × τ . We also define the generalized RSTZ-skeleton [RSTZ] by Λ ∞T , Θ := (cid:91) τ ∈ ∂ T { θ ∈ M T : (cid:104) α, θ (cid:105) = Θ( α ) for all vertices α ∈ τ } × τ (0.2) The case where Q is not full-dimension can be reduced to this one, by defining N (cid:48) R = span R ( Q ) ⊂ N R , and M R (cid:16) M (cid:48) R . The skeleton for ( M R , N R , Q ) would be that of ( M (cid:48) R , N (cid:48) R , Q ) times ( S k ) where d = dim M R − dim M (cid:48) R . AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Remark 0.1.
The definition of the original RSTZ-skeleton is for Θ | ∂ A = 0 and is living over theboundary of the Newton polytop | ∂ T | ⊂ ∂Q ⊂ N R . In this case Λ T , Θ is first defined by [FLTZ1].We will sometimes identify | ∂ T | with its projection to N ∞ R := ( N R \{ } ) / R > , then and Λ ∞T , Θ ishomeomorphic to the ’end-at-infinity’ of Λ T , Θ , hence the notation.For all large enough β > , we define the tropical polynomial as f β,h, Θ ( z ) = (cid:88) α ∈ A e − i Θ( α ) e − βh ( α ) z α . (0.3)where z α is a monomial on M C ∗ ∼ = ( C ∗ ) n . Let H β,h, Θ := { z ∈ M C ∗ | f R,h, Θ ( z ) = 0 } denote the complex hypersurface defined by f R,h, Θ . Theorem ([RSTZ]) . If Θ | ∂ A = 0 , then the skeleton Λ ∞T , Θ embeds into the hypersurface H R,h, Θ as a strongdeformation retract. We prove the following theorem, for general Θ . Main Theorem.
The hypersurface H R,h, Θ admits a Liouville structure such that its Liouville skeleton ishomeomorphic to Λ ∞T , Θ . Remark 0.2.
The deformation of f β,h, Θ by varying Θ continuously induces equivalences of cat-egories for Fukaya-Seidel category F S ( M C ∗ , f β,h, Θ ) . By realizing the dependence on Θ in La-grangian skeleto Λ T , Θ , we can prove [Z] equivalences of categories among the infinitesimallywrapped Fukaya categories Fuk ( T ∗ M T , Λ T , Θ ) used in [FLTZ1, FLTZ2].0.3. Sketch of Proof.
The idea of the proof is illstrated in the above example, that is, we projectthe hypersurface H β,h, Θ to M R , then use a distance function to a point to induce the psh function ϕ , which in turn induces a Liouville structure on H . However, there are two technical modificationused.(1) The first modification is is to ’straighten the tube’, or called tropical localization by Mikhalkinand Abouzaid [Mi, Ab]. In the defining Laurent polynomial f = f β,h, Θ , not all terms areof equal importance at all points on H = H β,h, Θ . We may drop the irrelevant terms andsimplify the defining equation for H locally.(2) The second modification is to find a convex function ϕ on M R ∼ = R n adapted to the ’tropicalamoeba’ of the hypersurface. Let C =: P is denote the convex polytope for the comple-ment of the tropical amoeba corresponding to vertex . The condition for ϕ is (Definition2.8) (1) ϕ ( λx ) = λ ϕ ( x ) for λ > , x ∈ M R \{ } , and (2) for each face F of P of positivedimension, we want ϕ | F to have a minimum in Int( F ) . See Figure 2 for an example, where | x | fails to be a good potential for case (b). Remark 0.3.
The use of ’non-standard’ Kahler potential ϕ on M C ∗ (non-canonically isomorphic to ( C ∗ ) n ) may be unorthodoxical, but it is natural in some sense. (1) The often used ’standard’ Kahlerpotential (cid:80) i (log | z i | ) on M C ∗ is not standard in the first place, since it depends of the choice ofbasis for M . (2) To identify the RSTZ-skeleton (0.2) that lives in M T × N R with the Liouvilleskeleton (3.1) that lives in M C ∗ ∼ = M T × M R , one needs to identify N R with M R . Here this is doneusing the Legendre transformation induced by ϕ . Equivalently, one fixes a (Finsler) metric onfibers of T M T and identify T M T ∼ = T ∗ M T . We thank Gabe Kerr for the suggestion to consider this freedom of coefficients.
PENG ZHOU (a) (b)F
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2. (a) Tropical amoeba of f = 1 + e − β ( x + y + 1 /xy ) (b) Tropical amoebaof f = 1 + e − β ( x + xy + 1 /xy ) . The function | x | on R is adapted to the amoebapolytope in (a) but not the one in (b), since the minimum of | x | on the top edge lieson the endpoint.0.4. Related works.
The study of skeleton for Liouville (or Weinstein) manifold was motivatedlargely by Homological Mirror Symmetry (HMS). It was Kontsevich’s original proposal, to com-pute the Fukaya category of a Weinstein manifold W by taking global section of a (co)sheaf ofcategories living on the skeleton. Following this approach, the (complex) 1-dimensional case hasbeen studied by [STZ, PS] and [DK]. In general, the category under consideration has two ver-sions, a microlocal sheaf version, and a Floer-theoretic version. The two versions are expected toagree by the on-going work of Ganatra-Pardon-Shende [GPS1, GPS2]. The microlocal sheaf ver-sion, originates from the seminal work of Nadler and Zaslow [NZ, N1], says the infinitesimallywrapped Fukaya category on T ∗ M with asymptotic condition of non-compact Lagrangian givenby a conical Lagrangian Λ , is equivalent to constructible sheaves on M with singular support in Λ . The wrapped Fukaya category also has a microlocal sheaf version, developed by Nadler [N4].The microlocal sheaf category for local Lagrangian singularities has been studied by Nadler. In[N2], Nadler defined a class of ’simple’ singularities, termed ’arboreal singularities’, and provedthat the microlocal sheaf category on arboreal singularity is equivalent to the category of represen-tation of quivers. In [N3], Nadler showed that one can deform an arbitrary Lagrangian singularityto an arboreal one, while keeping the microlocal sheaf category invariant. It is also expected thatsuch arborealization can be induced by a perturbation of Weinstein structure[St, ENS].The skeleton for n dimensional pair-of-pants P n has been studied by Nadler[N4], where ahigher dimensional analog of Figure 1 is constructued. A Σ n +2 -symmetric skeleton for P n is con-structed by Gammage-Nadler [GN], where Σ n +2 is the symmetric group. With the technique oftropical phase variety of Kerr-Zharkov [KeZh], we hope to find other Lagrangian skeleta Λ k for P n − , where k = 1 , · · · , n indicating the number of dominant terms in the defining equation of P n − .In Gammage-Shende [GS], as one ingredient in proving HMS for the toric boundary of a toricvariety, they constructed the Liouville skeleton for the same hypersurface as considered here.However, their results [GS, Theorem 3.4.2] depends on the following hypothesis that, there existssome tropicalization function h : A → R and some identification M ∼ = Z n , such that the tropical amoebapolytope P = { x ∈ M R : (cid:104) x, α (cid:105) ≤ h ( α ) , ∀ α ∈ ∂A } contains as an interior point, and | x | restricts toeach face F of P has a minimum in the interior of F . This hypothesis is true in two-dimension, and canbe verified in certain examples, e.g. mirror to weighted projective spaces. But in general, one doesnot know if it is always true, it would be interesting to find a proof or construct a counter-example.
AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Our approach here does not rely on this hypothesis, which is equivalent to ” | x | is adapted to thetropical amoeba polytope P ” for some choice of h . We avoid this by considering a more flexible choiceof Kahler potentials on than | x | , and our approach works for any choice of h compatible with T .0.5. Outline.
In section 1, we reviewed the tropical localization of Mikhalkin and Abouzaid. Weare careful in picking the cut-off functions such that the inner boundary of the amoeba remainsconvex (Section 1.4). In Section 2, we review how to identify M C ∗ with T ∗ M T by choosing a Kahlerpotential, and we introduce the key concept of Kahler potential adapted to a polytope in Section2.4. Then we state our main theorems in more detail in Section 3.0.6.
Acknowledgements.
I would like to thank my advisor Eric Zaslow for introducing the ideaof Lagrangian skeleton and the problem of finding Lagrangian embedding. The idea of usingLegendre transformation to identify N R and M R was inspired by a talk of Helge Ruddat. I thankVivek Shende and Ben Gammage for the clarification of their work. I also thank Nicol `o Sibilla,Gabe Kerr, David Nadler and Ilia Zharkov for many helpful discussions.1. T ROPICAL G EOMETRY
Triangulation and Amoeba.
We follow [GKZ, Chapter 7] and [Mi] to give background oncoherent triangulations and tropical amoeba.Let A ⊂ N ∼ = Z n and Q = conv( A ) its convex hull. Assume Q has full dimension. A coherenttriangulation ( T , ψ ) for pair ( Q, A ) is a triangulation T of Q with vertices in A and a piecewiselinear (PL) convex function ψ : Q → R , such that the maximal linear domains of ψ are exactlythe maximal simplices of T . For any assignment h : A → R , there exists a maximal PL convexfunction ˆ h : Q → R , such that ˆ h ( α ) ≤ h ( α ) for all α ∈ A . For generic choice of h , ˆ h inducesa triangulation T of ( Q, A ) . Since we work with a fixed triangulation instead of considering allpossible triangulations, we will assume A is the set of vertices of T by reducing A if necessary. If ( T , ψ ) is a coherent triangulation of ( Q, A ) , h = ψ | A , we may also denote ( T , ψ ) by ( T , h ) .Let ( T , ψ ) be a coherent triangulation of ( Q, A ) . We define the Legendre transformation of ψ as L ψ : M R → R , u ψ ( y ) = max x ∈ A (cid:104) x, y (cid:105) − ψ ( y ) , where (cid:104)− , −(cid:105) is the dual pairing M R × N R → R . One can show that L ψ is a PL convex functionon M R , inducing a cell-decomposition of M R dual to the triangulation of T on Q . If τ ∈ T is a k -simplex, then we use C τ or τ ∨ to denote the dual cell of dimension n − k in M R . In particular, C α are the n -dimensional cells of M R . The cells and simplices are closed in our convention. Definition 1.1.
