GEOMETRY OF AMOEBAS
(LECTURE NOTES, INSTITUT HENRI POINCARÉ, FALL 2013.)
GRIGORY MIKHALKIN
1
2
GRIGORY MIKHALKIN
1. Lecture 1: Introduction, definitions and first examples
1.1. Setting algebraic conventions in (C× )N : very affine varieties. We undertake geometric approach to algebraic geometry. Algebraic varieties are locally
given by system of algebraic equations in ambient spaces like CN , CPN and (C× )N .
They are closed subspaces inside them. Given a subspace in one of these ambient
spaces we may obtain a subspace in another one by either taking either a restriction
(if we pass to a smaller ambient space) or the topological closure (if we pass to a
larger space). The rest of this subsection is devoted to fixing algebraic conventions
resulting from this approach (and can be skipped by most readers).
Let fj (z1 , . . . , zN ) be polynomials in N variables. Their common set of zeroes
is an affine algebraic variety V ⊂ CN . If we need to define a projective algebraic
variety, we take the complex projective space
CPN = (CN +1 r {(0, ..., 0)}) / {(z0 , . . . , zN ) ∼ (λz0 , . . . , λzN ), λ 6= 0}.
It can be thought of as glued from N + 1 affine charts CN (each chart corresponding
to zk 6= 0 with the affine coordinates ( zzk0 , . . . , zk−1
, zk+1
, . . . , zzNk )). Each of this chart
zk
zk
gives us an embedding CN ⊂ CPN , (by default we’ll be using this embedding for
k = 0). If a subspace V ⊂ CPN (or rather its intersection with the corresponding
chart) is an affine algebraic variety in each of these charts we say that V is a projective
algebraic variety.
Another way to define projective varieties is to consider homogeneous polynomials
in N + 1 variables
z1
zN
(1.1)
Fj (z0 , . . . , zn ) = z dj fj ( , . . . , ).
z0
z0
If a positive integer dj is sufficiently large then Fj is a polynomial of degree dj . If
we take an affine variety V ⊂ CN and then take its closure in CPN ⊃ CN then the
result is a common zero of the system of homogeneous polynomials obtained in this
way (provided that dj is chosen to be the minimal positive integer such that (1.1) is
a polynomial). These are standard conventions for passing between CN and CPN .
In these lectures we shall be however, mostly concerned with varieties in
(C× )N ⊂ CN ⊂ CPN .
As usual C× = Cr{0} is a multiplicative group of non-zero complex numbers. Note
N
T
N
that (C× )N =
CN
k , where Ck stands for the affine chart with zk 6= 0.
k=0
Once again we say that a variety V ⊂ (C× )N is algebraic if it is given by a system
of polynomial equations fj (z1 , . . . , zn ) = 0. For each k there exists the minimal
integer dkj such that zkdk j fj (z1 , . . . , zn ) is a polynomial. Proceeding as above we see
that the topological closure of an algebraic variety V ⊂ (C× )N in CN is given by
GEOMETRY OF AMOEBAS
3
equations
(
N
Y
zkdk j )fj (z1 , . . . , zN ) = 0.
k=1
As an easy corollary we get the following proposition.
Proposition 1.1. A variety V ⊂ (C× )N is an algebraic variety if and only if there
exists a projective algebraic variety Vproj ⊂ CPN such that V = Vproj ∩ (C× )N .
Such a projective algebraic variety may be obtained by taking the topological closure
V̄ = Vproj of V in CPN (enhanced with Euclidean topology).
Note that Vproj associated to V with the help of taking a closure does not have
components contained in
∂CPN = CPN r (C× )N .
But if we restrict ourselves to irreducible varieties we do not need to worry about
extra components. An irreducible projective variety Vproj is uniquely determined by
an irreducible algebraic V ⊂ (C× )N , though different V may give the same Vproj .
Example 1.2. We may present any Riemann sphere punctured in four distinct
points as a line L ⊂ (C× )3 . Indeed, suppose that the three puncture are 0, 1, a, ∞
for a 6= 0, 1 ∈ C. Then such line can be given as an image of the map
C× 3 t 7→ (t, t − 1, t − a) ∈ (C× )3 .
Different a give different punctured spheres as the corresponding cross-ratio is a
biholomorphic invariant. In the same time, the corresponding projective variety
Lproj ⊂ CP3 is always the line and thus is independent of a up to a projective linear
transformation.
Definition 1.3 (cf. [?]). A variety V ⊂ (C× )N is called a very affine variety.
As usual, we can consider V independently of its embedding to (C× )N by iden0
tifying V ⊂ (C× )N with V 0 ⊂ (C× )N with the help of N 0 holomorphic functions
defined on V and landing on V 0 whenever the inverse map given by N holomorphic
function in a neighbourhood of V 0 exists. In particular, the lines from Example 1.2
remain to be different very affine variety.
E.g. (C× )2 as well as (C× )2 punctured in any number of points are very affine
varieties. In the same time neither CP1 nor C are very affine (though C is, clearly,
affine).
1.2. Definitions of amoebas and coamoebae. Each point z ∈ C× can be presented in polar coordinates z = reiα , r > 0 with z = rei(α+2π) , i.e. α ∈ R/2πZ ≈ S 1 .
The polar coordinates of this point is r = |z| and α = arg(z). Given a point
(z1 , . . . , zN ) ∈ (C× )N we may consider
Log(z1 , . . . , zN ) = (log |z1 |, . . . , log |zN |) ∈ RN
4
GRIGORY MIKHALKIN
and
Arg(z1 , . . . , zN ) = (arg(z1 ), . . . , arg(zN )) ∈ (R/2πZ)N ≈ (S 1 )N .
Definition 1.4 (Gelfand-Kapranov-Zelevinski [?], Passare [?]). The amoeba of a
(very affine) algebraic variety V ⊂ (C× )N is the set
A = Log(V ) ⊂ RN .
