NNON-COMMUTATIVE AMOEBAS
GRIGORY MIKHALKIN AND MIKHAIL SHKOLNIKOV
Abstract.
The group of isometries of the hyperbolic space H isthe 3-dimensional group PSL ( C ), which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroupSO(3) ⊂ PSL ( C ) naturally maps it back to H . Each fiber of thismap is diffeomorphic to the real projective 3-space RP .The resulting map PSL ( C ) → H can be viewed as the simplestnon-commutative counterpart of the map Log : ( C × ) n → R n fromthe commutative complex Lie group ( C × ) n with the Lagrangiantorus fibers that can considered as a Liouville-Arnold type inte-grable system. Gelfand, Kapranov and Zelevinsky [2] have in-troduced amoebas of algebraic varieties V ⊂ ( C × ) n as imagesLog( V ) ⊂ R n . We define the amoeba of an algebraic subvariety ofPSL ( C ) as its image in H . The paper surveys basic properties ofthe resulting hyperbolic amoebas and compares them against thecommutative amoebas R n . Introduction
Three-dimensional hyperbolic space.
The hyperbolic space H is a complete contractible 3-space enhanced with a Riemannianmetric of constant curvature −
1. Such a space is unique up to isometry.In the disk (Poincar´e) model we may represent H as the interior ofthe unit 3-disk D = { x ∈ R | || x || < } ⊂ R . The geodesics on H in this model are cut by planar circles (or straightlines) orthogonal to the boundary 2-sphere ∂D ⊂ R . This 2-sphere ∂D is called the absolute . Similarly, geodesic 2-planes in H are cutby 2-spheres (or planes) in R perpendicular to ∂D .The absolute ∂ H can be constructed intrinsically and independentof the choice of model. Let us choose a point x ∈ H . The set of geo-desic rays emanating from x can be identified with ∂ H by tracing theendpoint of the geodesic ray. On the other hand it can also be iden-tified with the unit tangent 2-sphere U T ( x ) ≈ S of the unit tangent Research is supported in part by the grants 182111 and 178828 of the SwissNational Science Foundation. a r X i v : . [ m a t h . C V ] F e b GRIGORY MIKHALKIN AND MIKHAIL SHKOLNIKOV vectors at x . This gives us a canonical identification U T ( x ) = ∂ H for any x ∈ H and, in particular, an identification U T ( x ) = U T ( y )for x, y ∈ H . It is easy to see that while this identification doesnot preserve the metric induced from the tangent space T H , it doespreserve the conformal structure associated to that metric. Thus theabsolute ∂ H comes with a natural conformal structure and can beidentified with the Riemann sphere ∂ H = CP . The Riemann sphere CP is obtained from the 2-dimensional vectorspace C by projectivization. The group GL ( C ) of linear transfor-mations of C acts also on CP so that a matrix (cid:18) a bc d (cid:19) acts by theso-called M¨obius transformations z (cid:55)→ az + bcz + d on CP = C ∪ {∞} . Theorem 1.1 (Classical) . Any isometry of H extends to the absolute ∂ H . Furthermore, there is a natural 1-1 correspondence between thegroup I of orientation-preserving isometries of the hyperbolic space H and the group PSL ( C ) of the M¨obius transformations of the absolute ∂ H ≈ CP . In other terms, I = PSL ( C ) = PGL ( C ) . Groups I = PSL ( C ) and ˜ I = SL ( C ) . The group of all M¨obiustransformation can be identified with the projectivization PGL ( C ) ofthe linear group GL ( C ) – two linear transformation induce the samelinear map if and only if one is a scalar multiple of another. Notethat as the complex numbers are algebraically closed, the projectiviza-tion PGL ( C ) of the general linear group GL ( C ) coincides with theprojectivization PSL ( C ) = SL ( C ) / {± } of the group SL ( C )The group SL ( C ) is the first non-trivial example of a complex sim-ple Lie group. Its maximal compact subgroup is the group of specialunitary matrices SU(2). We have a diffeomorphism SU(2) ≈ S asSU(2) acts transitively and without fixed points on the unit sphere S in C . We also have a diffeomorphismSL ( C ) ≈ T ∗ (SU(2)) ≈ S × R identifying SL ( C ) with the tangent space T ∗ (SU(2)) to its maximalcompact subgroup. ON-COMMUTATIVE AMOEBAS 3
In particular, SL ( C ) is simply connected and coincides with theuniversal covering ˜ I of the group I = PSL ( C ), while I is the quotientof ˜ I by its center (isomorphic to Z = {± } ). Topologically we have(1) I ≈ RP × R . Both spaces, I and its universal covering ˜ I will be important fur usas ambient spaces. They are complex 3-folds and contain differentsubvarieties, particularly, curves and surfaces. The goal of this paperis to look at geometry of these subvarieties in the context of hyperbolicgeometry, provided by presentation of I as the group of isometries of H .1.3. Compactifications I and ˆ I of the 3-folds I and ˜ I . Theorem1.1 provides a convenient way to compactly I . Indeed, the group I isidentified with the non-degenerate 2 × ad − bc vanishes. We get thefollowing proposition. Proposition 1.2.
We have I = CP (cid:114) Q, where Q is a smooth quadric { ad − bc = 0 } in the projective space CP (cid:51) [ a : b : c : d ] . We also use notation ∂ I = Q . Note that we can recover the computation π ( I ) = Z also from thisproposition as I is a complement of a smooth quadric in P . We set I = CP ⊃ I and use this space as the compactification of the group I . This view-point justifies the notation ∂ I = Q .In its turn, the group ˜ I = SL ( C ) of 2 × I = { ( a, b, c, d ) ∈ C | ad − bc = 1 } . Its topological closure ˆ I in CP ⊃ C is a smooth 3-dimensional projec-tive quadric. We use ˆ I as the compactification of the (simply connected)group ˜ I .Note that the projectivization CP of C can be identified with theinfinite hyperplane CP ∞ = CP (cid:114) C . Thus we may identify I = CP ∞ . Proposition 1.3. ˆ I ∩ CP ∞ = Q = ∂ I ⊂ I . GRIGORY MIKHALKIN AND MIKHAIL SHKOLNIKOV
Proof.
When we pass from C to CP we introduce a new coordinatethat vanishes on CP ∞ . For an equation of degree d in affine coordi-nates a, b, c, d ∈ C all monomials of order lower than d vanish when weapproach CP ∞ . In particular, the affine equation ad − bc = 1 in C becomes a homogeneous equation ad − bc = 0 in CP ∞ . (cid:3) Note that 0 / ∈ ˜ I ⊂ C , so the central projection from 0 defines a map π from the projective quadric ˆ I to the infinite hyperplane I . Proposition 1.4.
The map π : ˆ I → I is a double covering ramified along Q ⊂ I .Proof. We have π ( a, b, c, d ) = π ( − a, − b, − c, − d ), thus π − ([ a : b : c : d ])consists of two points, unless [ a : b : c : d ] ∈ Q . (cid:3) Map κ : I → H . Let us fix the origin point 0 ∈ H . As I is thegroup of isometries of H we can define the map(2) κ : I → H by I (cid:51) z (cid:55)→ z (0) ∈ H . Proposition 1.5.
The map (2) is a proper submersion. We have κ − ( x ) ≈ RP , x ∈ H .Proof. The fiber κ − (0) consists of isometries of H preserving the ori-gin 0 ∈ H . This group coincides with the group SO(3) ≈ RP ofisometries of the tangent space T ( H ). As H is a homogeneous spacefor the group I , the same holds for any other fiber κ − ( x ). (cid:3) Amoebas and coamoebas in H Amoebas.
Let V ⊂ I be an algebraic subvariety. This meansthat V = V (cid:114) ∂ I for a projective subvariety V ⊂ I = CP . Withoutloss of generality we may assume that V is the closure of V in I . Definition 2.1.
The amoeba A = κ ( V ) ⊂ H is the image of V under the map κ .This definition can be thought of as a hyperbolic (or non-commutative)counterpart of the amoebas of varieties in ( C × ) n defined in [2]. Theseamoebas are the primary geometric objects studied in this paper. ON-COMMUTATIVE AMOEBAS 5
The map κ can be extended to the compactification I ⊃ I once weextend the target H to its own compactification H = H ∪ ∂ H . Recall that an element of I = CP is a non-zero matrix (cid:18) a bc d (cid:19) up tomultiplication by a scalar. If a matrix is non-degenerate (rank 2) itis an element of I , while a degenerate non-zero (rank 1) matrix is anelement of ∂ I = I (cid:114) I .A matrix z ∈ ∂ I is a map C → C with a 1-dimensional kernel A ⊂ C and a 1-dimensional image B ⊂ C , thus A, B ∈ CP . Notethat a choice of such A and B uniquely determines z ∈ ∂ I as the onlyremaining ambiguity is a scalar factor. This gives us an isomorphismbetween the smooth projective quadric Q = ∂ I and CP × CP . Definethe projections to the first and second factors(3) π ± : Q → CP by π − ( z ) = A and π + ( z ) = B .For z ∈ ∂ I we define κ ( z ) = π + ( B ). For z ∈ I we define κ ( z ) = κ ( z ). Proposition 2.2.
The map κ : I → H is a continuous map from the complex projective 3-space.Proof. If z ∈ I is close to ∂ I then one of the two eigenvalues of thecorresponding unimodular linear map C → C is very small whilethe other is very large. Let us mark the points p, q corresponding tothese eigenspaces on the absolute ∂ H = CP . The projective-lineartransformation has these two points as its fixed points.Since the eigenvalue corresponding to q is very large, a small neigh-bourhood of q will contain the image of almost entire ∂ H (except fora small neighborhood of p ). The same holds for the extension of ourprojective-linear transformation to H . Thus the image of the originwill be contained in a small neighborhood of q , so that κ is continu-ous. (cid:3) Corollary 2.3.
The amoeba κ ( V ) ⊂ H is a closed set. Definition 2.4.
For a subvariety V ⊂ I ≈ CP we define its compact-ified amoeba ¯ A ⊂ H as the image of V under κ .The following proposition is straightforward. GRIGORY MIKHALKIN AND MIKHAIL SHKOLNIKOV
Proposition 2.5.
The left action of A ∈ I on V translates its amoebaby the isometry A, i.e. κ ( A · V ) = A ( κ ( V )) . The right action V (cid:55)→ V · A can significantly changes the shape of κ ( V ) ⊂ H . There is an obvious exception: the right action on asubvariety by a rotation A ∈ SO(3) around 0 ∈ H doesn’t change itsamoeba. If V is irreducible then its amoeba is connected.2.2. Coamoebas.