The tropical amoeba Π ψ ⊂ M R is defined as the singular loci of L ψ .The tropical amoeba is the limit of amoeba, which we now define. Given a coherent triangula-tion ( T, h ) of ( Q, A ) , h : A → R , we may define the patchworking polynomial f β,h ( z ) = (cid:88) α ∈ A e − βh ( α ) z α : M ∗ C → C . More generally, given a function
Θ : A → T , we have f β,h, Θ ( z ) = (cid:88) α ∈ A e − βh ( α ) e − i Θ( α ) z α : M ∗ C → C . We thank Gammage and Shende for this clarification.
PENG ZHOU
Definition 1.2.
The
Log amoeba Π β,h, Θ of f = f β,h, Θ is defined as the image of f − (0) under the(rescaled) logarithm map Log β : M ⊗ Z C ∗ → M ⊗ Z R , m ⊗ z (cid:55)→ m ⊗ β − log | z | . Mikhalkin proved the following convergence theorem:
Theorem ([Mi]) . The tropical amoeba Π h is the Hausdorff-limit of rescaled amoeba Π β,h, Θ as β → ∞ . Monomial cut-off functions.
The complements of the tropical amoeba has a one-to-one cor-respondence with the vertices of the triangulation T , M R \ Π h = (cid:71) α ∈ A C α , M R \ Π β,h = (cid:71) α ∈ A C α,β .C α are convex polyhedra, and C α,β are smooth strictly convex domains [GKZ, Chapter 6, Cor 1.6].The idea of introducing a monomial cut-off function χ α,β ( z ) is to turn off the term e − βh ( α ) z α if itis much smaller than the rest, thus straighting the hypersurface. The idea is first used in Abouzaid[Ab] to control the symplectic geometry of the hypersurface.We fix a cut-off function χ ( x ) on R with the following properties • χ ( x ) = x ∈ [0 , ∞ ) ∈ (0 , x ∈ ( − , x ∈ ( −∞ , − . • χ ( x ) exp( x ) is convex. Example 1.3.
One can check that the following specification of χ ( x ) on [ − , gives a C functionon R with the desired convexity. χ ( x ) = e − / ( x +2)+1 / − x/ x / . See Figure 3 for a plot of χ ( x ) e x . We have ( e x χ ( x )) (cid:48)(cid:48) e x χ ( x ) = 256 + 608 x + 576 x + 288 x + 85 x + 14 x + x , x ∈ ( − , . which can be verified to be positive on ( − , The non-smooth point of χ ( x ) is at x = 0 , and canbe mollified if needed. - - - - F IGURE
3. . We modify exponential function e x (dashed line) to e x χ ( x ) (solid line),such that e x χ ( x ) remains convex. AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n We use log coordinates ( ρ, θ ) ∈ M R × M T for point z ∈ M C ∗ . We also use β -rescaled logcoordinates ( u, θ ) = ( β − ρ, θ ) , thus u = Log β ( z ) . For each α ∈ A , define linear function on M R l α ( u ) := (cid:104) u, α (cid:105) − h ( α ) . Definition 1.4.
For any vertex α ∈ A , we define the monomial cut-off function as χ α,β ( u ) = (cid:89) α (cid:48) adjacent to α in T χ α,α (cid:48) ,β ( u ) where χ α,α (cid:48) ,β ( u ) = χ ( β ( l α ( u ) − l α (cid:48) ( u )) + (cid:112) β ) . We define a distance-like function to region C α , r α ( u ) := L ˆ h ( u ) − l α ( u )) = max α (cid:48) ∈ A ( l α (cid:48) ( u ) − l α ( u )) . Thus r α ( u ) is a non-negative PL convex function, vanishes only on C α . Proposition 1.5.
For all large enough β , χ α,β ( u ) satisfies the following property χ α,β ( u ) = (cid:40) r α ( u ) < β − / r α ( u ) > β − / + 2 β − Proof. If r α ( u ) < β − / , then for all α (cid:48) adjacent to α , we have l α (cid:48) ( u ) − l α ( u ) < β − / ⇒ β ( l α ( u ) − l α (cid:48) ( u )) + (cid:112) β ) > thus each factor in χ α,β ( u ) equals . The other case is similar to check, where β large enoughmeans r α ( u ) − ( β − / + 2 β − ) intersects all the neighboring cells C α (cid:48) for C α . (cid:3) Definition 1.6.
For each α ∈ A , we define the bad region as the open set B α,β = { u ∈ M R | β − / + 2 β − > r α ( u ) > β − / } . The (total) bad region B β is defined as the union of all B α,β . The good region is defined as the closedset G β := M R \ B β .On the good region, each χ α,β is either or , hence we have a partition labeled by cells of T : G β = (cid:71) τ ∈T G β,τ , where G β,τ = { u ∈ G β | χ α,β ( u ) = 1 ⇐⇒ α ∈ τ } is a closed convex polyhedron with non-emptyinterior.1.3. Tropical Localized Hypersurfaces.
Following [Ab, Section 4], we define a family of hyper-surfaces H s as the zero-loci of f s ( z ) : f s ( z ) := (cid:88) α ∈ A f α ( z ) χ α,s ( z ) , H s = f − s (0) , where f α = e − βh ( α ) − i Θ( α ) z α , χ α,s ( z ) = sχ α ( z ) + (1 − s ) . Here we have dropped the β, h, · · · subscripts from previous notations for clarity. These hyper-surfaces H s interpolates between the complex hypersurface H and tropical localized hypersurface (cid:101) H , where H := H , (cid:101) H := H . PENG ZHOU
The following proposition is a modification of Proposition 4.2 in [Ab].
Proposition 1.7.
Fix any identification M C ∗ ∼ = ( C ∗ ) n . Let ω be any toric Kahler metric on M C ∗ , compa-rable with ω = i (cid:80) i d log z i ∧ d log z i . Then, for all large enough β , the family of hypersurfaces H s aresymplectic with induced symplectic form from ω .Proof. We proceed as in Proposition 4.2 in [Ab]: to prove f − (0) is symplectic, it suffices to prove | ¯ ∂f ( z ) | ω < | ∂f ( z ) | ω . Since ω and ω are comparable, in the following the norm on M R or its dual N R are taken to be the Euclidean norm.Assume s > and z is in the ’bad region’(see Definition 1.6) of the hypersurface H α,s , since oth-erwise the hypersurface is holomorphic at z and there is nothing to prove. Let I ( z ) = { α , · · · , α k } ⊂ A be subset of vertices where r α i ( z ) ≤ β − / , thus I (cid:48) ( z ) is vertex set for a simplex τ ( z ) of T . Since f α ( z ) − f s ( z ) is an equally good defining equation for H s , hence without loss of generality, wemay assume that α = 0 . Let I ( z ) = I (cid:48) ( z ) \{ α } .Define F ( z ) = (cid:88) α ∈ A | f α ( z ) | = (cid:88) αinA e βl α ( u ) =: e βϕ β,h ( u ) . Then we have ϕ β,h ( u ) ≥ L h ( u ) for all u ∈ M R , and as β → ∞ we have ϕ β,h ( u ) → L h ( u ) in C .We first note that of derivatives for the cut-off functions χ α ( z ) have a uniform bound | dχ α ( ρ ) | = | d ( (cid:89) α (cid:48) χ ( (cid:112) β + l α ( ρ ) − l α (cid:48) ( ρ ))) |≤ (cid:88) α (cid:48) | d ( χ ( (cid:112) β + l α ( ρ ) − l α (cid:48) ( ρ ))) |≤ (cid:107) χ (cid:48) (cid:107) ∞ (cid:88) α (cid:48) | α (cid:48) − α | < C where the product or sum are over α (cid:48) adjacent to α in triangulation T , and we used the bound χ ≤ .We have for ¯ ∂f s ( z ) . | F ( z ) − ¯ ∂f s ( z ) | = | F ( z ) − (cid:88) α f α ( z ) ¯ ∂χ α,s ( z ) | ≤ s (cid:88) α e β ( l α ( u ) − ϕ β,h ( u )) | ¯ ∂χ α ( z ) |≤ (cid:88) α e β ( l α ( u ) − L h ( u )) | ¯ ∂χ α ( z ) | ≤ (cid:88) α e β ( − β − / ) | ¯ ∂χ α ( z ) |≤ Ce −√ β where we used ϕ β,h ( u ) ≥ L h ( u ) and on the support of dχ α ( z ) we have r α ( z ) = L h ( u ) − l α ( u ) > / √ β . AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Next, we compute ∂f s ( z ) . | F ( z ) − ∂f s ( z ) | = | F ( z ) − (cid:88) α ∂f α ( z ) χ α,s ( z ) + sf α ( z ) ∂χ α ( z ) | = F ( z ) − | k (cid:88) i =1 ∂f α i ( z ) | + O ( e −√ β )= F ( z ) − | k (cid:88) i =1 f α i ( z ) (cid:104) α i , d ( ρ + iθ ) | + O ( e −√ β )= F ( z ) − k (cid:88) i =1 k (cid:88) j =1 f α i ( z ) f α j ( z ) (cid:104) α i , α j (cid:105) / + O ( e −√ β ) > C F ( z ) − (cid:34) k (cid:88) i =1 | f α i ( z ) | (cid:35) / + O ( e −√ β ) > C F ( z ) − k (cid:88) i =1 | f α i ( z ) | + O ( e −√ β ) = C + O ( e −√ β ) where O ( e −√ β ) represent a remainder bounded by e −√ β , C is the smallest eigenvalue of the k × k matrix M ij = (cid:104) α i , α j (cid:105) , which is non-degenerate since { α = 0 , α , · · · } are vertices of a k -simplexin T . We also used that all the l p ( ≤ p ≤ ∞ ) norm on R k are equivalent.Thus, we have shown | ∂ ( f α ( z ) − f s ( z )) | > | ¯ ∂ ( f α ( z ) − f s ( z )) | for z ∈ f − s (0) , hence H s aresymplectic. (cid:3) Proposition 1.8 (Proposition 4.9 [Ab]) . The family of hypersurfaces H s are symplectomorphic for all s ∈ [0 , . Tropical Localization with Convexity .