The coamoeba (or alga (cf. [?]) is the set
B = Arg(V ) ⊂ (R/2πZ)N .
Note that the fiber of the map Log : (C× )N → RN is the torus (S 1 )N . This map
is a proper map, i.e. the inverse image of any compact set is compact. Restricting
Log to our algebraic variety V , which is a closed set in (C× )N , we get the following
statement.
Proposition 1.5. The amoeba A ⊂ RN is a closed set.
On the contrary, the fiber of the map Arg : (C× )N → (S 1 )N is the non-compact
space RN , so it is not a proper map. Thus there is no reason for the coamoeba B
to be closed inside (S 1 )N and we shall soon see examples of non-closed coamoebas.
The coamoebas do provide a complimentary viewpoint to that of amoebas.
Amoebas take into account the norms of coordinates and ignore the phases while
coamoebas take into account the phases and ignore the norms.
1.3. First examples: the amoeba and coamoeba of z + w + 1 = 0. First
we would like to set the convention for variables corresponding to the amoeba
and coamoeba projection. Recall that the coordinates in (C× )N are denoted with
zj ∈ C× , j = 1, . . . , N . For the general N we denote the corresponding projection
coordinates in the amoeba space RN and the coamoeba torus (S 1 )N with
xj = log |zj | ∈ R and βj = arg(zj ) ∈ S 1 = R/2π.
However for the special case of N = 2, we adopt the following (shortened) letter
choice for variables:
z = z1 , w = z2 ∈ C× ; x = x1 , y = x2 ∈ R; α = β1 , β = β2 ∈ S 1 ,
so that x, y are the amoeba coordinaes and α, β are the coamoeba coordinates in
the special case if we only have two complex variables z and w.
Let L ⊂ (C× )2 be the line given by equation z + w + 1 = 0. Its real part is
depicted on Figure 1.
Proposition 1.6. The amoeba A = Log(L) is given by the system of three triangle
inequalities for the three positive real numbers ex , ey and 1, namely
ex + ey ≥ 1, 1 + ex ≥ ey , 1 + ey ≥ ex .
see Figure 2.
GEOMETRY OF AMOEBAS
5
Figure 1. The line z + w + 1 = 0 in the real plane .
Figure 2. The amoeba of z + w + 1 = 0.
Proof. Existence of α and β such that ex+iα + ey+iβ + 1 = 0 is equivalent to existence
of the triangle with sides ex , ey , 1.
For comparison, Figure 3 depicts the set of pairs of positive real numbers (|z|, |w|)
such that |z|, |w| and 1 satisfy to the triangle inequalities. To get Figure 2 from
Figure 3 we just need to reparameterize the picture, namely take the logarithm
coordinatewise.
The coamoeba of L is depicted on Figure 4. If 0 < α < π then two of the angles
of the triangle are π − α and π + α − β. All angles of a non-degenerate triangle have
to be between 0 and π, while their sum have to be π. Thus β < α + π and α < β.
Similary we get β < α and α < β + π if 0 > α > π.
6
GRIGORY MIKHALKIN
Figure 3. Possible absolute values (|z|, |w|) for z + w + 1 = 0.
Figure 4. The coamoeba of z + w + 1 = 0.
There are three types of degenerate triangles depending on which side has the
maximal length. These types correspond to (α, β) = (π, 0), (α, β) = (π, 0) and
(α, β) = (π, π). Thus to two open triangles on Figure 4 we have to add three points
which are vertices of each of these two triangles on the torus (S 1 )2 .
1.4. Linkages. Let us fix three lengths l1 , l2 , l3 > 0 and then search for triangles
(with ordered sides) on the Euclidean plane (which we identify with C) of length
l1 , l2 , l3 (according to this order). Note that the order defines the orientation of the
boundary of the triangle and thus the orientation of its every side.
If l1 , l2 , l3 satisfy to strict triangle inequalities l1 + l2 > l3 (and two cyclic permutations of this inequality) then there are two such triangles up to an orientationpreserving Euclidean motion. If we set l3 = 1 and ask that the third side is an
GEOMETRY OF AMOEBAS
7
(oriented) interval from −1 to 0 (in C) then there are exactly two such triangles,
see Figure 5.
Figure 5. Two triangles in C with given length and of the sides and
two vertices fixed at −1 and 0.
If l1 , l2 , l3 does not satisfy to at least one triangle inequality, e.g. l1 + l2 < l3
then no such triangles exist. Finally, we may have a non-generic case when one of
the triangle inequality turns to an equality, e.g. e.g. l1 + l2 = l3 . The last case
corresponds to a degenerate triangle (contained in a line). If the third side is fixed
as above then we have a unique (degenerate) triangle with these lengths.
Summarising we get the following proposition for the amoeba A = Log(L) of the
line L = {z + w + 1 = 0} ⊂ (C× )2 . We denote the (topological) interior of A with
A◦ and its boundary with ∂A.
Proposition 1.7. If (x, y) ∈ A◦ then Log−1 (x, y) ∩ L consists of two points; if
(x, y) ∈ ∂A then Log−1 (x, y) ∩ L consists of a single point.
Recall that a linkage is a finite graph Γ enhanced with positive lengths associated to its edges and a partition of the set Vert(Γ) of its vertices into two classes
Verta (Γ) ∪ Vertm (Γ) of anchored and movable vertices. In addition a (planar) linkage is enhanced with the anchor map a : Verta (Γ) → R2 . For technical reasons we
require that Γ is connected and that there is at least one anchor, i.e. Verta 6= ∅.
We interpret this graph as a mechanism with bars (of specified length) corresponding to the edges of the graph. The bars are connected at joints which correspond
to the vertices of the graph. Anchored vertices are fixed in position determined by
the map a. The bars themselves are rigid, but can be freely rotated at the joints.
