Amoebas of complex curves and tropical curves

Mark Gross University of Warwick

1 Amoebas of complex curves and tropical curves (I)

In these two lectures we will talk about amoebas and tropical curves. We will begin by giving some basic definition of amoebas and tropical geometry. As we proceed, we shall see the applications of these things in enumerative geometry and mirror symmetry. The plan of the talk is as follows

I. Amoebas

II. Tropical geometry

III. Applications to quantum cohomology and mirror symmetry

1.1 Amoeba of a Laurent polynomial

We begin by defining the amoeba of a Laurent polynomial f.

Definition (Amoeba). (Gelfand-Kapranov-Zelevinsky) Let

f(x, y) = ∑_{i, j∈Z} a_{i j}x^{i} y^{j}

be a Laurent polynomial with only a finite number of the a_{i j}’s being non-zero. Define the set V_{f} and the function Log as follows

V_{f} = {(x, y) ∈ (ℂ^{∗})^{2}|f(x, y) = 0}

Log : (ℂ^{∗})^{2} ⟶ ℝ^{2}
(x, y) ⟼ (log |x|, log |y|)

The amoeba of f is the set

A_{f} = Log(V_{f})

To get an idea of what an amoeba looks like, we will now look at some examples

Example 1. The amoeba of f(x, y) = 1 + x + y. The set |V_{f}| is bounded by 3 straight lines, which is shown on the left, it is mapped to the the set A_{f}, which is the Amoeba of f, by the map Log. We can see that the set A_{f} looks like an amoeba. The set |V_{f}| in this case is homeomorphic to a sphere with 3 puncture.

[figure: |V_{f}|; |x|; |y|; x=0; y=0; 1; 1; A_{f}; Log(x = 0); Log(y = 0)]

Example 2. If we take f(x y) = x^{−1} y^{−1} + 1 + x + y, we get an amoeba with a ‘hole’ in it. In this case, |V_{f}| is homeomorphic to a torus with 3 puncture.

1.2 Newton polytope and components of ℝ^{2}\A_{f}

We will now introduce the Newton polytope of f. We shall see that the Newton polytope turns out to be a useful tool in counting the connected components of ℝ^{2}\A_{f}.

Definition (Newton polytope). The Newton polytope of a Laurent polynomial f △_{f} ⊂ ℝ^{2} is the convex hull of {(i, j)|a_{i j} ≠ 0} ⊂ ℤ^{2}

For example, the Newton polytopes of f = 1+x+y and f = x^{−1}y^{−1}+1+x+y looks like Example 3. (i) f(x, y) = 1 + x + y (ii) f(x, y) = x^{−1}y^{−1} + 1 + x + y

[figure: (i); (0, 0); (0, 1); (1, 0); (ii); (0, 0); (0, 1); (1, 0); (−1, −1); ·]

We now take (z_{1}, z_{2}) to be coordinates on (ℂ^{∗})^{2} and (x_{1}, x_{2}) to be Log(z_{1}, z_{2}). The Ronkin (Forsberg-Passare-Tsikh, Ronhlin, Passare-Rullgård) function on ℝ^{2} is then

N_{f}(x_{1}, x_{2}) = \frac{1}{(2π i)^{2}} ∫_{Log^{−1}(x_{1}, x_{2})} log |f(z_{1}, z_{2})| \frac{dz_{1}}{z_{1}} ∧ \frac{dz_{2}}{z_{2}}

We can think of this function as the average of log |f(z_{1}, z_{2})| over the fibre Log^{−1}(x_{1}, x_{2}). Note that this function is defined even on A_{f} despite the blowup of the integrand. In fact, we have

Theorem. N_{f} : ℝ^{2} ⟶ R is a convex function which is strictly convex over A_{f} and linear over each connected component of ℝ^{2}\A_{f}

This gives the following corollary

Corollary. Every connected component of ℝ^{2}\A_{f} is convex

The following proposition allows us to relate the Newton polytope with the connected components of ℝ^{2}\A_{f}.