For log amoeba Π h,β and tropical amoeba Π h , theirconnected components of complement C α,β and C α are convex. Let (cid:101) Π β,h = Log β ( (cid:101) H ) be the amoebaof the tropical localized hypersurface, and let (cid:101) C α denote the complements dual to vertex α ∈ A .We would like to show that (cid:101) C α are close to convex as well. Proposition 1.9.
The defining equation for (cid:101) C α is (cid:88) α (cid:48) adjacent to α e β ( l α (cid:48) ( u ) − l α ( u )) χ α (cid:48) ,β ( u ) = F α . and F α is convex in the good region, i.e., where all χ α,β ( u ) are constant with value or .Proof. The boundary of the complement (cid:101) C α is where the dominant term equals the sum of theother non-dominant term. By the tropical localization, there are at most n non-dominant termsfor a point z on the boundary (thanks to T being a triangulation). Hence the θ i can be chosen,such that the argument of the dominant and non-dominant terms are the same. For the secondstatement, we note that over the good region, F α is a sum of convex functions. (cid:3) Definition 1.10.
The convex model (cid:98) C α for (cid:101) C α is defined by { u ∈ C α | (cid:98) F α ( u ) = 1 } , where (cid:98) F α = (cid:88) α (cid:48) adjacent to α e β ( l α (cid:48) ( u ) − l α ( u )) χ α (cid:48) ,α,β ( u ) The two defining functions, F α and (cid:98) F α , differ by the cut-off functions: F α uses χ α (cid:48) ,β which cutsalong the boundary of C α (cid:48) , whereas (cid:98) F α uses χ α (cid:48) ,α,β which cuts along the hyperplane separating C α and C α (cid:48) . However, on C α the two functions and the hypersurfaces are very close, as the followingtwo propositions show. Proposition 1.11.
For all k ≥ , there are constant c k , c (cid:48) k , such that (cid:107) F (cid:107) C k ( C α ) + (cid:107) (cid:98) F (cid:107) C k ( C α ) ≤ c (cid:48) k β k and (cid:107) F − (cid:98) F (cid:107) C k ( C α ) < c k β k e −√ β . Proof.
First, we note that since all χ α ,α ,β ≤ , we have χ α (cid:48) ( u ) < χ α (cid:48) ,α ( u ) , thus (cid:98) F α ( u ) > F α ( u ) , (cid:98) C α ⊂ (cid:101) C α . For u ∈ C α and α (cid:48) adjacent to α , if χ α (cid:48) ( u ) − χ α (cid:48) ,α ( u ) (cid:54) = 0 , then l α (cid:48) ( u ) − l α ( u ) + √ β ∈ ( − , . Hence (cid:98) F α ( u ) − F α ( u ) = (cid:88) α (cid:48) adjacent to α e β ( l α (cid:48) ( u ) − l α ( u )) ( χ α ( u ) − χ α (cid:48) ,α ( u )) < Ce −√ β . Similarly, taking k -th derivative, we have | ∂ ku (cid:98) F α ( u ) − ∂ ku F α ( u ) | < C k β k e −√ β . where the norm are taken with respect to Euclidean norm on R n , after choosing an identification M R ∼ = R n . This finish the proof. (cid:3) Fix M R ∼ = R n and equip R n with Euclidean metric. Let S ∗ R n denote the unit cosphere bundle.If C is a domain with smooth boundary, we define Λ C = { ( p, ξ ) ∈ S ∗ R n | p ∈ ∂C, ξ ∈ ( T p ∂C ) ⊥ and points outward } Proposition 1.12.
We have the following convergence results: (1)
In the good region in C α , ∂ (cid:101) C α = ∂ (cid:98) C α . (2) The Hausdorff distance between ∂ (cid:101) C α and ∂ (cid:98) C α is O ( β − e −√ β ) . (3) The Hausdorff distance between Λ (cid:101) C α and Λ (cid:98) C α is O ( βe −√ β ) .Proof. We will write F = F α , (cid:98) F = (cid:98) F α and so on, omitting the α subscript when it is not confusing.(1) Since in the good region in C α , all the cut-off functions χ α (cid:48) and χ α (cid:48) ,α are equal.(2) Since (cid:98) F ≤ F , hence the domain (cid:98) C ⊂ (cid:101) C ⊂ C . If u ∈ ∂ (cid:98) C \ ∂ (cid:101) C , we take gradient flow of F ,starting from u and ending on u (cid:48) ∈ ∂ (cid:101) C . Let γ : [0 , t ] → C α denote this integral curve. Since around ∂ (cid:101) C and ∂ (cid:98) C , we have uniform lower bound for | dF | and | d (cid:98) F | by some constant cβ , hence cβ dist( u, u (cid:48) ) ≤ cβt < (cid:90) t |∇ F ( γ ( s )) | ds and (cid:90) t |∇ F ( γ ( s )) | ds = F ( γ ( t )) − F ( γ (0)) = 1 − F ( u ) = (cid:98) F ( u ) − F ( u ) < Ce −√ β hence dist( u, u (cid:48) ) = O ( β − e −√ β ) (1.1) AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Similarly, if we start from u (cid:48) ∈ ∂ (cid:101) C \ ∂ (cid:98) C we may find u ∈ (cid:98) C using gradient flow of (cid:98) F , with the samebound as above. This establishes the bound on Hausdorff distance(3) Let u ∈ ∂ (cid:98) C \ ∂ (cid:101) C , and u (cid:48) ∈ ∂ (cid:101) C constructed as in (2). We have | dF ( u (cid:48) ) − d (cid:98) F ( u ) | ≤ | dF ( u (cid:48) ) − dF ( u ) | + | dF ( u ) − d (cid:98) F ( u ) |≤ (cid:107) F (cid:107) C dist( u, u (cid:48) ) + O ( βe −√ β ) = O ( βe −√ β ) where we used the C bound of F in Proposition 1.11 and the distance bound in (1.1). (cid:3) Let Λ C α denote the Legendrian of the unit exterior conormal to C α . Proposition 1.13.
Let α ∈ A , (cid:98) C α be the tropical localized amoeba’s complement. We have the followingconvergence of the boundary of (cid:98) C α , and its Legendrian lifts: (1) The Hausdorff distance between ∂ (cid:98) C α and ∂C α is O (1 / √ β ) . (2) The Hausdorff distance between Λ (cid:98) C α and Λ C α is O (1 / √ β ) .Proof. Consider the simplices around vertex α in T . Let τ be such a k -dimensional simplex, withvertices α, α , · · · , α k . Denote the dual face by τ ∨ on the polytope C α . We also define an locallyclosed subset U τ ⊂ ∂ (cid:98) C α , such that z ∈ U τ iff the set I α ( z ) = { α , · · · , α k } . I α ( z ) := { α (cid:48) adjacent to α, χ α (cid:48) ,α,β ( z ) > } . Define the orthogonal projection projection map π τ : U τ → τ ∨ . If u ∈ U τ , u (cid:48) = π τ ( u ) , then since − / (cid:112) β − /β < l α i ( u ) − l α ( u ) < , and l α i ( u (cid:48) ) − l α ( u (cid:48) ) = 0 Hence dist ( u, u (cid:48) ) < O ( k (cid:88) i =1 | l α i ( u ) − l α ( u ) | ) = O (1 / (cid:112) β ) . Also τ ∨ is in O (1 / √ β ) neighborhood of Im ( π τ ) , thus the Hausdorff distance between τ ∨ and U τ is O (1 / √ β ) . Considering all faces τ ∨ of C α proves the first statement.For the second statement, we further note that, for any u ∈ U τ , the exterior unit conormal ξ of (cid:98) C α at u is contained in cone( α − α, · · · , α k − α ) = R + τ . Define Λ τ ∨ := τ ∨ × ( R + · τ ) ∩ S ∗ R n . Then the projection map π τ , lifts to (cid:101) π τ : Λ (cid:101) C α | U τ → Λ τ ∨ , ( u, ξ ) (cid:55)→ ( π τ ( u ) , ξ ) Since the fiber direction has distance zero, the similar argument as (1) proves the second statement. (cid:3)
2. L
EGENDRE TRANSFORMATION AND T ORIC K AHLER P OTENTIAL .In this section we use Legendre transform to define a diffeomorphism between ( C ∗ ) n and T ∗ T n ,and define a Kahler structure on ( C ∗ ) n . Legendre transformation.