Definition 1.8. A realisation of a planar linkage is a map h : Vert(Γ) → R2 with
the following properties.
• We have h|Verta (Γ) = a.
• For every edge E ⊂ Γ we have the distance between the images of its endpoints under the map h equal to the length of E.
8
GRIGORY MIKHALKIN
Configuration space of a linkage is the space of all its realisations with the given
length. Union of the configuration space for all possible (nonzero) lengths is called
the universal configuration space. We shall see that it is a very affine variety given
by a system of linear equations.
Note that the (fixed-length) configuration spaces are obtained as intersections of
quadrics coming from the Pythagorus theorem. Namely, each edge gives a constraint
that the sum of squares of the coordinate differences of its endpoints is equal to the
square of the length of the edge.
To see a connection between amoebas and linkages we orient each edge of Γ
and interpret the vector in R2 = C connecting its endpoints as a complex number
zE ∈ C× . As Γ is a finite graph, the homology group H1 (Γ, Verta ) is of finite rank
generated by relative cycles. Its generators can be chosen to be either circles or
intervals embedded to Γ.
We take NΓ to be the number of edges of Γ and interpret zE as a coordinate
in
P
(C× )NΓ . Every relative cycle [Z] ∈ H1 (Γ, Verta (Γ)) can be written as Z = mE E,
E
mE ∈ Z (recall
that the edges E are already oriented). Its boundary can be written
P
as ∂Z = mv v, mv ∈ Z, v ∈ Verta (Γ).
We associate to Z the linear equation
X
X
mE zE =
mv a(v) ∈ C
E
v
and define VΓ to be the locus of common solutions of these equations.
Theorem 1.9. The universal configuration space of a linkage Γ is given by the
very affine variety VΓ ⊂ (C× )N given by system of linear equations. Each bar of
Γ corresponds to a variable in CN while each relative cycle in (Γ, Verta (Γ)) gives a
linear equation on VΓ .
The variety VΓ maps to its amoeba A = Log(VΓ ) so that the inverse image over
each point {xE } ∈ RNΓ coincides with the configuration space of Γ with lengths exE .
Proof. This theorem is straightforward. A coordinate zE records the length |zE | and
the angle arg(zE ) for each bar E. To reconstruct a linkage from a point of VΓ we
add adjacent bars one by one starting from an anchor.
Remark 1.10. The construction of Theorem 1.9 is well-known in Combinatorics
under the so-called matroid language disguise. Let us recall that a matroid is a
combinatorial notion that mimics a hyperplane arrangement in CPn .
For simplicity let us assume that Verta (Γ) consists of a single point (so that all
cycles in (Γ, Verta (Γ)) are absolute cycles). Theorem 1.9 associates to Γ a linear
space in (C× )NΓ ⊂ CPNΓ , where the coordinates are in 1-1 correspondence with the
edges of Γ.
The result is a subvariety L ⊂ (C× )NΓ whose topological closure L̄ in CPNΓ can be
identified with CPn for some n (as it is given by linear equations). Each coordinate
GEOMETRY OF AMOEBAS
9
zE in CPNΓ (for an edge E ⊂ Γ) defines a hyperplane zE = 0. Thus all edges of Γ
define a hyperplane arrangement in L.
It is easy to see that the matroid corresponding to this hyperplane arrangement
coincides with the so-called graphics matroid associated to Γ.
Example 1.11. The linkage depicted on Figure 6 has two cycles in the graph Γ, it
is the well-known inversor linkage. Its vertex O is anchored at the origin and the
two bars adjacent to O have the same length equal to R. The other four bars have
the same length equal to r < R.
Moving the bars we may put the vertex A to any point of the annulus centered
at the origin with the outer radius R and the inner radius r. Then the point B
(assuming that it is different from the point A as of course there is also a somewhat
less interesting √
realization with A = B) is the result of the inversion of A at the
circle of radius R2 − r2 centered at the origin. This means that the points A and
B sit on the same ray emanating from O and |OA||OB| = R2 − r2 .
Figure 6. Inversor linkage.
Exercise 1.12. Prove that the linkage from Example 6 indeed performs the inversion
as described above.
One of the simplest class of linkages is given by k-polygons when Γ̂k is a cyclic
graph obtained by decomposing a circle into k edges, so that we have only one
cycle. The sides of the k-polygon are ordered and oriented as in Figure 7. The only
anchored vertex is the one where the first and the last edges are joined. We refer to
Kapovich and Millson [?] for configuration spaces k-polygon linkages.
Given a realisation of the k-polygon linkage we can obtain another one by rotation
around the anchor. To get rid of this obvious symmetry we may fix the last bar of
the linkage to occupy a given position in R2 . Clearly this is the same as anchoring
the second endpoint of the last bar (recall that on its endpoints is already fixed)
and removing this bar from Γ, see Figure 8.
10
GRIGORY MIKHALKIN
Figure 7. A k-polygon linkage with k = 5.
Figure 8. The linkage from Figure 7 after deprojectivisation.
By rescaling all lengths lj by the same positive factor we may assume that lk = 1.
In this way getting rid of the rotational symmetry of k-polygons amounts to taking
a graph Γk homeomorphic to an interval [−1, 0] subdivided into N = k − 1 edges
and anchoring the first endpoint at 0 and the second endpoint at −1. Note that
the passage to such linkage from a k-polygon linkage corresponds to passing from
projective to affine coordinates algebraically.
By Theorem 1.9, the universal configuration space of such linkages coincides with
N
V = {(z1 , . . . , zN ) ∈ (C)
| 1+
N
X
zj = 0}.
j=1
We have considered the case of N = 2 in the previous subsection to determine the
amoeba of the line z + w + 1 = 0.