The topological diffeomorphism (1) can be upgradedonce we recall that the factor RP is a compact real Lie group G =SO(3). The bi-invariant metric on G is well-defined up to a scalar. Itcoincides with the spherical metric on RP .Furthermore, the whole group I can be recovered as the complexi-fication G C of G . Geometrically, G C can be identified with the totalspace T ∗ G of the tangent bundle to G . Namely, the tangent space tothe unit element 1 ∈ G is the Lie algebra g of G and can be thoughtof as infinitely small elements of G . Any element of G C = T ∗ G can berepresented as a product of an element of G and an element of g . Thetotal space G C can be given complex and group structures and comeswith the map ι : G C → G .In algebraic terms, with the help of polar decomposition of matriceswe can uniquely write ˜ z = ˜ up for any ˜ z ∈ ˜ I , where ˜ u is a unitarymatrix and p is a non-negatively definite hermitian matrix. Up to signthe matrix ± ˜ z gives an element of I . Its polar decomposition definesthe unitary matrix ± ˜ u up to sign, which in its turn can be consideredas an element of G . We define ι ( ± ˜ z ) = ± ˜ u and thus(4) ι : I = G C → G = SO(3) ≈ RP is a continuous map. Definition 2.6.
For a subvariety V ⊂ I we define its coamoeba B ⊂ G ≈ RP as the image B = ι ( V ).Let z ∈ I be an arbitrary isometry of H with κ ( z ) = x . A paralleltransport along a geodesic path in H connecting 0 and x provides apreferred isometry between tangent spaces T H and T x H and thus anelement p ⊂ I corresponding to a unimodular positive-definite hermit-ian matrix with κ ( p ) = x . The isometry z can be obtained by takinga composition of p with a self-isometry of T H which corresponds toan orthogonal matrix u ∈ G . Thus we recover polar decomposition inhyperbolic geometry terms. ON-COMMUTATIVE AMOEBAS 7 Amoebas of curves
Amoebas of lines.
The shape of the amoeba A l = κ ( l ∩ I ) ofa line l ⊂ CP = I ∪ Q depends on the position of l with respectto the quadric Q : either l lies on Q , is tangent to Q or intersects ittransversally in two points.In the case when the line lies on the quadric, the amoeba and theintersection of l with I are both empty. To describe the image of l under κ consider the same identification of Q with CP × CP givenby (3). There are exactly two families of lines in Q appearing as fibersof π + and π − . If the line l is a fiber of π + then its image is a singlepoint. If l is a fiber of π − then κ projects l isomorphically to ∂ H . If Q doesn’t contain l then they intersect either at one or two points.This cases give amoebas of quite different shape. Consider first a line l which meets the quadric exactly at one point, i.e. l is tangent to Q .This implies that the amoeba of l is non-empty and touches ∂ H at asingle point.Recall that a horosphere is a surface in H such that it is orthogonalto any geodesic starting at some fixed point at the infinity ∂ H . Proposition 3.1.
If a line is tangent to Q then its hyperbolic amoebais a horosphere in H . Conversely, any horosphere in H is an amoebaof a line tangent to Q . One can give the following equivalent statement for this proposition.A hyperbolic amoeba of a line in the 3-dimensional quadric ˜ I is a horo-sphere. Indeed, a curve in ˜ I is a line if and only if its image underthe two-fold covering ˜ I → I is a line tangent to Q . We also note thatboth families of lines on Q give amoebas, which can be interpretedas infinitely small and infinitely large horospheres. The proof of thisproposition is given after the proof of Lemma 3.6.An amoeba of a line transverse to Q is generic. It must be nonempty,with two infinite points at its closure. A cylinder of radius r ≥ H is defined to be a locus of points that are at the same distance r froma given geodesic. The degenerate cylinder for r = 0 coincides with thegeodesic itself. Proposition 3.2.
If a line is not tangent to Q then its hyperbolicamoeba is a (possibly degenerate) cylinder in H . Conversely, any geo-desic in H as well as any cylinder of radius r > is the amoeba of aline in I . Both left and right actions of I on itself can be uniquely extended to I = CP . Clearly, Q is invariant under these actions. GRIGORY MIKHALKIN AND MIKHAIL SHKOLNIKOV
Lemma 3.3.
The left action of I on Q = CP × CP acts by M¨obiustransformations on the second factor and conserves the first factor.Namely, for A ∈ I we have A · ( α, β ) = ( α, A ( β )) where A ( β ) ∈ CP isthe image of β under the M¨obius action.Similarly, the right action of I on Q = CP × CP conserves thesecond factor and acts on the first factor by M¨obius transformations ( α, β ) · A = ( A − ( α ) , β ) .Proof. Recall that Q corresponds to rank 1 matrices P , the coordinate α is the projectivization of the 1-dimensional kernel of P while β is theprojectivization of the 1-dimensional image of P . (cid:3) Corollary 3.4.
For any two lines l, l (cid:48) ⊂ I transverse to Q there existelements A, B ∈ I such that l (cid:48) = AlB . The transformations
A, B areunique up to (left or right) multiplication by subgroups isomorphic to C × .Proof. The lines l and l (cid:48) are determined by the pairs of their intersec-tion points with Q . Since both lines are transverse to Q , each pairproduces a pair of distinct points in CP under π + and π − . M¨obiustransformations act transitively on pairs of distinct points in CP withthe stabiliser isomorphic to C × . (cid:3) Consider the space of all lines in CP tangent to Q . We have anaction of I × SO(3) on this space, where the first factor I acts on theleft and the second factor SO(3) ⊂ I acts on the right. Lines in thesame orbit of this action have congruent hyperbolic amoebas. We claimthat there are only three different orbits for the action. Lemma 3.5.
Two families of lines lying on Q and a set of all otherlines tangent to Q are the only orbits for the action of I × SO(3) on Q. Thus, if we show that an amoeba of some particular properly tangentline to Q is a horosphere then amoebas of all other lines of this kindwill be horosphere. Proof.
Let x be a point in Q . Take a stabiliser subgroup for x under theaction of I × SO(3) and consider its action on the tangent space to Q atthe point x . It is clear that the stabiliser has a subgroup isomorphic to C × × U (1) and acts separately on each multiplier in T x Q = C × C . Theaction of this subgroup on the projectivization P ( T x Q ) for the tangentspace can be obviously reduced to the standard action of C × on CP .The last is stratified on three orbits: two points and a torus. The pointscorrespond to the lines in the intersection of Q and a plane tangent to Q at x . The torus parametrizes the space of all lines properly tangentto Q at x . ON-COMMUTATIVE AMOEBAS 9
To finish the prove note that I × SO(3) acts transitively on Q andevidently preserves the stratification for projectivization of a tangentspace to Q at each point. (cid:3) Our goal now is to show that there exist a line tangent to Q withan amoeba equal to a horosphere in H . The main idea here is to useinteractions of some specific subgroups of I to produce extra symmetriesfor their amoebas.As we saw before, the group I can be interpreted as the group ofautomorphisms for CP = C ∪ {∞} . A group B of affine transforma-tions on the complex line C is a 2-dimensional subgroup of I . It can bealso defined to be a stabiliser of ∞ . In fact B can be described up toconjugation as a Borel subgroup of I . Each element of I can be seen aM¨obius transformation z (cid:55)→ az + bcz + d , ad − bc (cid:54) = 0 . Such transformation is affine if and only if c = 0 and is given by z (cid:55)→ az + b . So the closure of B is a plane in CP .Consider the following two subgroups in B : the subgroup l = { z (cid:55)→ z + b } of translations in C and the subgroup l = { z (cid:55)→ az } generatedby homotheties and rotations around 0 ∈ C . In the above notations l is given by the equation a = 1 and l is given by b = 0. So the closuresfor both subgroups are lines in CP . It is also clear that l is a normalsubgroup of B and so l acts on l by the conjugation, l is a maximalaffine subgroup and l is a maximal torus in I . Since l and l intersectby a unity and generate the whole group of affine transformations, B is a semi-direct product of l and l .Now we are going to describe amoebas for these groups. First wehave to note that an amoeba of any subgroup in I is smooth at eachpoint because the group acts on its amoeba transitively by isometries.We start with l . The point 0 and ∞ in ∂ H are the only pointsstabilized by l . Denote by γ ⊂ H the geodesic connecting these points. Lemma 3.6.
We have A l = γ .Proof. The elements of l correspond to isometries of H fixing thegeodesic γ . (cid:3) The only point in ∂ H which is preserved by l is the point ∞ . Theamoeba A l is the horosphere tangent to ∞ and passing through 0.This observation completes the proof of the Proposition 3.1. Proof of Proposition 3.2.
A line l not tangent to Q meets the quadricat two points ( q , p ) , ( q , p ) ∈ Q. Since l is not contained in Q , we have p (cid:54) = p ∈ CP and q (cid:54) = q ∈ CP . Thus there exist M¨obiustransformations sending pairs ( p , p ) and ( q , q ) to (0 , ∞ ). Note thatthe line l meets Q at the points (0 ,
0) and ( ∞ , ∞ ). By Corollary 3.4there exist A, B ∈ I such that l = Al B . Since l consist of isometries of H fixing γ , the amoeba A l B is a cylinder around γ of radius equal tothe distance between B (0) and γ . By Proposition 2.5, A l is a cylinderobtained as the image of A l B under the isometry A . (cid:3) Proposition 3.7.
Let l be the line connecting ( q , p ) and ( q , p ) ∈ CP × CP = Q with p (cid:54) = p ∈ CP = C ⊂ {∞} and q (cid:54) = q , ∈ CP = C ⊂ {∞} . (1) The amoeba A l is a geodesic line if and only if q = − / ¯ q . In this case κ | l ∩ I is a circle bundle over its image.Conversely, for each geodesic line γ ⊂ H and a point x ∈ κ − ( γ ) there exists a unique line l (cid:51) x such that A l = γ . (2) If q = − / ¯ q then A l is a non-degenerate cylinder whose radiusdepends only on the spherical distance between q and − / ¯ q (the antipodal point to q ∈ CP ). Dependance of the radiusof this distance is monotone and goes to infinity when q ap-proaches q . In this case κ | l ∩ I is a smooth proper embedding to H .Conversely, for any non-degenerate cylinder Z ⊂ H and apoint x ∈ κ − ( Z ) there exists a line l (cid:51) x such that A l = Z . The spherical distance on CP ≈ S is the distance in the metricpreserved by the subgroup SO(3) ⊂ I acting on CP by M¨obius trans-formations. Proof.
By the proof of Lemma 3.6, κ ( l B ) = γ if and only if B ∈ κ − ( γ ). This is the case when the right action of B decomposes toa transformation of H preserving the origin and a transformation of H preserving γ . Since the subgroup SO(3) ⊂ I consists of isometriesof H preserving the origin, the amoeba κ ( l ) is a geodesic if and onlyif q and q are antipodal points in the spherical metric on CP , i.e. q = − / ¯ q . By Proposition 3.2 any cylinder in H is an amoeba of a line passingthrough ( q , p ) , ( q , p ) ∈ Q . Multiplying this line by an appropriateelement of SO(3) on the right we may ensure that l contains any givenpoint over its amoeba. Distinct lines passing through ( q , p ) cannotintersect at a point x (cid:54) = ( q , p ). This implies uniqueness of a line withthe same amoeba up to the right action of SO(3). In its turn, the ON-COMMUTATIVE AMOEBAS 11 uniqueness implies monotonicity of the cylinder radius as a function ofspherical distance between q and q . (cid:3) Gauss maps γ ± . Let C ⊂ I = CP be an irreducible spatialprojective curve of degree d and genus g not contained in ∂ I = Q .Define C = C (cid:114) ∂ I ⊂ I and A C = κ ( C ∩ I ) ⊂ H . Since Q ⊂ CP is aquadric, we get the following proposition for the compactified amoeba A C . Proposition 3.8.