Let V be a real vector space of dimension n , and V ∨ be its dualspace. There is a natural identification of symplectic space T ∗ V ∼ = V × V ∨ ∼ = T ∗ V ∨ . Let π V and π V ∨ denote the projection of V × V ∨ to its first and second factor, respectively.Let ϕ be a smooth strictly convex function on V . The Legendre transformation for ϕ is definedas Φ ϕ : V → V ∨ , x (cid:55)→ dϕ ( x ) . We will always assume ϕ satisfies some growth condition such that the Legendre transformationis surjective. The Legendre dual ψ of ϕ is also a convex function defined as ψ : V ∨ → R , y (cid:55)→ sup x ∈ V (cid:104) x, y (cid:105) − ϕ ( x ) = (cid:104) Φ − ϕ ( y ) , y (cid:105) − ϕ (Φ − ϕ ( y )) . If we fix a linear coordinate ρ = ( ρ , · · · , ρ n ) on V and dual coordinate p = ( p , · · · , p n ) on V ∨ ,then the Legendre transformation can be written as p i = ∂ ρ i ϕ ( ρ ) . If p = dϕ ( p ) , then Legendre dual function ψ ( p ) = (cid:88) i ρ i p i − ϕ ( ρ ) . And the two matrices
Hess ϕ ( ρ ) = ∂ ij ϕ ( ρ ) and Hess ψ ( p ) = ∂ ij ψ ( p ) are inverse of each other. Thereis a metric on V induced by ϕ : g ϕ = ∂ ij ϕ ( ρ ) dρ i ⊗ dρ j . The above contruction can be interpreted symplectically. Consider the graph Lagrangian Γ dϕ in T ∗ V Γ dϕ = { ( x, y ) ∈ V × V ∨ | y = dϕ ( x ) } . Let L = Γ dϕ . Then the Legendre transform is Φ ϕ = π V ∨ | L ◦ π V | − L LV V ∨ π V π V ∨ .L as a section in T ∗ V ∨ is the graph of Γ dψ for the Legendre dual function ψ of ϕ .The following lemma says, gradient vector field on V and differential one form are related byLegendre transformation. Lemma 2.1.
Let ϕ be any smooth convex function on V , and let f : V → R be any smooth function. Forany ρ ∈ V , and p = Φ ϕ ( ρ ) ∈ V ∨ , then (Φ ϕ ) ∗ ( ∇ f | ρ ) ∈ T p V ∨ ∼ = V ∨ and df ( ρ ) ∈ T ∗ ρ V ∼ = V ∨ are equal. AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Proof.
We work with linear coordinates ( ρ , · · · , ρ n ) on V and dual coordinate ( p , · · · , p n ) on V ∨ .Let g ij = ( g ϕ ) ij = ∂ ij ϕ and g ij be the matrix inverse of g ij . (Φ ϕ ) ∗ ∇ f ( ρ ) = (cid:88) i,j,k ∂ ρ k f · g jk · ∂p i ( ρ ) ∂ρ j · ∂ p i = (cid:88) i,j,k ∂ ρ k f · g jk · g ij · ∂ p i = (cid:88) i,k ∂ ρ k f · δ ik · ∂ p i = df. (cid:3) Identification between M C ∗ and T ∗ M T . There is a canonical complex structure on M C ∗ ∼ = M R × M T , and a canonical symplectic structure on T ∗ M T ∼ = N R × M T . We will use notation θ ∈ M T , ρ ∈ M R and p ∈ N R . If we fix a Z -basis for M , then we have M C ∗ ∼ = ( C ∗ ) n = { ( e ρ i + iθ i ) i } and T ∗ M T ∼ = T ∗ T n = { ( θ i , p i ) i } .Let ϕ : M R → R be a smooth strictly convex function such that the Legendre transformation Φ ϕ : M R → N R is surjective. We abuse notation and also denote by ϕ the pullback via M C ∗ → M R ,and call ϕ a K¨ahler potential on M C ∗ . Then we may define Liouville one-form and symplectictwo-form on M C ∗ λ = − d c ϕ, ω = − dd c ϕ. In coordinate form, we have λ ϕ = (cid:88) i ∂ i ϕ ( ρ ) dθ i , ω ϕ = (cid:88) i,j ∂ ij ϕ ( ρ ) dρ i ∧ dθ j . The Riemannian metric can also be obtained by g ϕ ( X, Y ) = ω ϕ ( X, J Y ) , where J ∂ ρ i = ∂ θ i , J ∂ θ i = − ∂ ρ i , or in coordinate form g = (cid:88) i,j ∂ ij ϕ ( ρ )( dρ i ⊗ dρ j + dθ i ⊗ dθ j ) . If we equip T ∗ M T with the standard exact symplectic structure ( ω, λ ) : λ std = (cid:88) i p i dθ i , ω std = (cid:88) i dp i ∧ dθ i , then by Legendre transformation Φ ϕ × id : M C ∗ = M R × M T → N R × M T = T ∗ M T , we have (Φ ϕ × id) ∗ ( λ std ) = λ ϕ , (Φ ϕ × id) ∗ ( ω std ) = ω ϕ . Homogeneous K¨ahler potential.
Next we will restrict ourselves to homogenous convex func-tions as K¨ahler potential.
Definition 2.2.
A convex function ϕ on M R is said to be homogeneous of degree d for some d ≥ , iffor any (cid:54) = x ∈ M R and any λ > , we have ϕ ( λx ) = λ d ϕ ( x ) , (2.1)and Ω = { x : ϕ ( x ) ≤ } is a bounded strictly convex closed set with smooth boundary. Remark 2.3.
Any positive definite quadratic form on M R is a homogeneous degree two convexfunction. More generally, any bounded strictly convex subset Ω ⊂ M R with smooth boundary andcontaining as an interior point determines a homogeneous degree d convex function ϕ Ω ,d suchthat Ω = { x : ϕ ( x ) ≤ } . Proposition 2.4.
For any homogeneous convex function ϕ of degree d with d ∈ [0 , ∞ ) , we have(1) ϕ is smooth on M R \{ } .(2) ϕ is C k at where k is the largest integer less than d .(3) If d > , then ϕ is strictly convex.Proof. (1) and (3) are easy to verify. We only prove (2). Fix a linear coordinate x , · · · , x n on M R . Formulti-index α = ( α , · · · , α n ) , any point (cid:54) = x ∈ M R and λ > , we have ∂ αx ϕ ( λx ) = λ d −| α | ∂ αx ϕ ( x ) . Hence if in addition | α | ≤ k < d , then lim λ → ∂ αx ϕ ( λx ) = 0 . Hence all k -th derivative can becontinuated to x = 0 . (cid:3) Lemma 2.5. If ϕ is a homogeneous degree d convex function, then for λ > ϕ ( λρ ) = λ d − Φ ϕ ( ρ ) . Definition 2.6.
Let M ∞ R := ( M R \ / R > and N ∞ R := ( N R \ / R > . Then we define the projectiveLegendre transformation Φ ∞ ϕ : M ∞ R → N ∞ R . It is easy to check that Φ ∞ ϕ is an orientation perserving diffeomorphism from S n − to itself.Geometrically, if we take the level set S = ϕ − (1) , then each element in M ∞ R corresponds to apoint on S , and the outward conormal of S at the point is the element in N ∞ R obtained by Φ ∞ ϕ . Proposition 2.7.
Let ϕ be any homogeneous convex function on M R of degree k > , and equip M R withmetric g ϕ induced from Hessian of ϕ . Then the integral curves in M R \{ } of the gradient of ϕ are rays.Equivalently, ∇ ϕ ( ρ ) = C ( ρ ) (cid:88) i ρ i ∂ ρ i , C ( ρ ) > . Proof.
For any nonzero ρ ∈ M R , we have Φ ϕ ( ρ ) = dϕ ( ρ ) , also by Proposition 2.1 we have (Φ ϕ ) ∗ ( ∇ ρ ) = dϕ ( ρ ) , hence the gradient vector field ∇ ϕ on M R when pushed-forward to N R is exactly the radialvector field p∂ p whose integral curves are rays. Since ϕ is homogeneous, hence Φ ϕ takes ray toray, hence the integral curve of ∇ ϕ is the pull-back of integral curve of p∂ p , i.e. rays. (cid:3) K¨ahler potentials Adapted to a Polytope .
This is one of the key construction in this paper.We replace the Kahler potential (cid:80) i u i on ( C ∗ ) n where u i = log | z i | by any homogeneous degreetwo Kahler potential ϕ ( u ) .Let P be a convex polytope (possibly unbounded) in M R containing as an interior point. Wedefine a notion of convexity with respect to P . Definition 2.8.
A homogeneous convex function ϕ on M R is convex with respect to P , if for eachface F of P of positive dimension, the restriction ϕ | F has a unique minimum point in the interiorof F . A K¨ahler potential adapted to P is a homogeneous degree two convex function ϕ : M R → R that is convex with respect to P . Remark 2.9.
A homogeneous convex function ϕ on M R is convex with respect to P , if the increas-ing sequence of level sets { ϕ ( ρ ) < c } meet the faces of P in the interior first. AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Proposition 2.10.