We can use linkages as above, with two anchors and three bars to find the amoeba
of the plane 1 + z1 + z2 + z3 = 0. It is easy to see that there are three cases if the
GEOMETRY OF AMOEBAS
11
Figure 9. Realizations of a 3 bar linkage with different lengths and
different configuration spaces.
lengths 1, l1 , l2 , l3 are generic. There are no linkage realisation if one of these lengths
is strictly greater than the sum of three others. Otherwise there are two possibilities:
when the configuration space consists of a single circle, or of two circles, cf. Figure 9.
It is easy to see that the walls separated these possibilities are given by the equation
1 + l1 = l2 + l3 and the other two equations corresponding to permutations of lj .
Figure 10. Octahedron.
Exercise 1.13. Take a regular octahedron O (see Figure 10) and colour its faces to
black and white in the chessboard style, i.e. so that no faces of the same colour are
adjacent. Let us remove from O the four closed white faces and denote the result
with A. Show that there exists a homeomorphism between A and the amoeba of
A = A(1 + z1 + z2 + z3 = 0) ⊂ R3 such that ∂A is mapped to the three white facets
and such that the image of the three square diagonals of O separate A r ∂A into
eight regions: four with a single circle over it and four with two circles.
Remark 1.14. Theorem 1.9 provide a linkage interpretation for amoebas of some
varieties given by systems of linear equations. General polynomial equations contain
12
GRIGORY MIKHALKIN
monomials of arbitrary degree. If we represent each monomial with a bar of Γ
there will be some relations, particularly on the slopes of this bar. E.g. the bar
corresponding to z 2 has to be rotated twice as much as the bar corresponding to z.
The rotation of the bar corresponding to zw has to be a combination of rotation of
the bars corresponding to z and w, etc.
Problem 1.15. Can you invent reasonable mechanisms generalizing linkages so
that they would be capable to describe amoebas of varieties given by non-linear
equations?
1.5. Coamoebas of hyperplanes in CPN . To describe the coamoeba B of the
N
P
hyperplane determined by 1+ zj = 0 we may use the following simple observation.
j=1
Proposition 1.16. We have (0, . . . , 0) 6= (β1 , . . . , βN ) ∈
/ B if and only if there
exists a closed arc A ⊂ S 1 = R/2π of length π such that {0, β1 , . . . , βN } ⊂ A and
{0, β1 , . . . , βN } 6⊂ ∂A.
In other words, (β1 , . . . , βN ) ∈ B if they are distributed in the circle R/2πZ (which
we may view as the unit circle in C with the help of exponentiation) in a non-convex
way, see Figure 11.
Figure 11. A collection of points in the unit circle |z| = 1 arranged
in a non-convex way: some positive multiples of these points add up
to 0 in C.
Proof. If such an arc A exists then 1 +
N
P
rj eiβj must sit in the interior of the half-
j=1
space in RN passing through the origin and defined by A for any choice of rj > 0.
Otherwise we may choose the weights rj on the unit circle S 1 ⊂ C at the points
eiβ1 , . . . , eiβN so that together with the unit weight at 1 they have the center of mass
at the origin.
But such collections βj are easy to describe. We may parameterize an arc A of
length π by its most-clockwise endpoint α ∈ S 1 . Since 0 ∈ A we have −π ≤ α ≤ 0.
We can reformulate βj ∈ A as α ≤ βj ≤ α + π.
GEOMETRY OF AMOEBAS
13
If α = 0 such conditions on β determine a square 0 ≤ βj ≤ π. For other
values of α this square has to be translated by α(1, . . . , 1). Thus the closure of
the complement of B is the Minkowski sum I0 + I1 + . . . IN ⊂ (R/2π)N of the
closed intervals I0 , I1 ,. . . , IN , where I0 = {(−α, . . . , α) | − π ≤ α ≤ 0} and
Ij = {(0, . . . , 0, βj , 0, . . . , 0) | 0 ≤ βj ≤ π}, j = 1, . . . , N . Recall that Minkowski
sum of intervals is called a zonotope.
Note that if {0, β1 , . . . , βN } 6⊂ ∂A correspond to the case when ei βj = ±1 for each
βj , i.e. all coordinates zj are real. Thus we have the following proposition.
Theorem 1.17. The coamoeba B1+z1 +···+zN =0 ⊂ (S 1 )N is the union of 2N − 1 points
of the form βj = 0, π with at least one nonzero βj , and the complement of the
Minkowski sum I0 + I1 + . . . IN .
Figure 12. The coamoeba of z + w + 1 = 0 in a different frame.
Example 1.18. In Figure 12 we redraw the coamoeba of 1 + z + w choosing the
square frame for the torus so that the origin is in its center. In this frame we easily
see the hexagon I0 + I1 + I2 .
Remark 1.19. Note that in the case N = 2 there is a single point of V1+z+w ⊂ (C× )2
projecting to a point of the coamoeba unless this is one of the three real points
(π, 0), (0, π) and (π, π). These three points correspond to the three real quadrants
having non-empty intersection with V1+z+w . Thus the fiber over each of these special
points is an arc.
Exercise 1.20. Visualize topologically V1+z+w fibered over its coamoeba as two disks
(from triangles) after gluing three twisted ribbons (from three special points). Show
that topologically it is homeomorphic to a pair-of-pants, i.e. a sphere punctured
at three points. Compare this topological picture with the one coming from the
amoeba presentation.
14
GRIGORY MIKHALKIN
2. Lecture 2: Convexity and behavior at infinity
So far we did not justify the choice of taking the logarithm of the absolute value
of the coordinates in Definition 1.4. Topologically the pictures resulting from taking
just the coordinate absolute value as in Figure 3 or their logarithms as in Figure 2
are the same. In this section we start to see advantages of taking the logarithms.