The intersection A C ∪ ∂ H = κ ( C ∩ Q ) consists ofnot more than d points. We define l x ⊂ I as the tangent line to C at x ∈ C . The intersection l x ∩ Q is an unordered pair of (perhaps coinciding) points in Q = CP × CP . The projections of this pair under π − and π + define theelements γ − ( x ) , γ + ( x ) ∈ Sym ( CP ) = CP . By the removable singularity theorem, these maps uniquely extend tomaps γ − , γ + : C → CP called the right and left Gauss maps . Remark . The maps γ ± are non-commutative counterparts of thelogarithmic Gauss map [6]. They can be alternatively defined by takingthe tangent direction of C at the unit element E ∈ I after left or righttranslate of C by x ∈ I . To see this, we identify the projectivization T E ( I ) ≈ CP of the tangent space of I at the unit element E ∈ I withthe space of lines in I passing through E . The image π ± ( L ∩ Q ) can beconsidered as a point inSym ( CP ) = CP = T E ( I ) . By Lemma 3.3, the left action conserves π + while the right actionconserves π − . Proposition 3.10.
The degree of γ ± is d − g .Proof. Note that the image of the map CP → Sym ( CP ) = CP mapping z ∈ CP to the unordered pair consisting of z and a fixedpoint z ∈ CP is 1. Thus the degree of γ ± can be computed as thenumber of planes in CP in the pencil passing through a given line in Q that are tangent to C . The proposition follows from the Riemann-Hurwitz formula for the corresponding map from C to the pencil. (cid:3) Denote by R ⊂ CP the fixed locus of the antiholomorphic involution σ R ( u : v : w ) (cid:55)→ ( ¯ w : − ¯ v : ¯ u ) . The holomorphic change of coordinates ( u : v : w ) (cid:55)→ ( u + w : iv : i ( w − u )) identifies σ R with the involution of complex conjugation in CP and R with RP . In particular, R is a totally real surface in CP . The following lemma is a counterpart of [8, Lemma 3] for non-commutative amoebas. Lemma 3.11.
A smooth point x ∈ C is critical for κ | C if and only if γ − ( x ) ∈ R .Proof. The point x is critical for κ | C if and only if it is critical for therestriction of κ to the tangent line l x ⊂ I to C at x . By Proposition 3.7this is determined by the type of the amoeba of l x . If the amoeba of l x is a geodesic line then all points of l x are critical for κ | l x . If the amoebaof l x is a non-degenerate cylinder then all points of l x are regular for κ | l x .In the affine chart C (cid:51) ( a, b ) corresponding to u (cid:54) = 0 we have theidentification C = Sym ( C ) with a = z + z (cid:48) , b = zz (cid:48) for z, z (cid:48) ∈ C .The antiholomorphic involution σ R is given by the involution z (cid:55)→− / ¯ q . By Proposition 3.7 its fixed locus corresponds to the lineswhose amoebas are geodesic lines. (cid:3) Theorem 3.12.
Let Ω be a domain in C and φ : Ω → I be a holomor-phic embedding. If κ ◦ φ is critical at every point then φ parametrizesa part of a line in I projecting by κ on a geodesic. Corollary 3.13.
The amoeba A C ⊂ H of a non-singular irreduciblecurve C ⊂ CP is smoothly immersed at its generic point.Proof of Theorem 3.12. By Lemma 3.11, if κ ◦ φ is critical then itsimage is contained in the totally real surface R . Thus κ ◦ φ is constantand φ (Ω) is everywhere tangent to the same line. (cid:3) Tropical limits of amoebas of curves
Tropical limits for curves in ( C × ) n . Let us recall how tropicallimits appear in the context of conventional (commutative) amoebas,i.e. amoebas of algebraic varieties V ⊂ ( C × ) n , see [4]. The Lie group( C × ) n is commutative, its maximal compact subgroup is the real torus( S ) n . The mapLog t : ( C × ) n → R n , Log t ( z , . . . , z n ) = (log t | z | , . . . , log t | z n | )takes the quotient by this subgroup. The image Log( V ) ⊂ R n is calledthe amoeba of V for t = e , using the notation Log = Log e . ON-COMMUTATIVE AMOEBAS 13
Definition 4.1.
An unbounded family { t α } of real numbers t α > α ∈ A , is called a scaling sequence .Scaling sequences allow to define tropical limits lim trop for familiesof various geometric objects parameterised by A (here we particularlyrefer to [3] and [5] for examples and details). Such families are called scaled families . In the simplest situation, if { z α } α ∈ A is a family ofcomplex numbers we definelim trop z α = lim t α →∞ log t α | z α | . This limit may be finite, i.e. an element of R , infinite, i.e. −∞ or + ∞ ,or not exist at all. Since R ∪ {±∞} ≈ [0 ,
1] is compact, there exists anunbounded subfamily B ⊂ A such that lim trop z β ∈ R ∪ {±∞} exists, t β → + ∞ , β ∈ B .If V α ⊂ ( C × ) n , α ∈ A , is a family of closed (in the Euclidean topol-ogy) sets (e.g. algebraic curves of the same degree) we definelim trop V α = lim t α →∞ Log t α ( V α ) ⊂ R n , where the latter limit is considered in the sense of the Hausdorff metricin all compact subsets of R n .By an open finite graph we mean the complement Γ = ¯Γ (cid:114) ∂ Γ of asubset ∂ Γ of the set of 1-valent vertices of a finite graph ¯Γ.
Definition 4.2.
A tropical curve is a finite metric graph Γ with acomplete inner metric together with a choice of the genus function g : Vert(Γ) → Z ≥ from the set of vertices of Γ such that g ( v ) (cid:54) = 0 if v is a 1-valent vertex. A vertex v of Γ is called essential if its genus g ( v ) is positive or its valence is greater than 2. Two tropical curvesare considered to be the same if there exists an isometry between themsuch that a vertex of positive genus corresponds to a vertex of the samegenus. Clearly, essential vertices must correspond to essential verticesunder such a correspondence.Recall that a metric is inner if the distance between any two pointsis given by the smallest length of a connecting path. Such metriccarries the same information as specifying the lengths of all edges.Completeness of the metric is equivalent to the condition that the openleaf-edges, i.e. the edges adjacent to ∂ Γ ⊂ ¯Γ, have the infinite lengths.Note that the simplest tropical curve I ≈ R has a single (double-infinite) edge and no vertices. It is obtained from the interval graph (afinite graph with two vertices and a connecting edge) by removing both1-valent vertices. A circle E τ = R /τ Z of perimeter τ > vertices. It is called a (compact) tropical elliptic curve of length τ . Allother connected tropical curves have at least one essential vertex. Definition 4.3.
A parameterised tropical curve in R n is a continuousmap h : Γ → R n from a tropical curve Γ such that the following twoconditions hold. • The restriction h | E to each edge E ⊂ Γ is a smooth map whosedifferential sends the unit tangent vector to E (with a chosenorientation) to an integer vector u ( E ) ∈ Z n . • At every vertex v ∈ Γ the balancing condition (5) (cid:88) E u ( E ) = 0hold. Here the sum is taken over all edges E adjacent to v oriented away from v .The image Y = h (Γ) ⊂ R n can be viewed as a graph embedded to R n .For a generic point y of an edge E y ⊂ Y the inverse image h − ( y ) is afinite set contained inside the edges of Γ. We define the weight w ( E y ) = (cid:88) x ∈ h − ( y ) w ( E x ) , where E x is the edge containing x , and w ( E x ) ∈ Z ≥ is the largestinteger number such that u ( E x ) /w ( E x ) is integer. By the balancingcondition, w ( E y ) does not depend on the choice of y . The image Y ⊂ R n with the weight data is called the unparameterised tropical curve .For an algebraic curve V α ⊂ ( C × ) n we may define its degree deg( V α ) ∈ Z ≥ as the projective degree of its closure in CP n ⊃ ( C × ) n . There isa corresponding notion for tropical curves in R n . Namely, we orienteach unbounded edge E of an unparameterised tropical curve Y ⊂ R n toward infinity and define deg( E ) to be zero if all the coordinates of u ( E ) ∈ Z n are non-positive and to be the maximal value of these co-ordinates otherwise, see [11]. We define deg Y ∈ Z ≥ as the sum of thedegrees of all its unbounded edges. Theorem 4.4 (Unparameterised tropical curve compactness theorem[10]) . If V α ⊂ ( C × ) n , t α → ∞ , α ∈ A , is a scaled family of curves ofdegree d then there exists a scaling subsequence t β → ∞ , β ∈ B ⊂ A ,and an unparameterised tropical curve Y ⊂ R n of degree deg Y ≤ d such that lim trop V β = Y . Phase-tropical limits over tropical curves in R n . In thissetting it is convenient to present V α ⊂ ( C × ) n as the image of a proper ON-COMMUTATIVE AMOEBAS 15 holomorphic map(6) f α : C α → ( C × ) n from a Riemann surface of finite type C α , α ∈ A .Suppose that C α is connected for all α ∈ A , and that the genus g of C α and the number k of its punctures does not depend on α . Bya nodal curve of genus g with k punctures we mean a connected butpossibly reducible nodal curve, such that its topological smoothing is a(smooth) curve of genus g with k punctures. Assume that 2 − g − k <
0. A stable curve is a nodal curve such that each of its componentsis either of positive genus, or is a sphere adjacent to at least threenodes or punctures. In other words, a connected nodal curve is notstable if it contains a punctured spherical component passing througha single node or a non-punctured spherical component passing throughone or two nodes. Such components can be contracted while staying inthe class of connected nodal curves and thus any nodal curve can bemodified to a canonical stable curve. Given g and k , the space of stablecurves is compact, and admits a universal curve over it, see [1]. If wedistinguish different punctures, i.e. mark them by numbers 1 , . . . , k ,then the universal curve is known as U g,k , it admits a continuous map(7) U g,k → M g,k over the space of the stable curves so that the fiber over each curve C ∈ M g,k is this curve itself.By a closed nodal curve we mean a nodal curve without punctures.Given a (nodal) curve C of genus g with k punctures we may considerthe corresponding closed curve C with k marked points by filling thepoint at each puncture and marking it. The map f uniquely extendsto(8) ¯ f : C → CP n which we call nodal curve of genus g with k marked points. Its degreeis the sum of the degrees of its components. The curve (8) is stable ifevery component of C contracted to a point by ¯ f is either of positivegenus, or is a sphere adjacent to at least three nodes or punctures. Inother words, we do not put any restrictions on the components of C mapped nontrivially by f and put the same restrictions on componentscontracted to points as before.As it was observed in [7], all stable curves of a given degree in CP n form a compact space M g,k ( CP n , d ). In addition we have the universalcurve(9) F : U g,k ( CP n , d ) → CP n and the map(10) π : U g,k ( CP n , d ) → M g,k ( CP n , d )such that for any ¯ f ∈ M g,k ( CP n , d ) the inverse image π − ( ¯ f ) is a nodalcurve, and the restriction of F to π − ( ¯ f ) coincides with ¯ f . Furthermore,we have the continuous forgetting map(11) ˜ft : U g,k ( CP n , d ) → U g,k mapping a stable curve (8) in the fiber of (10) to the correspondingstable curve in the fiber of (7) obtained from the source nodal curve C after contracting all its non-stable spherical components. In particular,it induces the forgetting map(12) ft : M g,k ( CP n , d ) → M g,k . We use a combination of the universal curves (7) and (9) to definethe phase-tropical limit of the scaled sequence (6). This technique isborrowed from [5], where it is used also for definition of further tropicalnotions in the limiting tropical curve, in particular, differential forms.By a straight holomorphic cylinder in ( C × ) n we mean the subset Z = { b ( z a , . . . , z a n n ) | z ∈ C × } ⊂ ( C × ) n , where a = ( a , . . . , a n ) ∈ Z n and b ∈ ( C × ) n . This set is the mul-tiplicative translate bT a of the 1-dimensional torus subgroup T a = { ( z a , . . . , z a n n ) | z ∈ C × } of ( C × ) n . Lemma 4.5.