For any convex polytope P in M R containing as an interior point, there exists anon-empty contractible set of K¨ahler potential adapted to P .Proof. First, we show the existence of such potential ϕ . We will build the level set S = { ϕ = 1 } ,and show that as we rescale S to λS , for λ from to ∞ , S will meet the interior of each face F first.We will proceed by first build a polyhedral approximation of S , then smooth it.For each face F of P , we pick a point x F in the interior of F if dim F > , or x F = F if F isa point. Let T be the simplicial triangulation of P with vertices of F , then T is also a barycentricsubdivision of P . Let φ T : P → R a piecewise linear convex function on P , with maximal convexdomain the top-dimensional simplices of T , and such that for any ≤ d ≤ n − , and any face x F of dimension d , φ T ( x F ) = c d are the same for all such F . Such φ T can be constructed inductivelyfrom x F with dim F from to n − . Let φ T be extended to M R by linearity. Thus φ T has a uniqueminium point in each face F .Let η ∈ C ∞ c ( R n ) be a bump function with (cid:82) η = 1 , and let η (cid:15) ( x ) = η ( x/(cid:15) ) /(cid:15) n . Let φ T,(cid:15) = η (cid:15) (cid:63) φ T + (cid:15) | x | , where | x | is taken with respect to any fixed inner product on R n , then φ T,(cid:15) is a linearcombination of convex function hence still convex. Since φ T,(cid:15) → φ T as (cid:15) → , for (cid:15) small enough, φ T,(cid:15) still has a unique minimum point in each face F . And S T,(cid:15) = { φ T,(cid:15) = 1 } is a convex smoothboundary, such that S T,(cid:15) → S T = { φ T = 1 } as (cid:15) → . Then, for small enough (cid:15) , we can use S T,(cid:15) asthe contour of the homogeneous degree two convex function { ϕ ( x ) = 1 } .(2) Let K be the set of homogeneous degree two potential adapted to P . Then there is surjectivecontinuous map π : K → (cid:81) F, dim F > Int( F ) , by sending ϕ to its critical points on each face. Sinceif two convex functions ϕ , ϕ have the same critical points, then their convex linear combination tϕ + (1 − t ) ϕ for t ∈ [0 , are still homogeneous degree two and with the same critical points, wesee the fiber of map π is convex hence contractible. Since the base of the fibration Cr is contractibleas well, we have K contractible. (cid:3) Let P be a convex polytope in M R containing as an interior point. Recall the definition of thedual polytope P ∨ ⊂ N R P ∨ = { p ∈ N R | (cid:104) p, x (cid:105) ≤ ∀ x ∈ P } . (2.2)For any face F ⊂ P , there is dual face F ∨ ⊂ P ∨ , and dim R F + dim R F ∨ = n − . We define threesubsets of M R × N R L P = (cid:91) F cone( F ) × F ∨ , L P ∨ = (cid:91) F F × cone( F ∨ ) , Λ P = (cid:91) F F × F ∨ , (2.3)where F runs over the faces of P , and cone( F ) = R > · F . Remark 2.11. L P and L P ∨ are piecewise Lagrangians, and Λ P = L P ∩ L P ∨ is piecewise isotropic. L P is the exterior conormal of P ∨ in T ∗ N R , and L ∨ P is the exterior conormal of P in T ∗ M R . If welet ϕ P, be the piecewise linear function on M R , such that P = { x : ϕ P, ( x ) ≤ } , then L P morallyis Γ dϕ P, . Lemma 2.12.
Let ϕ be a homogeneous degree two convex function on M R . P, P ∨ be dual convex polytopesin M R and N R as above. Let F be a face of P . Then there is a natural bijection cone( F ) × F ∨ ∩ Γ dϕ ↔ F × cone( F ∨ ) ∩ Γ dϕ . (2.4) Proof. If ( λx, p ) ∈ cone( F ) × F ∨ ∩ Γ dϕ , where λ > and x ∈ F, p ∈ F ∨ , then by conic invariance of Γ dϕ , we have ( x, p/λ ) = 1 λ ( λx, p ) ∈ F × cone( F ∨ ) ∩ Γ dϕ . (2.5)Sending ( λx, p ) to ( x, p/λ ) is the desired bijection. (cid:3) Next, we give some equivalent characterization for convexity with respect to a polytope.
Proposition 2.13.
Let P be a convex polytope in M R containing as an interior point. Let ϕ be a homo-geneous degree two convex function on M R . The following conditions are equivalent:(1) ϕ is adapted to P .(2) For each face F of P , the smooth component Int( F × cone( F ∨ )) of L ∨ P has a unique intersection with Γ dϕ .(3) For each face F of P , the smooth component Int(cone( F ) × F ∨ ) of L P has a unique intersection with Γ dϕ .Proof. (2) is equivalent to (3) by Lemma 2.12.(2) ⇒ (1): since ϕ | F is still convex, hence as at most one minimum point in the interior, and anyinterior critical point is a minimum. Since ∅ (cid:54) = F × cone( F ∨ ) ∩ Γ dϕ ⊂ T ∗ F M R ∩ Γ dϕ (2.6)we see ϕ | F has a critical point.(1) ⇒ (2): for each face F of P , let x F be the critical point of ϕ | F , and let H F ⊂ M R be the affinehyperplane tangent to the contour of ϕ at x F . We claim that H F is a supporting hyperplane for P ,and P ∩ H F = F . Then p = dϕ | x F ∈ T ∗ x F M R ∼ = N R is in the exterior conormal of H F (exterior withrespect to P ), hence p ∈ cone( F ∨ ) . Thus, ( x, p ) ∈ F × cone( F ∨ ) . (cid:3) A consequence of the proposition is the compatibility of the ‘adaptedness’ with Legendre trans-formation.
Corollary 2.14.
Let P be a convex polytope in M R containing as an interior point and P ∨ the dualpolytope. Let ϕ be homogeneous degree two convex function, and ψ the Legendrian dual of ϕ . Then ϕ isadapted to P if and only if ψ is adapted to P ∨ .
3. M
AIN R ESULTS
Let Q be a convex lattice polytope in N R containing α . Let T be a coherent star triangula-tion of Q based at with integral vertices, and ∂ T be the subset of simplices not containing . Let A be the set of vertices of T , and let h : A → R induce T with h (0) = 0 . Fix a Θ : A → T with Θ( α ) = π .Let Π h ⊂ M R be the tropical amoeba of ( T , h ) , and P = C α be the connected component in M R \ Π h corresponding to α .Let ϕ : M R → R be a homogeneous degree two convex function (i.e. ϕ ( λx ) = λ ϕ ( x ) for all λ > ). We assume ϕ is adapted to P , i.e, every face of P contains a minimum of ϕ in its interior. We may smooth ϕ at an small neighborhood around ∈ M R , but this is irrelevant since we will use ϕ only as ϕ ( βu ) for β (cid:29) and u in a neighborhood of ∂P . AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Then the tropical localized polynomial is (cid:101) f ( z ) := (cid:88) α ∈ A e − βh ( α ) − i Θ( α ) χ α,β ( z ) z α where the monomial cut-off function χ α,β defined as in Definition 1.4. Denote the localized hy-persurface and amoeba as (cid:101) H := (cid:101) f − (0) and (cid:101) Π := Log β ( (cid:101) H ) respectively. We use (cid:101) C = (cid:101) C α to denotethe complement labelled by α . Theorem 1.
The critical points for ϕ | ∂ (cid:101) C on the boundary of amoeba ∂ (cid:101) C are indexed by simplices τ ∈ ∂ T .The critical point (cid:101) ρ τ for τ in ∂ T has Morse index dim | τ | . The unstable manifold (downward flowing) (cid:102) W τ of (cid:101) ρ τ contains (cid:101) ρ τ (cid:48) in its closure, if and only if τ ⊃ τ (cid:48) . The Liouville structure of (cid:101) H are induced from ( M C ∗ , ω, λ ) (see Section 2.2), where in coordinates ω = ∂ ij ϕ ( ρ ) dρ i ∧ dθ j , λ = ∂ i ϕ ( ρ ) dθ i . The (candidate for) Liouville skeleton is defined as S β,h, Θ = (cid:91) τ ∈T ( β · (cid:102) W τ ) × T τ, Θ ⊂ M R × M T ∼ = M C ∗ , (3.1)where (cid:102) W τ is the unstable manifold from (cid:101) ρ τ and T τ, Θ is the subtorus of M T defined by T τ, Θ = { θ ∈ M T : (cid:104) θ, α (cid:105) = Θ( α ) , for each vertex α in τ . } (3.2) Theorem 2. S β,h, Θ is the Lagrangian Liouville skeleton for ( (cid:101) H , ω | (cid:101) H , λ | (cid:101) H ) The Lagrangian skeleton defined here can be related with the RSTZ skeleton via the ’projective’Legendre transformation Φ ∞ ϕ : M ∞ R ∼ −→ N ∞ R , which is induced by homogeneous Legendre trans-formation Φ ϕ : M R → N R . Let q M : M R \{ } → M ∞ R and q N : N R \{ } → N ∞ R be quotient by R + .Recall the RSTZ-skeleton Λ ∞T , Θ is defined in the introduction (0.2). Let id denote the identity mapon M T , then we have: Theorem 3. Φ ∞ ϕ × id : M ∞ R × M T → N ∞ R × M T induces a homeomorphism between S β,h, Θ identified as ( q M × id )( S β,h, Θ ) and Λ ∞T , Θ identified as ( q N × id )(Λ ∞T , Θ ) . The main theorem then follows from Theorem 2 and 3, and the diffeomorphism of H with (cid:101) H from Proposition 1.8. 4. G RADIENT F LOW : P
ROOF OF T HEOREM τ in ∂ T = T ∩ ∂Q . We use τ to denote the simplexof conv( { } ∪ τ ) . We will sometimes omit β from the subscript to unclutter the notation. Convergence of smooth Convex Domain and Critical Points.