2.1. Convexity of amoebas. The following theorem was one of the motivation for
defining amoebas by Gelfand, Kapranov and Zelevinsky.
Theorem 2.1 ([?]). If V ⊂ (C× )N is a hypersurface then every (topological) connected component of RN r A is a convex open domain.
Proof. We already know that the complement of A is open since Log : (C× )N → RN
is a proper map. Convexity of components of RN r A is ensured by the next
lemma.
Suppose x, x0 ∈ RN . We denote by
[x, x0 ] = {λx + (1 − λ)x0 | 0 ≤ λ ≤ 1} ⊂ RN
the interval obtained by connecting x to x0 .
Lemma 2.2. Let A ⊂ RN be the amoeba of a hypersurface V ⊂ RN . If x, x0 ∈ RN
belong to the same connected component of RN r A then we have [x, x0 ] ⊂ RN r A.
Proof. Since RN r A is open it is sufficient to prove this lemma for the special case
when the slope of the interval [x, x0 ] is rational, i.e. we have c(x − x0 ) = p ∈ ZN for
some positive real scalar c > 0.
Let us consider a closed geodesic γ in the phase torus (S 1 )N = (R/2π/Z)N realising the homology class p ∈ ZN = H1 (R/2πZ)N . All such geodesic foliate the torus
γ and are parameterised by the (N − 1)-dimensional quotient torus (S 1 )N /γ.
The key observation for this lemma is that
[x, x0 ] × γ ⊂ RN × (S 1 )N = (C× )N
is a holomorphic annulus. Indeed, we have log |zj | = Re(log(zj )) while arg |zj | =
Im(log(zj )). Thus the multiplication by i takes a vector tangent to RN × {β} to
the corresponding vector tangent to {x} × (S 1 )N . The correspondence for tangent
vectors here is given by the tautological identifying the tangent vectors to (S 1 )N =
(R/2πZ)N with RN .
Suppose that [x, x0 ] ∩ A =
6 ∅. Then we have
([x, x0 ] × γ) ∩ V 6= ∅
for one of the closed geodesics γ in the phase torus (S 1 )N = (R/2πZ)N realizing the
homology class p ∈ ZN = H1 (R/2πZ)N (as such geodesics foliate the whole phase
torus).
GEOMETRY OF AMOEBAS
15
Varieties V and [x, x0 ] × γ are holomorphic varieties of complimentary dimension.
The boundary ∂([x, x0 ] × γ) = {x, x0 } × γ is disjoint from V . Thus the topological
intersection number V and [x, x0 ] × γ is a well-defined positive number.
But if x and x0 belong to the same component of RN r A then there exists a path
δ ⊂ RN r A connecting x with x0 . Clearly, δ × γ ∩ V = ∅, thus ([x, x0 ] ∪ δ) × γ has
a positive intersection number with V . But [x, x0 ] ∪ δ is a closed path in RN and
thus is null-homologous. Therefore, ([x, x0 ] ∪ δ) × γ is null-homologous as well and
its intersection number with any homology class with closed coefficients in (C× )N
(such as V ) must be zero. We get a contradiction.
Remark 2.3. As it was noticed by Kapranov, the convexity of amoebas can be viewed
as a generalisation of Hadamard’s three-circle theorem. Let us recall its classical
formulation (with three circles).
Given three concentric circle in the complex plane of radii R1 < R2 < R3 around
0 and a holomorphic function f in a neighbourhood of the closed annulus R1 ≤ |z| ≤
R3 we have
log( max |f (z)|) ≤
|z|=R2
log(R3 ) − log(R2 )
log(R2 ) − log(R1 )
log( max |f (z)|)+
log( max |f (z)|).
|z|=R1
|z|=R3
log(R3 ) − log(R1 )
log(R3 ) − log(R1 )
This theorem has another classical reformulation: the function
M (x) = maxx log |f (z)|
|z|=e
is convex on an open interval I ⊂ R for any holomorphic function defined in a
domain U ⊃ log−1 (I).
But, clearly, M (x) is nothing but the upper boundary of the amoeba of the curve
w = f (z) (defined in the strip I × R). Such convexity is assured by Theorem 2.1
(which clearly works equally well for amoebas defined not only in RN , but also in
any contractible domain in RN ).
P
Example 2.4. Let us consider a polynomial function f (z) =
aj z j where all
j=0
coefficients aj > 0 are positive. We note that the logarithmic image
C = {(x, y) ∈ R2 | ey = f (ex )},
of the graph of the function w = f (z) restricted to the positive value of z is a
component of the boundary of the amoeba A of
V = {(z, w) ∈ (C× )2 | w = f (z).
Indeed, we have f (ex+iα ) ≤ f (ex ) by the triangle inequality. Therefore, C ⊂ ∂A,
namely C is the upper boundary of A in R2 .
The curve C is also the boundary of the closed domain
D = {(x, y) ∈ R2 | ey ≥ f (ex )},
16
GRIGORY MIKHALKIN
which is a connected component of R2 rA and therefore must be convex by Theorem
2.1.
We deduce the following (classical) logarithmic convexity property for polynomials
f in one variable with positive coefficients.
p
u+v
(2.1)
f (e 2 ) ≤ f (eu )f (ev ).
Similar convexity holds also for polynomials in several variables with positive coefficients. (Deduction of this property from convexity of amoebas is due to Kapranov.)
Local positivity of intersection of complex varieties of complimentary dimension
in CN is one of the manifestation of the maximum principle. We may also deduce convexity of amoebas directly from the maximum principle as each coordinate
log |zj | = Re log(zj ) or their linear combination restricted to V is a real part of a
holomorphic function and thus a pluriharmonic function.