Let φ : A → ( C × ) n be a non-constant (parameterized)proper algebraic curve. Then the number of punctures of A is at leasttwo. Furthermore, if the number of punctures of A is two then theimage of A is a straight holomorphic cylinder in ( C × ) n .Proof. The closure φ ( A ) ⊂ CP n of φ ( A ) in CP n ⊃ ( C × ) n must intersecteach of the ( n + 1) coordinate hyperplanes in CP n . Thus φ ( A ) (cid:114) ( C × ) n contains at least two points. By the properness of φ each of these pointsmust correspond to a puncture.Suppose that A has two punctures. Denote by δ ∈ H (( C × ) n ) = Z n the class of the image of the simple positive loop going around one ofthe punctures. Then the similar loop for the other puncture gives isthe class − δ . Let π δ : ( C × ) n → ( C × ) n − be a surjective homomorphismwhose kernel is T δ . Then the image of a loop around each puncture of A under π δ ◦ φ is homologically trivial. Thus π δ ◦ φ ( A ) is constant. (cid:3) Corollary 4.6.
Suppose that the family (6) converges to a stable curve ¯ f : C → CP n when t α → ∞ . Then each spherical component K ⊂ C must be adjacent at least to two distinct marked or nodal points. ON-COMMUTATIVE AMOEBAS 17
Proof.
The restriction ¯ f | K is non-constant by the stability of ¯ f . Thus¯ f ( K ) ∩ ( C × ) l (cid:54) = ∅ for a coordinate subspace (intersection of n − l coordinate hyperplanes) CP l ⊂ CP n , 1 ≤ l ≤ n . By Lemma 4.5,¯ f − ( CP l (cid:114) ( C × ) l ) consists of at least two points. The loops going aroundthese points are non-trivial in H (( C × ) l ), and thus they cannot comeas limits of boundaries of non-trivial holomorphic disks in ( C × ) n . (cid:3) Corollary 4.7.
Let (cid:98) C be the stable curve obtained as the image by (12) of a stable curve ¯ f : C → CP n obtained as the limit of the family (6) when t α → ∞ . If K ⊂ C is a component which is either non-sphericalor is adjacent to at least three nodal or marked points then the map (11) maps K isomorphically to a component of (cid:98) C . In other words, themap (11) does not contract K to a point.Proof. By Corollary 4.6 the only components contracted by ˜ft | C arespherical components with two nodal or marked points. Their contrac-tion does not reduce the number of nodal or marked points adjacentto other components of C . (cid:3) Suppose that the family C α converges to (cid:98) C ∈ M g,k when t α → ∞ .Let z α ∈ C α be a family of points converging to z = lim t α →∞ z α ∈ (cid:98) C ⊂U g,k . Let(13) τ α = | f α ( z α ) | − ∈ R n> ⊂ ( C × ) n be obtained from f α ( z α ) by taking the absolute inverse value coordinate-wise. Then we have τ α f α ( C α ) ∩ Log − t α (0) (cid:54) = ∅ for the multiplicativetranslate τ α f α ( C α ) of f α ( K α ) in ( C × ) n by τ α . In particular, if thefamily(14) τ α f α : C α → ( C × ) n closes up to a converging family(15) τ α f α : C α → CP n in M g,k ( CP n , d ) then the limit(16) ¯ f τ : C τ → CP n of the family (15) when t α → ∞ has a component K ⊂ C τ containingan accumulation point z τ of z α ∈ C α inside U g,k ( CP n , d ). The point z τ is mapped to z under the map C τ → (cid:98) C induced by the map (11). ByCorollary 4.7, if z ∈ (cid:98) C is not a nodal or marked point then ˜ft | K is anisomorphism to its image. Thus z τ is the unique accumulation point,i.e. the limit of the points z α in U g,k ( CP n , d ). We get the followingproposition. Proposition 4.8.
Suppose that (6) is a scaled family such that C α converges to (cid:98) C ∈ M g,k , and z α ∈ C α is a family of points in (6) converging to a point z ∈ (cid:98) C in U g,k , and such that (15) converges to (16) , where τ α ∈ R n> are defined by (13) . If z ∈ (cid:98) C is neither nodal normarked then the lift of z α to U g,k ( CP n , d ) under (11) with the help of (15) also converges to a point of C τ . Proposition 4.9.
Under the hypotheses of the previous proposition,suppose in addition that w α ∈ C α is another family of points convergingto a point w ∈ (cid:98) C in U g,k . If z and w are non-nodal, non-marked,and belong to the same component K ⊂ (cid:98) C then σ α f α : C α → ( C × ) n , σ α = | f α ( w α ) | − , also converges to a stable curve ¯ f σ : C σ → CP n .Furthermore, the restrictions ¯ f σ | K ◦ σ and ¯ f τ | K ◦ τ both take value in ( C × ) n and one map is obtained from the other by multiplication by lim t α →∞ σ α /τ α (which exists). Here K σ ⊂ C σ and K τ ⊂ C τ are the com-ponents corresponding to K under (11) and the superscript K ◦ for acomponent K signifies the complement of the set of nodal and markedpoints.Proof. The limit lim t α →∞ σ α /τ α coincides with ¯ f τ ( w τ ), where w τ ∈ K τ isthe point corresponding to w under the isomorphism K τ ≈ K . Multi-plication by σ α /τ α ∈ R n> identifies τ α f α into σ α f α . (cid:3) Definition 4.10.
We say that a scaled family (6) of curves of degree d in CP n ⊃ ( C × ) n converges phase-tropically if the following conditionshold: • The family C α converges to a curve (cid:98) C ∈ M g,k , t α → ∞ . • For each component K ⊂ (cid:98) C there exists a point z ∈ K ◦ anda family z α ∈ C α converging to z ∈ (cid:98) C in U g,k such that (15)converges to a map in M g,k ( CP n , d ) and such that(17) h K = lim trop f α ( z α ) ∈ [ −∞ , ∞ ] n exists. Here K ◦ is the complement of the nodal and markedpoints in K .If h K ∈ R n then the component K is called tropically finite , other-wise infinite. The source C τ of the limiting map (16) of (15) has acomponent corresponding to K by Corollary 4.7. We refer to the map(18) φ K : K ◦ → ( C × ) n defined by φ K = ¯ f τ | K ◦ as well as its image(19) Φ( K ) = φ K ( K ◦ ) ⊂ ( C × ) n ON-COMMUTATIVE AMOEBAS 19 as the phase of the component K ⊂ (cid:98) C . Phases are defined up tomultiplication by an element of R n> in ( C × ) n .The following proposition shows independence of the phases fromthe choice of z α and thus justifies the notations free from z α or τ . Proposition 4.11.
In Definition 4.10 neither the tropical limit (17) nor the phase (18) of a component K ⊂ (cid:98) C depends on the choice of afamily z α ∈ C α converging to a point z ∈ K ◦ ⊂ (cid:98) C .Proof. For two choices z α → z and w α → w with z, w ∈ K ◦ the limit of | z α | / | w α | exists by Proposition 4.9. Thus the tropical limits (involvingrescaling by 1 / log t α ) of z α and w α must coincide. The phase of K does not depend on the choice of z α by Proposition 4.9. (cid:3) A nodal or marked point p ∈ K ⊂ (cid:98) C corresponds to a punctureof K ◦ . Define γ p ⊂ K ◦ to be a simple loop going around p in thenegative direction with respect to p (and thus in the positive directionwith respect to K (cid:114) { p } ), and set δ K ( p ) = [ φ K ( γ p )] ∈ H (( C × ) n ) = Z n . Compactifying φ K : K ◦ → ( C × ) n to ¯ φ K : K → CP n and expanding¯ φ at p to a series in the corresponding affine coordinates, we get thefollowing proposition. Proposition 4.12 (cf. [9]) . If δ K ( p ) (cid:54) = 0 then the limit Φ( K, p ) = lim s → + ∞ s δ K ( p ) Φ( K ) ,s ∈ R > , is a straight holomorphic cylinder given by { bz δ K ( p ) | z ∈ C × } for some b ∈ ( C × ) n . The notations s δ K ( p ) ∈ R > and z δ K ( p ) ∈ ( C × ) n refer to taking power coordinatewise by δ K ( p ) . If δ K ( p ) = 0 then Φ( K, p ) = Φ( K ) . Recall that Φ(
K, p ) as well as Φ( K ) is defined up to a multiplicativetranslation. The notations s δ K ( p ) ∈ R > and z δ K ( p ) ∈ ( C × ) n in theproposition above refer to taking power coordinatewise by δ K ( p ). Proposition 4.13.