We fix an identification of V ∼ = R n and take Euclidean metric on V and the induced metric on T ∗ V and S ∗ V . We identify thesphere compactification boundary T ∞ V = ( T ∗ V − V ) / R > with the unit cosphere bundle S ∗ V . If U ⊂ V open set with smooth boundary, then S ∗ U V is the one-sided unit conormal bundle of ∂U with covectors pointing outward. The generalization to open convex set U with piecewise smoothboundary is also straightforward. Proposition 4.1.
Let V ∼ = R n be a real vector space of dimension n , P ⊂ V a convex polytope containingthe origin, ϕ : V → R a potential adapted to P . Let { P j } be a sequence of convex bounded domains withsmooth boundaries, such that the exterior conormals L j := S ∗ P j V converges to L := S ∗ P V in the cospherebundle S ∗ V in the Hausdorff metric. Then for all large enough j , there is a one-to-one correspondencebetween face F of P and critical points of ϕ on ∂P j , denoted as (cid:101) ρ F , such that(1) (cid:101) ρ F has Morse index n − − dim F .(2) As β → ∞ , (cid:101) ρ F tends to the ρ F , the minimum of ϕ on the face F of P .Proof. (1) We express the critical point condition in terms of Legendrian intersection. Define theprojection image of Γ dϕ in T ∞ V as Γ ∞ dϕ = ( R > · Γ dϕ ) / R > ⊂ T ∞ V. (4.1)Then Γ ∞ dϕ is also the union of unit conormal for level sets of ϕ : Γ ∞ dϕ = (cid:91) c ∈ R S ∗{ ϕ ( ρ ) ≤ c } V. (4.2)The Legendrian L = S ∗ P V is a piecewise smooth C manifold, where the smooth components L F are labelled by faces F of P . If ρ F is a critical point of ϕ on F , then there is a unique unitcovector p F ∈ L F , such that x F = ( ρ F , p F ) ∈ L (cid:116) Γ ∞ dϕ , and the intersection is transversal.(2) Consider the unit speed geodesic flow Φ tR on the unit cosphere bundle S ∗ V . Fix any smallflow time (cid:29) (cid:15) > , since Φ (cid:15)R : S ∗ V → S ∗ V is a diffeomorphism, Φ (cid:15)R ( L j ) still converges to Φ (cid:15)R ( L ) in Hausdorff metric. For any subset A ⊂ V , define A (cid:15) := { x : dist( x, A ) < (cid:15) } to be the (cid:15) -fattening of A . If A is a convex set, we have Φ tR ( S ∗ A V ) = S ∗ A (cid:15) V. Hence ∂P (cid:15) is a C hypersurface, and ∂P (cid:15)j → ∂P (cid:15) in Hausdorff metric as j → ∞ . Define L t = Φ tR ( L ) , L tj = Φ tR ( L j ) . The geodesic flow applied to Γ ∞ dϕ can be understood as follow Φ (cid:15)R (Γ ∞ dϕ ) = (cid:91) c ∈ R Φ (cid:15)R ( S ∗{ ϕ ( ρ ) ≤ c } V ) = (cid:91) c ∈ R S ∗{ ϕ ( ρ ) ≤ c } (cid:15) V. Define function (cid:101) ϕ (cid:15) , such that { (cid:101) ϕ (cid:15) ( ρ ) < c } = { ϕ ( ρ ) ≤ c } (cid:15) , then (cid:101) ϕ (cid:15) is a levelset convex function. ByLemma 2.7 of [CE], there exists a strictly increasing function f : R → R , such that ϕ (cid:15) = f ◦ (cid:101) ϕ (cid:15) is aconvex function. Thus, we have Φ (cid:15)R (Γ ∞ dϕ ) = Γ ∞ dϕ (cid:15) ϕ (cid:15) is convex . Let x (cid:15)F = Φ (cid:15)R ( x F ) , ρ (cid:15)F = π ( x (cid:15)F ) in the expanded face F (cid:15) = π (Φ (cid:15)R ( L F )) . Then x (cid:15)F is still theintersection of Γ ∞ dϕ (cid:15) and S ∗ P (cid:15) V , and ρ (cid:15)F is the unique Morse critical points of ϕ (cid:15) restricted on F (cid:15) ,and ρ (cid:15)F is in the interior of F (cid:15) . One may easily check that the Morse index of ρ (cid:15)F is n − − dim F . AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n (3) We now prove that for large enough j , for each F , there is a unique critical points ρ (cid:15)F,j of ϕ (cid:15) on ∂P (cid:15)j approaching ρ (cid:15)F .Fix a small neighborhood W F ⊂ ∂P (cid:15) near ρ (cid:15)F , and for small enough δ , let (cid:102) W F ∼ = W F × ( − δ, δ ) bethe flow-out of W F under the Reeb flow for time in ( − δ, δ ) , with projection map π W : (cid:102) W F → W F .We claim that for large enough j , ∂P (cid:15)j ∩ (cid:102) W F projects bijectively to W F , since otherwise this con-tradicts with P (cid:15)j being convex and the fiber of π W being straight-line segments Reeb trajectories.Thus, we have a sequence of smooth sections ι j : W F → (cid:102) W F for large enough j , such that ι j converges to the zero section in C .Let f j = ι ∗ j ϕ (cid:15) | (cid:102) W F ∈ C ∞ ( W F , R ) , and a smooth function f ∞ = ι ∗∞ ϕ (cid:15) | (cid:102) W F , where ι ∞ : W F (cid:44) → (cid:102) W F is the identity map of zero section. Since ι j → ι ∞ in C , f j → f ∞ in C . Since f ∞ has a non-degenerate critical point, by stability of critical points under C -perturbation, f j has a uniquecritical point of the same index as f ∞ .(4) Finally, we show that there are no other critical points. Let U F be the preimage of (cid:102) W F under S ∗ V → V . Let U be the union of all such (cid:101) U F . If δ > is small enough, such that dist(Γ ∞ dϕ (cid:15) \ U, L (cid:15) ) > δ. Then by the assumption that L (cid:15)j converges to L (cid:15) in Hausdorff metric, we make take j largeenough, such that for all j > j and all x ∈ L (cid:15)j , dist( x, L (cid:15) ) < δ . This shows dist(Γ ∞ dϕ (cid:15) \ U, L (cid:15)j ) ≥ dist(Γ ∞ dϕ (cid:15) \ U, L (cid:15) ) − dist( L (cid:15)j , L (cid:15) ) > δ, hence there is no intersection between L (cid:15)j and Γ ∞ dϕ (cid:15) away from U .(5) Since Φ (cid:15)R is a diffeomorphism, the result about L (cid:15)j ∩ Γ ∞ dϕ (cid:15) implies the same result about L j ∩ Γ ∞ dϕ , and we finish the proof of the proposition. (cid:3) Proof of Theorem 1: Critical Points and Unstable Manifolds.Proposition 4.2.
For large enough β , there is a one-to-one correspondence between simplices τ ∈ ∂ T andcritical points of ϕ on ∂ (cid:101) C , denoted as (cid:101) ρ τ , such that(1) (cid:101) ρ τ has Morse index dim τ .(2) As β → ∞ , (cid:101) ρ τ tends to the ρ τ , the minimum of ϕ on the face τ ∨ of P .Proof. First we approximate ∂ (cid:101) C by its convex model (cid:98) C (see Definition 1.10). Then by Proposition4.1, we have critical points { (cid:98) ρ τ } on (cid:98) C indexed by τ ∈ ∂ T . Then, a perturbation argument shows ϕ has critical points on ∂ (cid:101) C as { (cid:101) ρ τ = (cid:98) ρ τ } . (cid:3) Next, we prove that the unstable manifold (cid:102) W τ for critical point (cid:101) ρ τ are cells of a dual polyhedraldecomposition of ∂P . This is true not only in the combinatorial sense, but in a more refinedgeometrical sense. Proposition 4.3.
For large enough β , and for any τ ∈ ∂ T , the unstable manifold (cid:102) W τ is a smooth manifoldof dimension dim τ , and (cid:101) ρ τ (cid:48) ∈ ∂ (cid:102) W τ if and only if τ (cid:48) ⊂ τ .Proof. The statement of dim (cid:102) W τ follows from the Morse index of (cid:101) ρ τ . For any critical point ρ τ , takea small enough ball B of radius (cid:15) around it, then B can be stratified by the limit of gradient flow.For each facet σ ∨ of the polytope P adjacent to τ ∨ , there is an open ball U σ in ∂B whose points flows to critical point ρ σ . If a face τ (cid:48)∨ adjacent to τ ∨ can be written as τ (cid:48)∨ = σ ∨ ∩ · · · ∩ σ ∨ k for facets σ ∨ i , then points in the relative interior of ∩ ki =1 U σ i will flow to ρ τ (cid:48) . (cid:3) Now we give an explicit description of the unstable manifold. Let Φ ∞ ϕ , q N and q M be as in thestatement of Theorem 3. Proposition 4.4.