The following statement holds without assuming that V is a hypersurface and
is a variation of Theorem 2.1 Let us however keep the assumption that V is irreducible. Let H be a (closed) half-space in RN whose boundary ∂H ⊂ RN is an affine
hyperplane.
Theorem 2.5. Suppose that ∂H is a locally supporting hyperplane for A at x, i.e.
there exists an open convex domain U ⊂ RN such that A ∩ U ⊂ H then A ⊂ ∂H.
Proof. The half-space H is given by a vector u = (u1 , . . . , uN ) ∈ RN and a number
a ∈ R through the inequality u.x ≤ a. The linear combination
N
X
uj log |zj |
j=1
is a harmonic function on V and by assumption a is its local maximum. By the
maximum principle this means that the function is constant so that A ⊂ ∂H. Suppose now that dim V = n, so that its codimension is k = N − n. We may
refine Theorem 2.5 once k is known.
Let L ⊂ RN be a k-dimensional affine space and S ⊂ L is a closed ((k − 1)dimensional) hypersurface in L. As L is contractible the hypersurface S can be
presented as the boundary S = ∂D for a closed domain D ⊂ L.
Theorem 2.6 (cf. [?],[?]). If L ∩ A = ∅ and [S] is trivial in Hk−1 (RN r A) then
D ∩ A = ∅.
Proof. We proceed as in the proof of Theorem 2.1. First we note that we may
perturb the affine space L to ensure that it has rational slope, i.e. there exist k
linearly independent integer vectors tangent to L. Since A is closed for a sufficiently
small perturbation a hypersurface S disjoint from A will be retained.
Thus we may assume that the slope of L ⊂ RN is rational and thus its quotient
in L ⊂ RN /(2πZN ) is a closed k-dimensional subtorus of (S 1 )N . All k-subtori γ
GEOMETRY OF AMOEBAS
17
parallel to it foliate (S 1 )N with the quotient homeomorphic to (S 1 )N −k . Note that
L × γ is a k-dimensional complex subvariety for any γ from this foliation.
Suppose that D ∩ A =
6 ∅. Then D × γ ∩ V 6= ∅ for a k-subtorus γ in this foliation.
As ∂D × γ is disjoint from V the intersection number of D × γ and V is welldefined and must be positive as intersection number of two complex subvarieties of
complementary dimension.
But if S = ∂D bounds a k-cycle δ in the complement of A the closed cycle
(D∪δ)×γ has the same positive intersection with V as D×γ. We get a contradiction
as D ∪ δ is a trivial k-cycle in L (as any other positive-dimensional cycle in a
contractible space).
Remark 2.7. Theorem 2.6 says that it is not possible to find a k-plane L that cuts
A inside a domain D ⊂ L so that ∂D is disjoint from A and homologically trivial in
RN r A. In classical geometry such a domain is sometimes called a cutting k-cap, so
Theorem 2.6 says that it is not possible to cut a k-cap from the amoeba of a variety
of codimension k.
The condition that ∂D is homologically trivial in RN r A is necessary as it can
be seen already for amoebas of spatial curves, cf. the next example.
Example 2.8. Let V ⊂ (C× )3 be the twisted rational cubic curve given by the
parameterisation
t 7→ (t, t − 1, t − 2), t ∈ C× r {1, 2},
see Figure ??. Since the parametrisation is defined over R, the conjugate points of
C× r {1, 2} are mapped to the same point of the amoeba of V .
Thus the amoeba A is the image of the quotient of C× r {1, 2} under the involution of complex conjugation, which is topologically the disk minus four points at
its boundary. It is easy to see that near three of these punctures (namely, 0, 1, 2)
the amoeba A has asymptotes in the three coordinate directions. It goes to (−∞)direction as the corresponding coordinate before taking the logarithm vanishes. Near
the fourth puncture (which corresponds to ∞) the amoeba A has a diagonal asymptote, it is parallel to the vector (1, 1, 1).
Let us consider the intersection of A by the plane L = {(x1 , x2 , x3 ) ∈ R3 | x1 +x2 =
0}. The intersection points correspond to solutions of log |t| + log |t − 1| = 0 or
|t(t − 1)| = 1. It is easy to see that a circle of sufficiently large radius in L centered
at the origin (0, 0, 0) is disjoint from A. In the same time the disk bounded by this
circle intersect A. This circle is not nomologically trivial in H1 (R3 r A) as e.g. its
linking number with the path connecting the punctures 0 and ∞ is non-zero.
In the same time, in accordance with Theorem 2.5 the amoeba A ⊂ R3 is hyperbolically embedded to R3 : its Gaussian curvature is nowhere positive.
2.2. Behavior of amoebas at infinity. As we have seen, the amoeba A is closed.
But it can never be compact for positive-dimensional very affine varieties V as such
18
GRIGORY MIKHALKIN
varieties themselves are non-compact while Log |V is proper. Thus A has to approach
infinity at RN .
One way to see a neighborhood of infinity at (C× )N is to compactly it to CPN and
then look at the neighborhood of ∂CPN = CPN r(C× )N . By the very construction of
the projective space the boundary ∂CPN consists of N + 1 coordinate hyperplanes:
N corresponding to the coordinate zeroes zj = 0 which we have to put back to
recover CN from (C× )N . The remaining hyperplane in ∂CPN corresponds to the
infinite plane CPN r CN in projective geometry. It can also be seen as a coordinate
plane once we change the coordinates in (C× )N according to another coordinate
chart in CPN , e.g. in coordinates z1−1 , z2 z1−1 , . . . , zN z1−1 . These coordinate charts
can be identified with each other with the help of a natural multiplicative-linear
action of GL(N, Z) on (C× )N .