Suppose that (6) converges phase-tropically and p α ∈ C α is a family of points converging to a point p ∈ K ⊂ (cid:98) C whichis either nodal or marked. Then for a sufficiently small neighborhood U ⊂ U g,k of p ∈ (cid:98) C ⊂ U g,k the limit Φ( p ) = lim t α →∞ | f α ( p α ) | − f ( U ∩ C α ) ⊂ ( C × ) n (with respect to the Hausdorff metric on neighborhoods of compacts)exists and coincides with Φ( K, p ) if δ K ( p ) (cid:54) = 0 . Furthermore, if K is tropically finite then any accumulation point of Log t α ( f α ( p α )) ∈ R n is contained in the ray R K,p ⊂ R n emanating from h K ∈ R n in the direction of δ K ( p ) .Proof. To prove the convergence it suffices to show that each subse-quence has a subsequence convergent to Φ(
K, p ). Passing to a subse-quence, we may assume that | f α ( p α ) | − f ( C α ) yields a convergent familyin M g,k ( CP n , d ) with the limiting curve ¯ f p : C p → CP n , π : C p → (cid:101) C ,such that z α converges to z ∈ C p in U g,k ( CP n , d ). Since δ K ( p ) (cid:54) = 0,the point z cannot be a nodal or marked point of C p and must becontained in a component K z ⊂ C p contracted by π . By Lemma 4.5,¯ f p ( K ◦ z ) ⊂ ( C × ) n is a straight holomorphic cylinder. To see that itcoincides with Φ( K, p ) it suffices to change the coordinates in ( C × ) n (and accordingly, the toric compactification CP n ⊃ ( C × ) n ) so that δ K ( p ) = (0 , . . . , , − n ), with n ∈ Z > . Then the projectivization¯ φ K : K → CP n of (18) maps p to a point inside the n th coordinatehyperplane, and the argument of ¯ φ K ( p ) ∈ ( C × ) n − × { } determinesthe argument of Φ( p ).To locate accumulation points of the sequence Log t α ( f α ( p α )) we com-pare it against the sequence Log t α ( f α ( z α ) in (17). The ratio f α ( p α ) /f α ( z α )converges to ¯ φ K ( p ) ∈ ( C × ) n − × { } and thus the first ( n −
1) coor-dinates of Log t α ( f α ( p α ) /f α ( z α )) go to zero while the n th coordinate isessentially non-positive. (cid:3) Let (6) be a phase-tropically convergent family with the limitingcurve (cid:98) C ∈ M g,k . Let (cid:98) C ◦ ⊂ (cid:98) C be the union of tropically finite compo-nents of (cid:98) C . (Note that (cid:98) C ◦ may be disconnected or empty.)We define the extended dual subgraph ˜Γ of (cid:98) C ◦ to incorporate notonly its nodal, but also its marked points in the following way. The vertices v K ∈ ˜Γ correspond to the components K ⊂ (cid:98) C ◦ . Boundededges E p ⊂ ˜Γ correspond to the nodal points p ∈ (cid:98) C ◦ , they connectthe vertices corresponding to the adjacent tropically finite components(could be the same component). Vertices together with bounded edgesform the dual graph Γ( (cid:98) C ◦ ). To get the open finite graph ˜Γ we attachto Γ( (cid:98) C ) the leaves , or half-infinite edges, E q ≈ [0 , + ∞ ) correspondingto marked points q ∈ (cid:98) C contained in tropically finite components K ,and also to nodal points q ∈ (cid:98) C adjacent simultaneously to a tropicallyfinite component K and to an infinite component of (cid:98) C . We attach E q ≈ [0 , + ∞ ) to Γ( (cid:98) C ◦ ) by identifying 0 ∈ [0 , + ∞ ) with v K ∈ Γ( (cid:98) C ◦ ).Our next goal is to define ˜ h : ˜Γ → R n . ON-COMMUTATIVE AMOEBAS 21
We set ˜ h ( v K ) = h K ∈ R n using (17). Let p ∈ (cid:98) C be a nodal pointbetween components K and K (cid:48) . If v K = v K (cid:48) we define ˜ h | E p to bethe constant map to h K . If v K (cid:54) = v K (cid:48) then y Proposition 4.13 ˜ h ( K (cid:48) ) − ˜ h ( K ) = sδ p ( K ) with s >
0. In particular, in this case δ K ( p ) = δ K (cid:48) ( p ) (cid:54) =0. We identify a bounded edge E p with the Euclidean interval of length s , and define ˜ h | E p to be the affine map to the interval [˜ h ( K ) , ˜ h ( K (cid:48) )].We identify a leaf E q with the Euclidean ray [0 , + ∞ ), and define ˜ h | E q tobe the affine map to the ray emanating from ˜ h ( K ) in the direction of δ q ( K ) stretching the length | δ q ( K ) | times.Define Γ to be the (open finite) graph obtained from ˜Γ by contractingall edges collapsed to points by ˜ h and(20) h : Γ → R n to be the map induced by ˜ h . Each vertex v ∈ Γ corresponds to aconnected subgraph ˜Γ v ⊂ ˜Γ. We define the genus function g ( v ) to bethe sum of the genera of all components of (cid:101) C corresponding to thevertices of ˜Γ v and the number of cycles in ˜Γ v . Proposition 4.14.
The map (20) is a parameterised tropical curve.Proof.
Each edge E p ⊂ Γ corresponds to a marked or nodal point p of (cid:98) C and thus to an embedded vanishing circle γ p of C α for large t α . Thecircle γ p is oriented by the choice of a component K ⊂ (cid:98) C containing p , and thus by a choice of the vertex v K adjacent to E p . This choiceis equivalent to the orientation of the E p , and thus to the choice ofthe unit tangent vector to E p . The image u ( E p ) ∈ Z n of this vectorunder dh is given by δ K ( p ) The balancing condition (5) follows fromthe homology dependance of γ p given by K ◦ . (cid:3) Definition 4.15.
The phase-tropical limit of a phase-tropically con-verging family (6) consists of the parameterised tropical curve (20) aswell as the phases (19) for the components K ⊂ (cid:98) C . Remark . Consideration of Γ instead of ˜Γ allows us to ignore trop-ical lengths of the edges of ˜Γ collapsed by ˜ h . It is possible to define thelimiting tropical length also on these edges. The resulting edge mightappear to be not only finite, but also zero or infinite, see [5].For the following definition we use the identification( C × ) n = R n × ( S ) n given by the (logarithm) polar coordinates identification of z ∈ ( C × ) n with (Log( z ) , Arg( z )), where Arg refers to the map of taking the argu-ment coordinatewise. The closure of the imageArg K = Arg(Φ( K )) ⊂ ( S ) n . is known as the closed coamoeba of Φ( K ) ⊂ ( C × ) n . Since Φ( p ) ⊂ ( C × ) n is a straight holomorphic cylinder, the image Arg p = Arg(Φ( p )) is ageodesic circle in the flat torus ( S ) n . Definition 4.17.
The unparameterised phase-tropical limit of a phase-tropically converging family (6) is the set(21) Ψ = (cid:91) v K { ˜ h ( v K ) } × Arg K ∪ (cid:91) p { ˜ h ( E p ) } × Arg p , where K runs over all components of (cid:98) C while p runs over all nodal andmarked points of (cid:98) C ◦ . Note that Y = Log(Ψ) is an unparameterisedtropical curve with the weight data coming from (20). Remark . Loci (21) are complex tropical curves in the terminologyof [10]. In more modern terminology, complex tropical are replaced with phase-tropical . Theorem 4.19.
If a scaled family of holomorphic curves f α : C α → ( C × ) n converges phase-tropically then for any family z α ∈ C α such that lim trop f α ( z α ) ∈ R n and lim t α →∞ Arg( z α ) ∈ ( S ) n exist we have (22) (lim trop f α ( z α ) , lim t α →∞ Arg( z α )) ∈ Ψ . Conversely, any point of Ψ can be presented in the form (22) for somefamily z α ∈ C α .Proof. Passing to a subfamily if needed, we may assume that z α con-verge to a point z ∈ (cid:98) C ◦ . If z ∈ K ◦ for some component K ⊂ (cid:98) C ◦ then by Proposition 4.11 lim trop f α ( z α ) = ˜ h ( v K ), while lim t α →∞ Arg( z α ) =Arg( φ K ( z )). Conversely, to present a point (˜ h ( v K ) , Arg( φ K ( z ))) in theform (22), it suffices to approximate z by z α ∈ C α . To present a point(˜ h ( v K ) , ξ ), ξ ∈ Arg K (cid:114) Arg(Φ( K )), we first approximate ξ with pointsfrom Arg(Φ( K )), approximate them as above, and then use the diago-nal process.If z ∈ (cid:98) C ◦ is a nodal or marked point then by Proposition 4.13(lim trop f α ( z α ) , lim t α →∞ Arg( z α )) ∈ { ˜ h ( E p ) }× Arg(Φ( p )). Consider a smallneighborhood W (cid:51) p in the universal curve U g,k such that W ∩ (cid:98) C con-sists of one or two disks (depending on whether p corresponds to a leaf ON-COMMUTATIVE AMOEBAS 23 or to a bounded edge of ˜Γ). Then the image (Log t α , Arg)( f α ( W ∩ C α ))is a connected annulus converging to { ˜ h ( E p ) } × Arg p . This implies thatany point in { ˜ h ( E p ) } × Arg p is presentable in the form (22). (cid:3) Theorem 4.20 (Phase-tropical limit compactness theorem) . Let f α : C α → ( C × ) n ⊂ CP n , t α → + ∞ , α ∈ A , be a scaled family of curves ofdegree d , where the source curves C α are Riemann surfaces of genus g with k punctures. Then there exists a scaling subsequence t β → + ∞ , β ∈ B ⊂ A , such that the subfamily f β : C β → ( C × ) n converges phase-tropically.Proof. By compactness of M g,k we may ensure convergence of C α to (cid:98) C ∈ M g,k after passing to a subfamily. For each component K ⊂ (cid:98) C we choose z ∈ K ◦ and a small transversal disk ∆ z ⊂ U g,k to (cid:98) C at z .Then for large t α the intersection ∆ z ∩ C α ⊂ U g,k consists of a singlepoint, and defines a family z α ∈ C α converging to z . Compactness of M g,k ( CP n , d ) and that of R ∪ {±∞} ensures convergence of the family(15) and the limit (17) after passing to a subfamily. (cid:3) Tropical limits of non-commutative amoebas in H . It turnsout that non-commutative amoebas, considered in the first three sec-tions of the paper, also admit interesting tropical limits. But, due tonon-commutativity of hyperbolic translations, passing to such limit isonly possible once we distinguish the origin point 0 ∈ H . This choicedetermines the rescaling map H (cid:51) x (cid:55)→ sx ∈ H for every s > H to itself. This map extends to a home-omorphism H ≈ → H , x (cid:55)→ sx, by setting it to be the identity on ∂ H . We define(23) κ t : I → H , κ t ( z ) = 1log t κ ( z ) ,t >
1, as the rescaling of the map (2), and denote by(24) κ t : I → H , the corresponding compactified map. The images κ ( V ) ⊂ H of alge-braic varieties V ⊂ I are nothing but their rescaled hyperbolic amoebas.In particular, they are closed sets.In this section we show that when t → ∞ these rescaled amoebas ofcurves converge to the H -tropical spherical complexes we define below. Definition 4.21.