Proposition. For every connected component E of ℝ^{2}\A_{f},

∇N_{f}|_{E} ∈ △_{f} ∩ ℤ^{2}

This gives way of associating connected components of ℝ^{2}\A_{f} to the points on the Newton polytope △_{f}. Namely, we have an injective map

{connected components of ℝ^{2}\A_{f}} ⟶ △_{f} ∩ Z^{2}

In particular, this gives an upper bound to the number of connected components of ℝ^{2}/A_{f}.

(number of connected components of ℝ^{2}\A_{f}) ≤ #(△_{f} ∩ ℤ^{2})

1.3 Spine of an Amoeba

We will now define the spine of an amoeba. We first define a function N_{f}^{∞} which has the effect of ‘putting sharp corners’ in the function N_{f}.

For each connected component E of ℝ^{2}\A_{f}, let N_{E} be the linear extension of N_{f}|_{E} to ℝ^{2}, and define N_{f}^{∞} by

N_{f}^{∞}(x_{1}, x_{2}) = max_{E} N_{E}(x_{1}, x_{2})

This looks like the following

[figure]

As we can see, the function N_{f}^{∞} is like the function N_{f} ‘with corners’. We can now define the spine of A_{f}

Definition (Spine). (Passare-Rullgård)The spine S of A_{f} is the space

S = {(x_{1}, x_{2}) ∈ ℝ^{2} | N_{f}^{∞} is not linear at (x_{1}, x_{2})}

Clearly S ∈ A_{f}.

The following theorem shows that the topology of an amoeba is essentially captured by its spine

Theorem. S is a strong deformation retract of A.

As an example, we look at the spines of the amoebas of examples 1 and 2.

[figure: Ex (i); Ex (ii)]

Where the spines are indicated by the straight lines in the figures.

1.4 Non-Archimedean Amoebas

Let K be a field equipped with a valuation, i.e., a map V : K^{∗} ⟶ ℝ such that

(i) V (x y) = V (x) + V (y)
(ii) V (x + y) ≥ min{V (x), V (y)}
⟺ − V (x + y) ≤ max{−V (x), −V (y)}
⟺ e^{−V (x+y)} ≤ max{e^{−V (x)}, e^{−V (y)}}

Given a valuation V, we define |x| = e^{−V (x)}. This is is called a nonarchimedean absolute value. As we can see, this satisfies the triangle inequality

|x + y| ≤ max{|x|, |y|} ≤ |x| + |y|

and thus gives an absolute value on the field.

We now look at some examples of valuations

Example 4. (i) K = ℂ[[t]] : field of Laurent series ∑_{i∈ℤ} a_{i} t^{i}, a_{i} ∈ ℂ (a_{i} = 0 for i << 0)

V (∑_{i∈ℤ} a_{i} t^{i}) = min{i | a_{i} ≠ 0}

(ii)We now take K = \overline{ℂ[[t]]}, the algebraic closure of K, which is the field of Puiseux series.

An element of K can be represented by

∑_{i∈I} a_{i} t^{i} (I ⊂ ℝ)

such that I has a lower bound, we define V to be

V (∑_{i∈I} a_{i} t^{i}) = inf(I) (if a_{i} ≠ 0 ∀i ∈ I)

We can now define the map Log and the amoeba A_{f} for a f with coefficients in a general field with a valuation.

Log : (K^{∗})^{2} ⟶ ℝ^{2}
(x, y) ⟼ (log |x|, log |y|) = (−V (x), −V (y))
f(x, y) = ∑ a_{i j}x^{i} y^{j} a_{i j} ∈ K
V_{f} ={(x, y) ∈ (K^{∗})^{2} | f(x, y) = 0}
A_{f} =Log(V_{f})

We will now construct the amoeba of f = 1 + x + y with an non-archimedean absolute value.