Namely, for a matrix m = (mjk ) ∈ GL(N, Z) we define
(2.2)
N
N
Y
Y
m
m1j
m(z1 , . . . , zN ) = ( zj , . . . ,
zj N j ).
j=1
j=1
Proposition 2.9. The action (2.2) of GL(N, Z) on (C× )N is consistent with its
tautological action on RN . Namely, the diagram
(C× )N
/
(C× )N ,
Log
RN
/
Log
RN
where the horizontal arrows correspond to the actions my any element m ∈ GL(N, Z),
is commutative.
Proof. The proposition is a straightforward corollary of the logarithm property
log |zw| = log |z| + log |w|.
Thus any primitive integer vector v ∈ ZN can be thought of as a coordinate
direction (0, . . . , 1) in RN after a suitable GL(N, Z) coordinate change.
Let us consider the slices
At = A ∩ {xN = t}
of A by the horizontal hyperplanes in RN . Forgetting the coordinate xN we may
interpret At as a subset of RN −1 depending on the parameter t ∈ R.
Let us consider the topological closure V̄ in (C× )N −1 × C of V ⊂ (C× )N . Denote
V−∞ = V̄ ∩ (C× )N −1 × {0} = (C× )N −1 .
We may consider the amoeba A−∞ of V−∞ as a subset in RN −1 .
Proposition 2.10 (cf. [?]). We have
A−∞ = lim At .
t→−∞
GEOMETRY OF AMOEBAS
19
Furthermore, we may describe A−∞ as the set of points (x1 , . . . , xN −1 ) ∈ RN −1 such
that for any neighborhood U ⊂ RN −1 , U 3 (x1 , . . . , xN −1 ) and M ∈ R, we have
U × (−∞, M ] 6= ∅.
In particular, A−∞ is a well-defined set in the quotient of RN by the (0, . . . , 0, −1)direction and is invariant under a coordinate change in (C× )N as long as the direction (0, . . . , 0, −1) is constant in the first N − 1 new coordinates and decreases the
value of the last coordinate.
Similarly, we may describe V−∞ as the set of points (z1 , . . . , zN −1 ) ∈ (C× )N −1
such that for any neighborhood U ⊂ (C× )N −1 , U 3 (z1 , . . . , xN −1 ) and M ∈ R, we
have
U × {0 < |zN | < eM } =
6 ∅.
It is also invariant with respect to coordinate changes described above.
The limit here is taken in the sense of Hausdorff metric on compact sets in RN .
This means that for every compact set K ⊂ RN we have
lim d(K ∩ At , A−∞ ) = 0,
t→−∞
where d stands for the Hausdorff distance between the sets, i.e. the maximum
possible distance between one set and a point of the other.
Proof. The proposition follows from the similar statement about the limit of V ∩
{zN = et+αi } ⊂ (C× )N −1 . The limit of these sets when t → −∞ is V̄ ∩ {zN = 0}
since V̄ is algebraic and is obtained as the topological closure of V .
For any vector v ∈ RN we may define the affine space RN /v ≈ RN −1 as the
quotient space of RN by all lines parallel to v. Similarly, we define (C× )N /v ≈
(C× )N −1 as the quotient space of (C× )N by C× -subtori parallel to v, so that Log
provides a well-defined map from (C× )N /v to RN /v. We may define the v-infinite
direction amoeba as
N
(2.3) A∞
| {U + (M + t)v | t > 0} =
6 ∅ ∀U 3 x, ∀M ∈ R}/v ⊂ RN /v.
v = {x ∈ R
Here U is any open neighborhood of x in RN .
∞
Clearly, A∞
(0,...,0,−1) = A−∞ from Proposition 2.10. We define the variety Vv
as V−∞ under a suitable coordinate change from Proposition 2.9. We call it the
v-infinite variety for V ⊂ (C× )N .
Definition 2.11. We say that v ∈ RN is a A-limiting direction if A∞
v 6= ∅. The
union Log of all limiting directions and {0} is called the limiting set, or the Bergman
fan (see Theorem 2.13) of V ⊂ (C× )N .
Definition 2.12. A set C ⊂ RN is called a polyhedral cone if it is the intersection
of finitely many closed half-spaces Hj passing through the origin 0 ∈ RN . Each
Hj is defined by the inequality uj .x ≥ 0 for the scalar product uj .x with a vector
20
GRIGORY MIKHALKIN
uj ∈ RN . A polyhedral cone C is called Z-polyhedral if uj ∈ ZN for all j. A face of
C is cut on C by a system of equations uj .x = 0 for some j.
A Z-polyhedral fan F is a collection of Z-polyhedral cones such that
• whenever a cone C is contained in F all of its faces are also contained in F,
• the intersection of any subcollection of cones in F is the common face of all
cones in the subcollection.
We say that F is n-dimensional if it is a finite union of n-dimensional Z-polyhedral
cones and their faces. We often identify the fan F (as a collection of cones) with
the set
[
F =
C ⊂ RN
C∈F
which we also call a fan.
Recall that the basic construction of toric geometry (see e.g. [?], [?]) provides a
partial compactification TF of (C× )N for a Z-polyhedral fan F. Each cone C ∈ F
corresponds to stratum TC ⊂ TF .
Theorem 2.13 (cf. Bergman [?]). Let V ⊂ (C× )N be a n-dimensional algebraic variety. The limiting set L ⊂ RN of all A-limiting direction is a (finite) n-dimensional
Z-polyhedral fan.
Furthermore, the topological closure V̄ of V in TF is compact and the variety V̄ is
transversal to the stratum TC for each cone C ∈ L. If a vector v ∈ RN belongs to the
relative interior of C then Vv∞ ⊂ (C× )N /v is invariant with respect to multiplication
by any vector parallel to C. The quotient of Vv∞ by the subtorus formed by such
vectors coincides with V̄ ∩ TC (note that TC is the quotient of the whole (C× )N by
this subtorus).