For all large enough β , and τ ∈ ∂ T , Φ ∞ ϕ induces a homeomorphism Φ ∞ ϕ : q M ( (cid:102) W τ ) ∼ −→ q N ( τ ) . Proof.
Without loss of generality, we set h (0) = 0 . We will first do the computation on the convexmodel (cid:98) C , then state the necessary modifications for (cid:101) C .Then the defining function for (cid:98) C can is (cid:88) α ∈ A \{ } e βl α ( u ) χ ( βl α ( u ) + (cid:112) β ) =: (cid:98) F ( u ) For any point u ∈ (cid:98) C , we define the simplex τ ( u ) = conv { α ∈ A \{ } : χ ( βl α ( u ) + (cid:112) β ) > } . Then the gradient of ϕ on (cid:98) C can be expressed as ∇ ( ϕ | (cid:98) C ) = ∇ ϕ − c ∇ (cid:98) F where c = (cid:104)∇ ϕ,d (cid:98) F (cid:105)(cid:104)∇ (cid:98) F ,d (cid:98) F (cid:105) . Since by Proposition 2.7, ∇ ϕ is in the outward radial direction, and (cid:98) F is aconvex function with bounded sub-level set, hence (cid:104)∇ ϕ, d (cid:98) F (cid:105) > . Combining (cid:104)∇ (cid:98) F , d (cid:98) F (cid:105) > , wehave c > .For u on the unstable manifold (cid:99) W τ , we have τ ( u ) ⊂ τ . The defining function (cid:98) F for a neighbor-hood of u can be written as (cid:98) F τ ( u ) = (cid:88) α ∈ τ e βl α ( u ) χ ( βl α ( u ) + (cid:112) β ) thus d (cid:98) F τ = (cid:88) α ∈ τ ( e x χ ( x + (cid:112) β )) (cid:48) | x = βl α ( u ) · α ∈ Int cone( τ ) . And at the critical point dϕ ( (cid:98) ρ τ ) = c d (cid:98) F τ ∈ Int cone( τ ) . If γ : ( −∞ , + ∞ ) → (cid:98) C is an integral curve for −∇ ( ϕ | (cid:101) C ) with lim t →−∞ γ ( t ) = (cid:98) ρ τ , then underLegendre transformation we have a curve η ( t ) , such that lim t →−∞ η ( t ) = dϕ ( (cid:101) ρ τ ) ∈ Int cone( τ ) and using Lemma 2.1 and Proposition 2.7 ddt η ( t ) = (Φ ϕ ) ∗ ( −∇ ( ϕ | (cid:101) C )) = (Φ ϕ ) ∗ ( −∇ ϕ + c ∇ F ) ∈ R ( p∂ p ) + Int cone( τ ) , where p∂ p is the radial vector field on N R . Thus η ( t ) is within the cone Int cone( τ ) for all t ∈ R .This shows that Φ ∞ ϕ ( q M ( (cid:99) W τ )) ⊂ q N ( τ ) . Using induction on dimension of τ from to n − , we can show the image is onto. AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Now consider (cid:101) C . One need to replace χ ( βl α ( u ) + √ β ) by χ α, )( u ) in defining (cid:101) F . And one canstill show that d (cid:101) F τ ( u ) ∈ Int cone( τ ) , the rest is the same as (cid:98) C . (cid:3)
5. L
IOUVILLE F LOW : P
ROOF OF T HEOREM AND (cid:101) H . We showthat they are exactly the preimage of critical points of ϕ | ∂ (cid:101) C under Log β , which are tori of variousdimensions. The more difficult part is to show there are no other critical points.Then, we study the Liouville flow trajectory from these critical manifolds. There are two keypoints:(1) We write the fiberwise Liouville vector field as the ambient Liouville vector field in M C ∗ subtract its symplectic orthogonal component, then show that on the ’positive loci’ (Defini-tion 5.1), the symplectic orthogonal component is proportional to the Hamiltonian vectorfield X Im (cid:101) f .(2) We show that the unstable manifold correponding to the critical manifold indexed by τ ∈ ∂ T , is geometrically identified with the simplex τ under the projective Legendre trans-formation Φ ∞ ϕ : M ∞ R → N ∞ R . This determines the unstable manifolds.Recall our notations: • The complex hypersurface H defined by f ( z ) = 0 : f ( z ) = (cid:88) α ∈ A f α ( z ) = (cid:88) α ∈ A z α e − βh ( α ) − i Θ( α ) . • The tropical localized hypersurface (cid:101) H , defined by (cid:101) f ( z ) = 0 : (cid:101) f ( z ) = (cid:88) α ∈ A (cid:101) f α ( z ) = (cid:88) α ∈ A f α ( z ) χ α ( u ) , • The real-valued functions, F α ( z ) = | f α ( z ) | and (cid:101) F ( z ) = − (cid:88) (cid:54) = α ∈ A F α ( u ) χ α ( u ) . The proof of Theorem 2 follows from Propositions 5.7, 5.8 and 5.9. The proof of Theorem 3follows from Propositions 4.4 and 5.7.5.1.
Liouville Vector Field.
Take any point z ∈ (cid:101) H , we have X λ ( z ) = X (cid:107) λ ( z ) + X ⊥ λ ( z ) . where X ⊥ λ ( z ) is symplectically orthogonal to T z (cid:101) H . We note that X (cid:107) λ ( z ) = X λ H ( z ) , since for any v ∈ T z (cid:101) H , ω H ( X (cid:107) λ ( z ) , v ) = ω ( X λ ( z ) − X ⊥ λ ( z ) , v ) = ω ( X λ ( z ) − X ⊥ λ ( z ) , v ) = λ ( v ) = λ H ( v ) . And X ⊥ λ ( z ) is the symplectic horizontal lift of (cid:101) f ∗ ( X λ ( z )) ∈ T C . Definition 5.1.
The positive loci (cid:101) H + is the subset of (cid:101) H where (cid:101) f = − and (cid:101) f α ≥ for all α (cid:54) = 0 . Remark 5.2.
An equivalent definition is that (cid:101) H + = Log − β ( ∂ (cid:101) C ) . Proposition 5.3.
For all z ∈ (cid:101) H + , we have X Im (cid:101) f ( z ) positively proportional to X ⊥ λ ( z ) .Proof. Since both vectors are symplectic orthogonal to (cid:101) H , we only need to check their image under (cid:101) f ∗ are positively proportional to each other.First we study X Im (cid:101) f ( z ) . On (cid:101) H + , we have (cid:101) f = (cid:101) F . We also have d (cid:101) f = (cid:88) α ∈ A (cid:101) F α ( u ) (cid:104) α, d ( ρ + iθ ) (cid:105) + (cid:88) α ∈ A f α ( z ) dχ α ( u )= d (cid:101) F ( u ) + i (cid:88) α ∈ A (cid:101) F α ( z ) (cid:104) α, dθ (cid:105) Hence d ( Im f ) = Im df = (cid:88) α ∈ A (cid:101) F α ( u ) (cid:104) α, dθ (cid:105) Thus X Im f = (cid:88) α ∈ ∂A (cid:101) F α ( u ) (cid:88) i,j α i g ij ( ρ ) ∂ ρ j (5.1)compare with ∇ ( (cid:101) F ) = g − ( d (cid:101) F ) = g − ( (cid:88) α ∈ ∂A F α ( u ) χ α ( u ) (cid:104) α, dρ (cid:105) + F α dχ α ) = X Im f + O ( e −√ β ) . We thus have (cid:104) d (cid:101) f , X Im (cid:101) f (cid:105) = (cid:104) d (cid:101) F , X Im (cid:101) f (cid:105) = (cid:107)∇ (cid:101) F (cid:107) + O ( e −√ β ) > . Next, we study X ⊥ λ ( z ) . We have (cid:104) d (cid:101) f , X ⊥ λ ( z ) (cid:105) = (cid:104) d (cid:101) f , X λ ( z ) (cid:105) = (cid:104) d (cid:101) f , ∇ ϕ (cid:105) = (cid:104) d (cid:101) F , ∇ ϕ (cid:105) Since ∇ ϕ is positively proportional to the radial vector field u∂ u by Proposition 2.7, and (cid:104) d (cid:101) F , u∂ u (cid:105) > . We have also (cid:104) d (cid:101) f , X ⊥ λ ( z ) (cid:105) > .Since (cid:101) f ∗ ( X ⊥ λ ( z )) and (cid:101) f ∗ ( X Im f ) are both in the positive direction of T C , X ⊥ λ ( z ) is positivelyproportional to X Im (cid:101) f ( z ) . (cid:3) Critical Manifolds.
Recall from the previous section, that on the boundary of the amoeba ∂ (cid:101) C , the critical points of ϕ are indexed by τ ∈ ∂ T as (cid:101) ρ τ . Proposition 5.4.
The preimages
Crit τ := Log β | − (cid:101) H ( (cid:101) ρ τ ) are critical manifolds.Proof. Since the critical points (cid:101) ρ τ are in the ’good’ region (cid:101) H good ⊂ (cid:101) H , where the monomial cut-offfunctions χ α are either zero or one, hence the hypersurface (cid:101) H good is holomorphic. Thus, zero of dϕ | (cid:101) H is also zero of d c ϕ | (cid:101) H . (cid:3) Proposition 5.5.