The H -floor diagram ∆ of degree d > • There is a map r : Vert(∆) → [0 , ∞ ]called the vertex width . We distinguish the subsetVert (∆) = r − (0) , Vert + (∆) = r − (0 , ∞ ) , Vert ∞ (∆) = r − ( ∞ )of vertices of zero, positive and infinite width. • There is a map ϕ : Edge(∆) → ∂ H = S called the edge angle . • There is a map( d + , d − ) : Vert + (∆) ∪ Vert ∞ (∆) → Z ≥ called the vertex bidegree as well as a degree map δ : Vert (∆) → Z ≥ . • There is an edge weight map w : Edge(∆) → Z > . This data is subject to the following properties. • No edge may connect vertices of the same width. In particular,∆ is loop-free. • (25) (cid:88) v ∈ Vert (∆) δ ( v ) + (cid:88) v ∈ Vert + (∆) ( d + ( v ) + d − ( v )) = d. • For every v ∈ Vert(∆) we define div( v ) to be the sum of theweights of the edges connecting v to vertices whose width islarger than v minus the sum of the weights of the edges con-necting v to vertices whose width is smaller than v and requirethat(26) 2( d + ( v ) + d − ( v )) = div( v ) , and 2 δ ( v ) = div( v )for v ∈ Vert + (∆) and v ∈ Vert (∆). • If d + ( v ) = 0, v ∈ Vert + (∆) ∪ Vert(∆) ∞ , then we have(27) ϕ ( E ) = ϕ ( E (cid:48) )whenever E, E (cid:48) are two edges of ∆ adjacent to v . In this casewe set ϕ ( v ) = ϕ ( E ). ON-COMMUTATIVE AMOEBAS 25
Given 0 ≤ r ≤ ∞ and ϕ ∈ ∂ H = S we define ( r, ϕ ) ∈ H to bethe point on the compactified geodesic ray R ϕ ⊂ H connecting 0 ∈ H to φ such that the distance between ( r, ϕ ) and 0 equals to r . For0 ≤ r (cid:54) = r ≤ ∞ we define R ϕ [ r , r ] ⊂ R ϕ to be the interval of pointswhose distance to 0 is between r and r . Denote by S ( r ) ⊂ H thesphere of radius 0 ≤ r ≤ ∞ and center 0, in particular, S (0) = { } , S ( ∞ ) = ∂ H . The coordinates ( ρ, ϕ ) can be thought of as polarcoordinates in H . The first coordinate gives the map(28) ρ : H → [0 , ∞ ]measuring the distance to the origin 0 ∈ H . The second coordinategives the map(29) ϕ : H (cid:114) { } → ∂ H = S corresponding to the projection from the origin 0 to the absolute ∂ H .Let v ∈ Vert(∆). If d + ( v ) > v ) = S ( r ( v )). If d + ( v ) = 0 we define Θ( v ) = { ( r ( v ) , ϕ ( v )) } . For E ∈ Edge(∆) an edgeconnecting vertices v and v we define Θ( E ) = R ϕ ( E ) [ r ( v ) , r ( v )]. Definition 4.22.
The H -tropical spherical complex associated to an H -floor diagram ∆ is the set(30) Θ(∆) = (cid:91) v ∈ Vert(∆) Θ( v ) ∪ (cid:91) E ∈ Edge(∆) Θ( E ) ⊂ H . Example . Figure 1 depicts a H -tropical spherical complex of de-gree 3. It consists of two concentric circles (representing spheres inthe actual 3D picture) corresponding to two vertices of ∆ of bidegree(1 , ,
1) and thus gives a point ratherthan a sphere in H . This vertex has a single adjacent edge of weight2 and thus conforms to (26) . All other edges have weight 1. The sixoutermost edges end at the absolute at six vertices of bidegree (0 , P = κ − (0) = SO(3) ≈ RP ⊂ I . It is the fixed locus of the antiholomorphic involution(31) (cid:18) a bc d (cid:19) = A (cid:55)→ ¯ A ∗ = (cid:18) ¯ d − ¯ b − ¯ c ¯ a (cid:19) In homogeneous coordinate ( a + d : i ( a − d ) : ib : ic ) this involution isthe standard complex conjugation, so P ≈ RP can be thought of thereal locus of I = CP . In particular, after this coordinate change, the Figure 1. A H -tropical spherical complex is a limit ofamoebas of curves.lines in CP invariant with respect to the involution (31) are nothingbut the lines defined over R . We call such lines P -real lines. Theirintersection with P is diffeomorphic to the circle RP . Lemma 4.24.
A line l ⊂ I = CP is P -real if and only if its amoeba κ ( l ∩ I ) is a geodesic line passing through ∈ H .Proof. Suppose that κ ( l ∩ I ) is a geodesic line passing through 0 ∈ H .Then, by Proposition 3.7, κ − (0) ∩ l ≈ S . But l is real if and only if l ∩ RP is infinite.Conversely, if a line l is P -real then by Proposition 3.7 its amoebais a geodesic containing the origin. (cid:3) For A ∈ CP (cid:114) P there exists a single P -real line l A passing through A . It is the line passing through A and ¯ A ∗ . Define the map(32) π P : I (cid:114) P → Q by setting π P ( A ) to be one of the two points of intersection l A ∩ Q .Namely, we note that l (cid:114) P consists of two open half-spheres in the ON-COMMUTATIVE AMOEBAS 27 sphere l ≈ CP , and set π P ( A ) to be the unique point in l A ∩ Q containedin the same component of l (cid:114) l ∩ P as A . Corollary 4.25.
The map (32) is a continuous map that agrees withthe map (29) under κ , i.e. κ ◦ π P = ϕ ◦ κ .Proof. By Lemma 4.24, the amoeba of the fiber of (32) is the fiber of(29). (cid:3)
Remark . Clearly, the map (32) is not only continuous, but alsosmooth in the sense of (real) differential topology. In particular, itpresents the space I as a tubular neighborhood of the hyperboloid Q in CP . All fibers of (32) are open hemispheres in some lines in CP . Inparticular, they are holomorphic curves. Nevertheless, the map (32) isnot holomorphic.To see this, consider a line l close to a generator { z } × CP of thehyperboloid Q = CP × CP , but intersecting Q at two distinct points.Its image π P ( l ) ⊂ Q must be homologous to { z } × CP . If it wereholomorphic then π P ( l ) would have to be a generator itself. But thenwe get a contradiction with the inclusion Q ∩ l ⊂ π P ( l ).Suppose that V α ⊂ CP , t α → ∞ , α ∈ A , is a scaled familyof irreducible algebraic curves and Θ(∆) is the H -tropical spheri-cal complex associated to an H -floor diagram ∆. For an interval I = [ r , r ] ⊂ [0 , ∞ ] we defineΘ( I ) = Θ(∆) ∩ ρ − ( I ) ⊂ H , and ∆( I ) ⊂ ∆to be the open subgraph consisting of all vertices v ∈ ∆ such thatΘ( v ) ⊂ Θ( I ) as well as all open (i.e. not including adjacent vertices)edges E ⊂ ∆ such that Θ( E ) ∩ Θ( I ) (cid:54) = ∅ . Given a component K ∆ ⊂ ∆( I ) we define Θ I ( K ∆ ) = Θ( I ) ∩ ( (cid:91) v Θ( v ) ∪ (cid:91) E Θ( E )) , where v (resp. E ) goes over all vertices (resp. edges) of ∆ contained in∆( I ).We define V α ( I ) to be the normalization of( ρ ◦ κ t α ) − ( I ) ∩ V α , i.e. the proper transform of ( ρ ◦ κ t α ) − ( I ) under the normalization map˜ V → V . For t > H t : H → H , by the properties • κ ( H t ( z )) = ( t ρ, ϕ ) if H t ( z ) = ( ρ, ϕ ), z ∈ H , • H t ( z ) = z if z ∈ Q , • ι ( H t ( z )) = ι ( z ) if z ∈ H (cid:114) Q .Here ι : I → P = SO(3) ≈ RP is the coamoeba map (4). We have(34) κ ◦ H t = κ t . Definition 4.27.
Let V α ⊂ CP , t α → ∞ , α ∈ A , be a scaled familyof irreducible algebraic curves and Θ(∆) be the H -tropical sphericalcomplex associated to an H -floor diagram ∆. We say that the family V α κ -tropically converges to Θ(∆) if κ t α ( V α ) → Θ(∆) ⊂ H when t α → ∞ (in the Hausdorff metric on subsets of H ), and forevery interval [ r , r ] = I ⊂ [0 , ∞ ] such that ∂I ∩ r (Vert + (∆)) = ∅ there is a 1-1 correspondence between the components K ∆ ⊂ ∆( I ) andthe components K α ⊂ V α ( I ) for sufficiently large t α with the followingproperties. • The amoebas κ t ( K α ) of the subsurfaces K α ⊂ V α converge toΘ I ( K ∆ ). • If K ∆ is vertex-free, i.e. consists of a single (open) edge E ⊂ ∆,then the following conditions hold. – The subsurface K α is homeomorphic to an annulus. – There exists a point q ( E ) ∈ Q such that K α converges to { q } , t α → ∞ , in the Hausdorff metric on subsets of I . Thepoint q ( E ) depends only on the edge E and not on theinterval I (as long as E ⊂ ∆( I ) is a component so that q ( E ) is defined). – The annuli Ψ( K α ) = H t α ( K α ) converge to the annulusΨ I ( K ∆ ) = π − P ( q ) ∩ κ − (Θ I ( K ∆ ))in the Hausdorff metric on subsets of I while going w ( E )times around it, i.e. so that in a small regular neighbor-hood W ⊃ Ψ I ( K ∆ ) the cokernel of the homomorphism Z ≈ H (Ψ( K α )) → H ( W ) = H (Ψ I ( K ∆ )) ≈ Z induced by the inclusion Ψ( K α ) ⊂ W for large t α is ofcardinality w ( E ). ON-COMMUTATIVE AMOEBAS 29 • If K ∆ contains a single vertex v then each component of theboundary ∂K α corresponds to an edge E adjacent to v . Define U ( v ) to be the union of small ball neighborhoods of the points q ( E ). – If v ∈ Vert + (∆) ∪ Vert ∞ (∆) then the fundamental class[ K α ] is an element of H ( I (cid:114)
P, U ( v )) = H ( I (cid:114) P ) = H ( Q ) = Z and we require that [ K α ] = ( d + ( v ) , d − ( v )) for sufficientlylarge t α . – If v ∈ Vert (∆) then the fundamental class [ K α ] is an ele-ment of H ( I , U ( v )) = H ( I ) = H ( CP ) = Z and we require that [ K α ] = δ ( v ) for sufficiently large t α . Theorem 4.28.
Let V α ⊂ I = CP , t α → ∞ , α ∈ A , be a scaled familyof reduced irreducible algebraic curves of degree d not contained in thequadric Q ⊂ I . Then there exists a subfamily t β → ∞ , β ∈ B ⊂ A ,such that V β κ -tropically converge to the H -tropical spherical complex (35) Θ(∆) = lim t β →∞ κ t β ( V β ) ⊂ H associated to an H -floor diagram ∆ . The limit in (35) is taken in the Hausdorff metric on subsets of H . Proof.