Example 5. Consider f = 1 + x + y, since |x| = e^{−V (x)}, we have

Log(x, y) = (−V (x), −V (y))

since V (x + y) ≥ min{V (x), V (y)}, if (x, y) is a solution of 1 + x + y = 0, then at least one of the V (x), V (y) must be non-positive to cancel with 1. We then have the following three cases:

(i) V (x) ≤ 0, V (y) ≤ 0 then V (x) = V (y)

(ii) V (x) ≤ 0, V (y) > 0 ⟹ V (x) = 0

(iii) V (y) ≤ 0, V (x) > 0 ⟹ V (y) = 0

The amoeba then looks like

[figure: (ii); (iii); (i)]

where the numbers near the lines indicate the case that corresponds to the line. We see that this construction in fact gives the spine of the amoeba.

Kapranov has observed that the non-archimedean amoeba is given by the non-linear locus of the function N_{f}^{K}, where

N_{f}^{K}(x_{1}, x_{2}) = max_{(i_{1}, i_{2})|a_{i_{1} i_{2}} ≠0}{i_{1} x_{1} + i_{2} x_{2} − V (a_{i_{1} i_{2}})}

and f(z_{1}, z_{2}) = ∑ a_{i j}z_{1}^{i} z_{2}^{j}, (a_{i j} ∈ K).

1.5 Spine and triangulation of the Newton polytope

We now consider the map (i_{1}, i_{2}) ⟼ V (a_{i_{1} i_{2}}). We can take the graph of this function. The projection in one direction looks like

[figure: R; ·; ·; ·; ·; ·; ·; ·; ·; i_{1}; ⋯; i_{n}; i_{2}; △_{f}]

The faces of the lower convex hull of the graph is projected to a decomposition of the Newton polytope △_{f}. For example,

Example 6. 1+x+y+t x y [figure: (0, 1); (0, 0); (1, 1); (1, 0)]

An interesting question would be whether all triangulations of △_{f} using only vertices appearing on f can be obtained in this way. The answer to this is no, and we shall call the triangulations which arise in this way regular triangulations . Notice that the field K is not essential in this construction, we can use any field K.

An interesting relation between the amoeba and the Newton polytope is that the amoeba is homeomorphic to the 1-skelton of the dual cell convex of the decomposition.

Example 7. 1 + x + y [figure]

1+x+y+t x y [figure: (0, 2); (0, 0); (2, 0); 1 + x + y + txy + t^{3}y^{2} + t^{3}x^{2}] t^{−1} + x + y + x^{−1}y^{−1} [figure]

2 Amoebas of complex curves and tropical curves (II)

We begin the lecture by defining tropical curves. The tropical semi-ring in (ℝ, ⊕, ⊙) is defined by

x ⊕ y = max(x, y)
x ⊙ y = x + y

It is called tropical because of a Brazilian computer scientist who finds it useful.

If we use ⊕ for addition and ⊙ for multiplication, a tropical polynomial in n variables is then given by

f = ∑ a_{I}x_{1}^{i_{1}} ⋯ x_{n}^{i_{n}}

= max_{(i_{1},··· ,i_{n})}{a_{I} + i_{1} x_{1} + ⋯ + i_{n} x_{n}}

and its zero set is V_{f} = locus where f is not linear

For example,

Example 8. Linear tropical polynomial f = a + bx + cy.

[figure]

These are called tropical curves.

2.1 Curve counting and tropical geometry

Mikhalkin and Kontsevich suggested that (Mikhalkin 2003, Kontsevich 2001) counting complex curves can be achieved by counting tropical curves.

For example, in ℂℙ^{2}, we want to know how many rational curves of degree d are passing through 3d − 1 general points. We first consider some simple cases. (Recall that a rational curve in ℂℙ^{2} is the image of ℂℙ^{1} under a map ℂℙ^{1} ⟶ ℂℙ^{2})

[figure: 1 d = 1; ×; ×; 1 d = 2; ×; ×; ×; ×; ×; ×; d = 3 8 points \underline{12}]

The general answer to this problem was given by Vafa using the WDVV equation, (Kontsewich, Rian-Tian ...).