For each τ ∈ ∂ T , Log β | − (cid:101) H ( (cid:101) ρ τ ) = { β (cid:101) ρ τ } × T τ, Θ , where T τ, Θ is defined in (3.2) .Proof. Since (cid:101) ρ τ is on the boundary ∂ (cid:101) C , we have (cid:80) α ∈ τ F α ( z ) . Comparing with the definingequation of (cid:101) H in a neighborhood of Log β | − (cid:101) H ( (cid:101) ρ τ ) , we have (cid:80) α ∈ τ f α ( z ) . Hence f α ( z )) = (cid:104) α, θ (cid:105) − Θ( α ) for each vertex α in τ . Thus the fiber is the torus T τ, Θ . (cid:3) AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n Unstable Manifolds.Proposition 5.6.
The Liouville vector field X λ H on the positive loci (cid:101) H + does not change the θ coordinate.In particular, the positive loci (cid:101) H + is preserved under the Liouville flow.Proof. Since X λ H = X (cid:107) λ = X λ − X ⊥ λ , suffice to check that X λ and X ⊥ λ does not change θ coordinates.We have X λ ∝ ρ∂ ρ , and X ⊥ λ ∝ X Im (cid:101) f . From Eq. (5.1), we see X Im (cid:101) f has no θ -component. Hence (cid:104) X λ H , dθ (cid:105) = 0 . (cid:3) Proposition 5.7.
For any τ ∈ ∂ T , the unstable manifold for Crit τ is (cid:102) W τ × T τ, Θ .Proof. From Proposition 5.6, we see the flowout of
Crit τ by the Liouville flow does not affect the M T component. Thus Louville flow X λ, (cid:101) H on (cid:101) H induces a flow on (cid:101) H + , and it descend to ∂ (cid:101) C , forwhich we denote as X λ,∂ (cid:101) C .On ∂ (cid:101) C good , X λ,∂ (cid:101) C agrees with ∇ ( ϕ | ∂ (cid:101) C ) . And they have the same critical points set. On ∂ (cid:101) C bad ,we have (cid:107) X λ,∂ (cid:101) C − ∇ ( ϕ | ∂ (cid:101) C ) (cid:107) = O ( e −√ β ) . Despite individual flowlines for the two vector fields with the same starting point in the goodregion may be split after flow through a bad region, we claim that for each critical point (cid:101) ρ τ , theunstable manifolds (cid:102) W X λ τ and (cid:102) W ∇ ϕτ for the two flows are the same.Let τ ∈ ∂ T have vertices { α , · · · , α k } . Then ∇ ( ϕ | (cid:101) C ) = ∇ ϕ − c ∇ (cid:101) F ∈ R · ρ∂ρ + (Φ ϕ ) − ∗ (Int cone τ ) and X λ,∂ (cid:101) C = X λ − X ⊥ λ = X λ − c ( u ) X Im (cid:101) f ∈ R · ρ∂ρ + (Φ ϕ ) − ∗ (Int cone τ ) where we used X ⊥ λ positively proportional to X Im (cid:101) f , and X Im (cid:101) f is given by Eq. (5.1). By similarargument in Proposition 4.4 that W ∇ ϕτ is dual to τ via Φ ∞ ϕ , we have W X λ τ is dual to τ via Φ ∞ ϕ . Thus (cid:102) W ∇ ϕτ and (cid:102) W ∇ ϕτ has to be the same. We drop the superscripts and denote both as (cid:102) W τ . (cid:3) Proposition 5.8.
For each τ ∈ ∂ T , the unstable manifold (cid:102) W τ × T τ, Θ is a Lagrangian in (cid:101) H .Proof. One can use the property of the Liouville flow to show the unstable manifold is isotropic,and then counting dimension dim R (cid:102) W τ × T τ, Θ = (dim τ ) + n − (dim τ + 1) = n − R (cid:101) H . We give an alternative proof. By Proposition 4.4, we have Φ ϕ (cone (cid:102) W τ ) × T τ, Θ = cone τ × T τ, Θ . However, cone τ × T τ, Θ is part of the conormal Lagrangian T ∗ T τ, Θ M T for the submanifold T τ, Θ in M T . Since Φ ϕ × id is a symplectomorphism between M C ∗ and T ∗ M T , we get cone (cid:102) W τ × T τ, Θ is aconical Lagrangian in M ∗ C ∗ . Finally, a Lagrangian restricts to a symplectic submanifold is isotropic.Thus by dimension counting, (cid:102) W τ × T τ, Θ = (cid:16) cone (cid:102) W τ × T τ, Θ (cid:17) (cid:92) (cid:101) H is a Lagrangian in (cid:101) H . (cid:3) No other Critical Points.Proposition 5.9.
There are no other zero of the Liouville vector field away from { Crit τ } .Proof. Suffice to prove that there are no zero of the Liouville vector field outside of the positiveloci (cid:101) H + . Here we only give the sketch the proof. We look at the good region first. Then d c ϕ | H = 0 is equivalent to dϕ | H = 0 , we only need to check there are no critical point for ϕ .Suppose there is a critical point ϕ at z ∈ (cid:101) H good , the terms labeled by α , · · · , α k are non-zero,i.e., near z , (cid:101) H is defined by k (cid:88) i =1 f α i ( z ) = 0 . Let τ ∈ T be the simplex with vertices { α , · · · , α k } . Let τ ∨ be the cell in the tropical amoeba Π ,and U τ ⊂ (cid:101) H good where the defining equation is as above. We split in to two cases below. Recall P is the polytope corresponding to vertex ∈ T . Let g denote the Euclidean metric on M R afteridentification M R ∼ = R n . (1) The case / ∈ τ . Then τ ∨ is a non-compact cell in Π , and intersect the amoeba polytope P atface F τ = P ∩ τ ∨ .Let u = Log β ( z ) , and let u (cid:48) denote the orthogonal projection w.r.t g to the cell τ ∨ . Then dist g ( u, u (cid:48) ) = O (1 / √ β ) . Let u (cid:48)(cid:48) denote the minimum of ϕ on F τ . We claim that ϕ ( u (cid:48)(cid:48) ) < ϕ ( u (cid:48) ) , since the increase level set of ϕ meet the convex cell τ ∨ first at u (cid:48)(cid:48) .Let v = u (cid:48)(cid:48) − u (cid:48) ∈ M R . If we view v as a tangent vector at u (cid:48)(cid:48) , then (cid:104) dϕ ( u (cid:48) ) , v (cid:105) < . Since u and u (cid:48) are O (1 / √ β ) close, we also have (cid:104) dϕ ( u (cid:48) ) , v (cid:105) < . Finally, one can check v can be lifted as a tangentvector to T z (cid:101) H , hence dϕ (cid:54) = 0 at z . (2) The case ∈ τ . Without loss of generality, we may assume
Θ(0) = π, h (0) = 0 , and α k = 0 .Thus, the defining equation of (cid:101) H near z can be written as k − (cid:88) i =1 f α i = k − (cid:88) i =1 e − i Θ( α i ) − βh ( α i ) e β (cid:104) α i ,u (cid:105) + i (cid:104) α i ,θ (cid:105) =: F ( u, θ ) Suppose z is a critical point of ϕ | { F =1 } , then there exists c , c , such that dϕ ( ρ ) = c d Re F ( ρ, θ ) + c d Im F ( ρ, θ ) . However, since dϕ ( ρ ) has no dθ component, hence dθ on the RHS need to be cancelled out. Usingall the α i are linearly independent, we can check this is only possible if all arg( f α i ) are equal ordiffer by π . Since (cid:80) i f α i = 1 , we get all f α i ∈ R , and at least one is positive.If all of f α i ( z ) are positive, then there is nothing to show, since we want to prove all the criticalpoints lies on the positive loci.If not all of f α i ( z ) are positive, say for i = 1 , · · · , m , f α i ( z ) < , then u lies on the real hypersur-face − e βl α ( u ) − · · · − e βl αm ( u ) + · · · + e βl αk +1 ( u ) =: H ( u ) . near the face τ ∨ on P . If we further require dϕ to be in the R -span of α , · · · , α k − , then u hasto be near the critical point of ϕ on face τ ∨ . One can show that dϕ has to be in the R + -span of α , · · · , α k − . Hence, there does not exists c ∈ R , such that dϕ ( u ) = cdH ( u ) . AGRANGIAN SKELETA OF HYPERSURFACES IN ( C ∗ ) n This concludes the discussion for z in the good region. If z is in the bad region, where at leastone < χ α ( z ) < , we will approximate the bad region using good region in the following way.Define a different set of cut-off functions, by changing the cut-off threshold from −√ β to − √ β ,ie. redefine χ α,α (cid:48) ,β ( u ) = χ ( β ( l α ( u ) − l α (cid:48) ( u )) + 10 (cid:112) β ) in Definition 1.4. Denote the new tropical localized hypersurface (cid:101) H . We claim the Hausdorffdistance between (cid:101) H and (cid:101) H in M C ∗ is O ( e − c √ β ) for some c > . Furthermore, their unit conormalbundles S ∗ (cid:101) H M C ∗ and S ∗ (cid:101) H M C ∗ should have distance O ( e − c √ β ) as well. A zero of d c ( ϕ | (cid:101) H ) corre-sponds to an intersection of Γ ∞ d c ϕ ⊂ S ∗ M C ∗ with S ∗ (cid:101) H M C ∗ , where Γ ∞ d c ϕ = (Γ d c ϕ ∩ ˙ T ∗ ( M C ∗ )) / R + ⊂ T ∞ ( M C ∗ ) ∼ = S ∗ ( M C ∗ ) . (cf Definition (4.1) and (4.2)). 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