After passing to a subsequence, we may assume that V α ⊂ CP converge to a (possibly reducible or non-reduced) curve V ∞ ⊂ CP ofdegree d . Choose a point p ∈ Q (cid:114) V ∞ so that the two lines passingthrough p and contained in Q are disjoint from the components of V ∞ not contained in Q . Then the projection from p defines a holomorphicmap π α : V α → Q such that π α ( z ) = z if z ∈ Q ∩ V α . Using (3) we define the holomorphicmap ϕ α = π + ◦ π α : V α → CP . Clearly, if z α ∈ V α is a family such that lim t α →∞ z α = z ∞ ∈ Q then wehave κ ( z ∞ ) = ( ∞ , lim t α →∞ ϕ α ( z α )) ∈ H in the polar coordinates presentation. Our next objective is to compute the tropical limit of the ρ -coordinateof κ ( z α ) ∈ H . Recall (see [12, Proposition 2.4.5]) that(36) H = { ( x , x , x , x ) ∈ R | x x − x − x = 1 } with the metric d H (( x , x , x , x ) , ( y , y , y , y )) = arccosh( x y + x y − x y − x y ) . Furthermore, see [12, Chapter 2.6]), the points of H may be identifiedwith the unitary Hermitian matrices (cid:18) x x + ix x − ix x (cid:19) , while theabsolute ∂ H can be identified with the rank 1 Hermitian matricesup to scalar multiplication. Any Hermitian 2 × ∈ H can be identified with the unit matrix. Thuswe have the following presentation of the map κ for a matrix A ∈ CP (viewed up to a complex scalar multiplication):(37) κ ( A ) = AA ∗ ∈ H . In particular,(38) ρ ( κ t ( A )) = 1log t d H ( AA ∗ ,
0) = 1log t arccosh || A || | det( A ) | , where || A || = | z | + | z | + | z | + | z | for A = (cid:18) z z z z (cid:19) . Note thatfor x ≤ e x ≤ x ) ≤ e x + 1 and thus for t ≤ t − ≤ arccosh( t ) ≤ log(2 t ). This implies that for any scaledsequence y α ∈ R ≥ we havelim trop y α = lim t α →∞ t α arccosh( y α ) , so that the limiting value of (38) is given by the tropical limit of thequantity(39) | z | + | z | + | z | + | z | | z z − z z | = (cid:88) j =0 | z j z z − z z | ≥ , where each individual term z j z z − z z is a rational function in the coor-dinates z , z , z , z .Let ν α : C α → V α ∈ ¯ I = CP be the normalization of V α . Define¯ f Qα : C α → ( CP ) as the composition of ν and the map( z z z − z z , z z z − z z , z z z − z z , z z z − z z ) ∈ ( CP ) . ON-COMMUTATIVE AMOEBAS 31
Denote ¯ f ∂ H α = π q ◦ ν α and¯ f α = ( ¯ f Qα , ¯ f ∂ H α , ν α ) : C α → ( CP ) × CP . Set C α = f − α (( C × ) ). The surface C α is obtained from C α by removing k points. After passing to a subsequence, the number k as well as thegenus g of C α may be assumed to be independent of α . The map ρ α : C α → [2 , ∞ ]defines as the composition of f α and (39) is a real algebraic function onon C α whose degree is bounded. Thus, after yet another passing to asubfamily, our curves C α may be assumed to have a finite number k c ofsuch critical points, so that we can mark these points on C α . Passingto a phase-tropically converging subfamily provided by Theorem 4.19we get the limiting curve (cid:98) C ∈ M g,k + k c with the tropical limits (17) andphases (18) for each component K ⊂ (cid:98) C . We also consider the limitingcurve f ∂ H ∞ : (cid:98) C ∂ H → CP in M g,k + k c ( CP ) as well as the forgettingmap (cid:98) C ∂ H → (cid:98) C contracting some components of (cid:98) C ∂ H . In addition, weconsider the limiting map ν ∞ ∈ M g,k + k c ( CP , d ) of ν α and the limitingmap f ∂ H ∞ ∈ M g,k + k c ( CP , d ) of f ∂ H α .We build the limiting H -floor diagram ∆ with the help of (cid:98) C ∂ H .As in the proof of Theorem 4.19 we first build a finer graph ˜∆ → ∆.Namely, we define a vertex of ˜∆ as either a component K ⊂ (cid:98) C ∂ H or a marked point p ∈ (cid:98) C ∂ H corresponding to an intersection point of¯ f α ( C α ) ∩ Q for large t α . We set r ( v p ) = ∞ and d + ( v p ) = d − ( v p ) = 0 if v p ∈ Vert( ˜∆) comes from such a marked point p .Since the tropical limit of the sum equals to the maximum of thetropical limits of the summands, the limit ρ K = lim trop ρ α ( z α ) ∈ [0 , ∞ ] , where z α ∈ C α ⊂ U g,k + k c ( CP ) is a family converging to a non-nodalnon-marked point of K depends only on K , and must be equal to ∞ for any non-tropically finite component K . Namely, ρ K coincides withthe maximum of the first four coordinates of h K ∈ [ −∞ , ∞ ] in (17).If v K ∈ Vert( ˜∆) corresponds to a component K ⊂ (cid:98) C ∂ H then we set r ( v K ) = ρ K . If ρ K > ν ∞ ( K ) ⊂ Q , and we set( d + ( v K ) , d − ( v K )) = [ ν ∞ ( K )] ∈ H ( Q ) = Z . If ρ K = 0 then we set δ ( v K ) = [ ν ∞ ( K )] ∈ H ( CP ) = Z . We define the edges of ˜∆ by means of the nodal points of (cid:98) C ∂ H aswell as those marked points p ∈ (cid:98) C ∂ H which correspond to intersectionpoints ¯ f α ( C α ) ∩ Q for large t α . In the latter case the edge E p connects v p and v K if p ∈ K . If p ∈ (cid:98) C ∂ H is a nodal point then the edge E p connects the vertices corresponding to the components adjacent to p .It might happen that E ∈ Edge( ˜∆) connects two vertices v , v with r ( v ) = r ( v ). In such case we contract E to the vertex v E of the newgraph, and set r ( v E ) = r ( v ) and d ± ( v E ) = d ± ( v ) + d ± ( v ). Afterperforming all such contractions we get the graph ∆.We define φ ( E p ) = f ∂ H ∞ ( p ). If E p ∈ Edge(∆) we consider the vanish-ing circle γ p ⊂ ¯ C α for large t α as in the proof of Proposition 4.14. Theimage ν α ( γ p ) is disjoint from Q and converges to ν ∞ ( p ) ∈ Q , since oneof the adjacent vertices v to E has positive width r ( v ) >
0. We define w ( E p ) as the local linking number in the small 6-ball around ν ∞ ( p ) of ν α ( γ p ) and Q .The identity (25) holds since d is the degree of the limiting curve ν ∞ : (cid:98) C ∂ H → CP . If v ∈ Vert(∆) comes from a marked point then v is 1-valent, and the condition (27) is vacuous. If v = v K comes froma component K ⊂ (cid:98) C ∂ H then ν ∞ ( K ) ⊂ Q and π + ( ν ∞ ( K )) ∈ CP is apoint if d + ( v ) = 0, and thus the condition (27) also holds in this case.To see that ∆ is a H -floor diagram we are left to verify (26).Suppose that v ∈ Vert + (∆) corresponds to K ⊂ (cid:98) C ∂ H . Here K maybe a single component or a a connected union of several componentsof the same width (resulting from the graph contraction ˜∆ → ∆).For large t α we define K ◦ α ⊂ (cid:98) C ∂ H as the subsurface bounded by thevanishing cycles γ p ⊂ (cid:98) C ∂ H corresponding to the edges E p ⊂ ∆ adjacentto v . The image ν α ( K ◦ α ) is contained in a small neighborhood of Q butdisjoint from the quadric Q ⊂ CP itself. Furthermore, the boundarycomponents of ν α ( ∂K ◦ α ) corresponding to the edges E p are containedin small ball neighborhoods of ν ∞ ( p ) ∈ Q and correspond to w ( E p )times the boundary of a small disk transversal to Q in these balls. Theequation (26) follows from the computation of the Euler class of thenormal bundle of Q in CP (equal to twice the plane section of Q ). If v ∈ Vert (∆) corresponds to K ⊂ (cid:98) C ∂ H then (26) follows since theintersection number of Q and the result of attaching of small disks to ν α ( ∂K ◦ α ) in the corresponding small balls is twice the degree of ν ∞ ( K ).We are left to prove the κ -tropical convergence of V α to Θ(∆) ⊂ H .For a vertex v ∈ Vert(∆) corresponding to K ⊂ (cid:98) C ∂ H we form K ◦ α asabove. Given a small (cid:15) > I v,(cid:15) = [ r ( v ) − (cid:15), r ( v ) + (cid:15) ] ON-COMMUTATIVE AMOEBAS 33 if r ( v ) < ∞ and I v,(cid:15) = [1 /(cid:15), ∞ ] if r ( v ) = ∞ . For a large t α we mayassume ρ ◦ κ t α ( γ p ) ⊂ [0 , ∞ ] (cid:114) I v, (cid:15) for the vanishing circles γ p adjacent to v . The surfaces K ◦ α ( v ; (cid:15) ) = K ◦ α ∩ ( ρ ◦ κ t α ) − ( I v,(cid:15) )are smooth surfaces with boundaries corresponding to the edges of ∆adjacent to v , and any critical point of ρ ◦ κ t α ◦ ν α : C α → (0 , ∞ ) iscontained in the surface K ◦ α ( v ; (cid:15) ) for some v . We havelim t α →∞ K ◦ α ( v ; (cid:15) ) = Θ( v ) ∪ (cid:91) E + R φ ( E + ) [ r ( v ) , r ( v )+ (cid:15) ] ∪ (cid:91) E − R φ ( E − ) [ r ( v ) − (cid:15), r ( v )] , where E + (resp. E − ) goes over edges connecting v to vertices of higher(lower) width. The complement C α (cid:114) (cid:83) v K ◦ α ( v ; (cid:15) ) consists of annuli K α ( E ; (cid:15) ) for edges E ⊂ ∆. If E connects vertices v and v with r ( v ) < r ( v ) < ∞ (resp. r ( v ) < r ( v ) = ∞ ) then κ t α ◦ ν α ( K α ( E ; (cid:15) ))converges to Θ [ r ( v )+ (cid:15),r ( v ) − (cid:15) ] ( E ) (resp. Θ [ r ( v )+ (cid:15), /(cid:15) ] ( E )) , since ρ ◦ κ t α ◦ ν α is critical point free on K α ( E ; (cid:15) ). Convergence of ν α ( K α ( E ; (cid:15) )) to q ( E ) = ν ∞ ( p ) ∈ Q implies convergence of H t α ( K α ( E ; (cid:15) ))to π − P ( q ( E )) ∩ κ − (Θ I ( E )) . (cid:3) Amoebas of surfaces
Convexity of the complement.