We postpone the proof of this theorem until later in these lectures. We’ll also see
that the Z-polyhedral fan L in this theorem satisfies to a certain balancing condition
(which is a fundamental property in tropical geometry).
Remark 2.14. We may rephrase the last part of Theorem 2.13 without use of toric
geometry language as follows. If v ∈ RN belongs to the relative interior of a kdimensional cone C ∈ F then Vv∞ ⊂ (C× )N /v is invariant with respect to multiplication by any vector parallel to C/v. If v 0 ∈ C 0 , where C 0 is a face of C, then we may
consider the v-infinite variety Vv∞,∞
⊂ ((C× )N /v)/v 0 for Vv∞
0 (taking the quotient of
0 ,v
N
0
∞
v in R /v ). This variety coincides with the quotient of Vv in ((C× )N /v)/v 0 .
2.3. Behavior of coamoebas at infinity. The map Arg |V : V → (S 1 )N is not
proper, but its target is compact. Thus the limit of the ends of V should give us a
potentially interesting subset in (S 1 )N .
GEOMETRY OF AMOEBAS
21
Namely, let us consider an exhausting family of compact sets Kj ⊂ V , Kj ⊂ Kj+1 ,
N
S
j ∈ {1, . . . , n},
Kj = V and the topological closures Arg(V r Kj ) in (S 1 )N of
j=1
the images of their complements.
Definition 2.15 (also called the phase-limit set, see [?]). We define the infinity
locus of coamoeba as
∞
\
∞
B =
Arg(V r Kj ) ⊂ (S 1 )N .
j=1
It is convenient to consider the topological closure B̄ of B in (S 1 )N . We call it the
closed coamoeba of V .
As the restriction of the map Arg to a compact set Kj is proper for any j, we get
the following proposition.
Proposition 2.16. The set B r B ∞ is closed in (S 1 )N r B ∞ . In other words,
∂B r B ⊂ B ∞ .
Corollary 2.17.
B̄ = B ∪ B ∞ .
The corollary follows from Proposition 2.16 since B ∞ ⊂ B̄ by the definition of
B∞ .
The complement of Kj in V for large j can be thought of as a neighbourhood of
V ∞ = V̄ r V
in the compactification V̄ (defined in Theorem 2.13) after removing the set V ∞ itself
from this neighborhood. For each cone C ∈ L we consider a family of open sets
∞
T
C
, j ∈ {1, . . . , n}, such that
UjC = C. We define
UjC ⊂ V , UjC ⊃ Uj+1
j=1
BC∞
=
∞
\
Arg(UjC ) ⊂ (S 1 )N .
j=1
As in the previous subsection we may consider the linear projection
πC : (C× )N → (C× )N /C
obtained by taking the quotient of (C× )N by the subtorus parallel to C. (Clearly,
(C× )N /C ≈ (S 1 )N −k , where k is the dimension of C.) Furthermore, we may consider
the coamoeba BC ⊂ (C× )N /C of VC∞ .
Proposition 2.18. We have
BC∞ = πC−1 (B̄C ).
22
GRIGORY MIKHALKIN
This proposition follows from Proposition 2.10. As a neighborhood of V ∞ decomposes into the union of neighborhoods of its strata VC∞ we get the following
statement.
Corollary 2.19 (cf. [?], [?]). The infinity locus of B is a closed set in (S 1 )N that
can be decomposed in the following way:
[
B∞ =
πC−1 (B̄C ).
C
Example 2.20. We see the three geodesics on the torus that form B ∞ in the case
of the line {z + w + 1 = 0} depicted on Figure 4.
2.4. Convexity of coamebas. Outside of its infinity locus B ∞ the coamoeba possesses the same convexity properties as the amoeba. In fact, its proof is even easier
as the fiber of the map Arg is contractible.
As usual, we start from the case when V is a hypersurface.
Theorem 2.21 (cf. [?]). For every connected component U of (S 1 )N r B ∞ the set
U rB is convex. This means that if I ⊂ U is a geodesic interval such that ∂I ∩B = ∅
then I ∩ B = ∅.
Proof. Let I ⊂ (S 1 )N be such an interval and L ⊂ RN be a line parallel to I (as
usual, we identify RN with the universal covering space of (S 1 )N ). There is a (N −1)dimensional family of such lines parameterized by RN /I ≈ RN −1 that provides a
foliation of RN . For any leaf L of this foliation the product
L × I ⊂ RN × (S 1 )N = (C× )N
is a holomorphic annulus.
Since I is disjoint from B ∞ , the set Arg−1 (I) ∩ V is compact. Therefore, (L ×
I) ∩ V = ∅ if L is sufficiently close to infinity in RN /I, but it is non-empty for some
leaves L if I ∩ B 6= ∅. We get a contradiction to positivity of intersection of complex
subvarieties of complimentary dimensions in (C× )N .
Now we treat the general case when dim V = N − k. Suppose that K ⊂ RN is
a k-dimensional linear space, S̃ ⊂ K is a closed hypersurface and D̃ is the closed
domain in K bounded by S̃. We denote by S and D the images of S̃ and D̃ in
RN /2πZN ≈ (S 1 )N .
Theorem 2.22. If D ∩ B ∞ = ∅ and S ∩ B̄ = ∅ then D ∩ B̄ = ∅.
In other words, there is no cutting k-cap for the coamoeba of a variety of codimension k in (C× )N . Note that unlike the case with Theorem 2.6 we do not require
any homological triviality for S.
Proof. We consider all k-dimensional linear spaces L that are parallel to K. As
Arg−1 (D) ∩ V is compact we have L × D ∩ V = ∅ if L is close to infinity in RN . In
GEOMETRY OF AMOEBAS
23
the same time there exists a choice of L with L × D ∩ V 6= ∅ (and thus a positive
intersection number) if D ∩ B ∞ 6= ∅.