Let S ⊂ CP be a surface. Wemay compare its hyperbolic amoeba A S = A H ( S ) = κ ( S (cid:114) Q ) ⊂ H against its conventional amoeba A R ( S ) = Log( S ∩ ( C × ) ) ⊂ R . Each amoeba is closed in its ambient space. The complement R (cid:114) A R ( S ) of the Euclidean amoeba is always non-empty, see [2]. Butthere exist surfaces S such that A H ( S ) = H . Example . The Borel subgroup B ⊂ I = PSL ( C ) consisting ofupper-triangular matrices (cid:18) a b d (cid:19) coincides with S (cid:114) Q for a coordinate plane in CP = I . It is generated by two 1-dimensional subgroups, l = { (cid:18) s (cid:19) | s ∈ R } and l = { (cid:18) t t − (cid:19) | t ∈ R × } . As we have seen in Section 3.1, the amoeba κ ( l ) is a horosphere whilethe amoeba κ ( l ) is a geodesic serving as the symmetry axis of thehorosphere κ ( l ). Thus the images of the products of elements of l and l under κ fill the entire space H , i.e. we have A B = H . Proposition 5.2.
There exist surfaces S such that A S = H as wellas surfaces such that H (cid:114) A S (cid:54) = ∅ .Proof. In view of Example 5.1 it suffices to find a surface such that0 / ∈ A S . Recall that by (31), the inverse image κ − (0) can be identifiedwith the real projective space RP ⊂ CP after a suitable change ofcoordinates. Thus any surface in S ⊂ CP with the empty real locus S ∩ RP = ∅ , e.g. { z + z + z + z = 0 } provides a required exampleafter a change of coordinates. (cid:3) Proposition 5.3.
For any surface S ⊂ CP we have κ ( S ) ⊃ ∂ H .Proof. Recall that κ ( Q ) = ∂ H . If Q ⊂ S the proposition is immediate.Otherwise, the intersection S ⊂ Q is a curve of bidegree ( d, d ), where d is the degree of S . We have κ ( S ∩ Q ) = π + ( S ∩ Q ) = ∂ H . (cid:3) It is interesting to note that there exist surfaces S (cid:54) = Q such that H (cid:114) A S is unbounded. Example . Choose a point x ∈ ∂ H . The inverse image κ − ( x ) is aline on the quadric Q . Let S x ⊂ CP be the surface obtained from Q by a small rotation around the line κ − ( x ), i.e. a quadric S x , such that S x ∩ Q consists of this line and an irreducible curve of bidegree (1 , Proposition . The complement H (cid:114) A S x in this example is un-bounded. The geodesic ray R x emanating from the origin ∈ H inthe direction of x ∈ ∂ H is disjoint from the amoeba A S x .Proof. Recall that P = κ − (0) is the fixed locus of the antiholomorphicinvolution (31) after a coordinate change. For each z ∈ κ − ( x ) thereexists a unique P -real line L z ⊂ CP passing through z and the imageof z under (31). The real locus R L z = L z ∩ P divides the line L z into two half-spheres. Denote with H z the closure of the half-spherecontaining z . By Proposition 2.5 and Lemma 3.3, we have κ − ( R x ) = (cid:91) z ∈ κ − ( x ) H z (cid:114) Q. ON-COMMUTATIVE AMOEBAS 35
Since H z and Q intersect at z ∈ Q ∩ S x = κ − ( R x ) transversally, wehave H z ∩ S x = { z } . Thus H z (cid:114) Q is disjoint from S x . (cid:3) Theorem 5.6. If S ⊂ CP is a surface of odd degree then A S = H .Proof. Note that κ − ( x ), x ∈ H , is isotopic to κ − (0) which in its turncan be identified with RP ⊂ CP . Thus H ( κ − ( x ); Z ) = Z and therestriction homomorphism H ( CP ; Z ) → H ( κ − ( x ); Z )is an isomorphism. Non-vanishing of the class in H ( CP ; Z ) whichis Poincar´e dual to [ S ] = d [ CP ] (cid:54) = 0 ∈ H ( CP ; Z ) implies that S ∩ κ − ( x ) (cid:54) = ∅ . (cid:3) In the general case we have the following theorem. Recall that aconvex set K ⊂ H is a set such that x, y ∈ K implies that the geodesicsegment [ x, y ] between x and y is also contained in K . Theorem 5.7.
For any surface S ⊂ CP different from the quadric Q = { ( z : z : z : z ) | z z − z z } the complement H (cid:114) A S is anopen convex set in H . In particular, it is connected.Proof. Openness of H (cid:114) A S is implied by Corollary 2.3. Let x, y ∈ H (cid:114) A S be distinct points. Each oriented geodesic γ is an amoeba of aline L γ ⊂ CP by Proposition 3.7. If x ∈ γ then κ − ( x ) ∩ L γ is a circledividing the line L γ to two half-spheres H + γ and H − γ . The intersectionnumbers r + x (resp. r − x ) of S and H + γ (resp. of S and H − γ ) do not dependon the choice of L γ since x / ∈ A S . Since we may continuously deform γ to itself with the reversed orientation via rotations around x , we have r + x = r − x . Similarly, we have r + y = r − y . Furthermore, r + x + r − x = r + y + r − y since both quantities coincide with the degree d of S in CP . Thus(40) r + x = r − x = r + y = r − y = d/ . Let γ [ x,y ] ⊂ H be the geodesic passing through x and y and L be aline such that γ [ x,y ] = A L . The complement L (cid:114) κ − ( { x, y } ) consistsof two half-spheres and a cylinder Z with κ ( Z ) = [ x, y ]. Since theintersection number of S and each of the two half-spheres is d/ Z and theholomorphic surface S is zero, thus they are disjoint. Lemma 3.3 andProposition 3.7 imply that κ − ([ x, y ]) is fibered by such holomorphiccylinders. Thus [ x, y ] ∩ A S = ∅ . (cid:3) Note that the identity (40) provides another proof of Theorem 5.6.
Left
PSL -Gauss map. Multiplication from the left defines aholomorphic action of I on itself, and thus also a trivialization of thetangent bundle T I ≈ I × C . Let S ⊂ CP be a complex surface and S ◦ ⊂ S be the smooth locus of S ∩ I . We define the left PSL -Gaussmap(41) γ − S : S ◦ → CP as the map sending A ∈ S ◦ to A − T A S , i.e. to the tangent spaceto the surface S after bringing it to the unit element e ∈ I by leftmultiplication. In this way we get a plane in the tangent space T e I which can be viewed as an element of P ( T ∗ e I ) = CP . Similarly we candefine the right PSL -Gauss map γ + S : S ◦ → CP .It is easy to compute γ − S in terms of the homogeneous polynomial p defining S = { ( a : b : c : d ) | p ( a, b, c, d ) = 0 } . For this purpose itsuffices to choose three linearly independent tangent vectors in T e I . It isconvenient to choose them tangent to P = κ − (0), i.e. in the real partof the Lie algebra of SO(3) = P which coincides with that of SU(2).We can take (cid:18) i − i (cid:19) , (cid:18) − (cid:19) and (cid:18) ii (cid:19) for the basis of T e P .Taking partial derivatives of p with respect to images of these vectorsunder the differential of the left multiplication by A = (cid:18) a bc d (cid:19) ∈ S weget the following expression for γ − S ( A ) in coordinates:( i ( ap a − bp b + cp c − dp d ) : − bp a + ap b − dp c + cp d : i ( bp a + ap b + dp c + cp d )) ∈ CP , where p a , p b , p c , p d stand for the corresponding partial derivatives of p at A . Thus it is an an algebraic map on S ◦ whose degree cannot begreater than d . Proposition 5.8.
Let N ⊂ CP be a plane tangent to the quadric Q .Then γ − N is a constant map.Proof. Since N is tangent to Q the intersection Q ∩ N is a union of twolines. By Lemma 3.3 the image BN ⊂ CP , B ∈ I , is a plane containingone of these lines. Thus either N = BN or N ∩ BN (cid:114) Q = ∅ . Thusa vector tangent to N at A ∈ N must remain tangent to N under theleft multiplication by B if BA ∈ N . (cid:3) Proposition 5.9.
Let R ⊂ CP be a plane not tangent to the quadric Q . Then γ − R is a map of degree 1.Proof. Since the coordinate γ − R are given by linear functions for theplane R , it suffices to prove that γ − R is not constant if R is transversalto Q . Assuming the contrary, for any line l ⊂ R the image of l under ON-COMMUTATIVE AMOEBAS 37 the Gauss map γ − would be contained in the line corresponding to theimage of γ − S ( R ) in CP , see Remark 3.9. But if R ∩ Q is a smooth curveof bidegree (2 ,
2) then any pair of points in Sym ( CP ) = CP appearsin this image for one of the lines as we may choose a pair of points on R ∩ Q with arbitrary images under π − . (cid:3) The map (31) leaves P = κ − (0) fixed. Thus it defines the realstructure on T e I = CP such that T e P = RP . We refer to this realstructure as the P -real structure on CP . Theorem 5.10.
Let S ⊂ CP be a surface and z ∈ S be its smoothpoint. Then z is a critical point for κ | S if and only if γ − S ( z ) ∈ RP ,i.e. γ − S ( z ) is a P -real point.Proof. Note that a plane R ⊂ CP always intersect P = RP transver-sally. It is P -real if and only if R ∩ P is real 2-dimensional. Otherwise, R ∩ P is 1-dimensional. If z = e ∈ S then e is critical for κ | S if andonly if the dimension of the kernel of the differential of κ is greaterthan one, i.e. if T e S is P -real. Left multiplication by z extends thisargument for an arbitrary smooth point z ∈ S . (cid:3) Consider the locus C N consisting of all points z ∈ CP such thatthere exists a plane N ⊂ CP tangent to Q such that γ − N ( N ) = { z } asin Proposition 5.8. Proposition 5.11.
The locus C N ⊂ CP is a non-degenerate coniccurve invariant with respect to the P -real structure without P -real points.Proof. The locus C N is parametrized by one of the factors in Q = CP × CP and thus is a holomorphic curve. It must intersects anypencil of planes through e ∈ I at two points since Q is a quadric, andso is its projective dual surface. If N ⊂ CP were a plane such that γ − N is constant and real then by Theorem 5.10 its amoeba would have tobe 2-dimensional. Therefore this assumption is in contradiction withTheorem 5.6. (cid:3) The locus C N has the following special property with respect to theleft Gauss map in I . Proposition 5.12.
Let w ∈ C N and S ⊂ CP be a generic surface ofdegree d . Then γ − S ( w ) consists of d ( d − points.Proof. Let N w be a plane tangent to Q such that γ − N w ( N w ) = { w } .As in the proof of Proposition 5.8, Lemma 3.3 implies that we have w = γ − S ( z ) for a point z ∈ S if and only if S is tangent at z to a surface BN z for some B ∈ I . By the hypothesis, S is generic and thus there exist d ( d − planes passing through the line κ − ( w ) and tangent to S as d ( d − is the class of the smooth surface of degree d . (cid:3) In particular, in the case of d = 1, the locus γ − S ( w ) is empty for w ∈ C N even for the case when S is a generic plane and the degree of γ − S is one. References [1] P. Deligne and D. Mumford. The irreducibility of the space of curves of givengenus.
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