We now look at how this can be achieved using tropical curves. A tropical curve of degree d is a tropical curve in ℝ^{2} where the defining equation has a Newton polytope of the form Conv{(0, 0), (d, 0)(0, d)}. One can define the notion of a tropical rational curve. We now want to ask: How many tropical rational curves are passing through 3d − 1 points in ℝ^{2}? In the simplest case, where the tropical curve is defined by the polynomial a+bx+cy and two general points [figure: ×; ×]

As we can see, provided the two points do not lie on a same ‘leg’ of the tropical curve, we only have one curve passing through these two points. This coincides with the number of complex curves passing through 2 points in ℂℙ^{2}. This is by no means a coincidence. In fact, if we assign a multiplicity of μ(C) to each each tropical curve C, (This was suggested by Mikhalkin) we then have

Theorem. For 3d − 1 general points {x_{1}, …, x_{3d−1}} in ℝ^{3}, we have

Σμ(C) = N

where the summation runs through the degree d rational tropical curves passing through the 3d-1 points x_{i} and N is the number of complex rational curves passing through these points.

2.2 Application to mirror symmetry

We will first define an affine manifold.

Definition. An affine manifold B (dim B = n) is a manifold with an open cover {U_{i}} and coordinate charts ψ_{i} : U_{i} ⟶ ℝ^{n} with ψ_{i} ∘ ψ_{j}^{−1} ∈ Aff(ℝ^{n}) = ℝ ⋊ GL_{n}(ℝ). We say that B is tropical if ψ_{i} ∘ ψ_{j}^{−1} ∈ ℝ ⋊ GL_{n}(ℝ).

Tropical automorphisms ℝ^{n} ⟶ ℝ^{n} are given by

(\begin{matrix}b_{1} ∏ x_{i}^{a_{1}} \\ ⋯ \\ b_{n} ∏ x_{i}^{a_{n}} \end{matrix} ) ⟶ x ⟼ Ax + b in non − tropical sense

where A ∈ GL(ℤ). Given a tropical manifold B, there exists a family of lattices Λ ⊂ T B generated locally by ∂/∂y_{1}, ⋯, ∂/∂y_{n}, where y_{1}, ⋯, y_{n} are local affine coordinates. It is important that B is tropical because it ensures that Λ is well-defined. We now take the quotient of the tangent bundle with this lattice and define a torus bundle over B

X(B) := T B/Λ ⟶ B

This manifold carries a natural complex structure by the identification

T U_{i} = ψ_{i}(U_{i}) × iℝ^{n} ⊂ ℂ^{n}

If B is orientable, then X(B) has a nowhere vanishing holomorphic n-form. In this case X(B) is Calabi-Yau.

We can also define a ‘dual space’ to X(B) by considering the lattice Λ̌ ⊂ T^{∗}B generated dy_{1}, ⋯, dy_{n}. The ‘dual space’ X̌(B) is then

X̌(B) = T^{∗}B/Λ̌

The symplectic form ω on T^{∗}B descends to X̌(B) and makes it a symplectic manifold. The ‘dual pair’ X(B), X̌(B) therefore consists of a complex manifold and a symplectic manifold. Also, X(B) ⟶ B, X̌(B) ⟶ B forms a pair of dual torus bundles. The philosophy behind this construction is that X(B) and X̌(B) forms a ‘mirror pair’ in the sense of Strominger-Yau-Zaslow, and that the tropical manifold B is the key to mirror symmetry. The structures on B, X(B) and X̌(B) should correspond to one another.

[figure: structures on B; structures on X(B); structures of X̌(B)]

For example, the affine structure on B corresponds to the complex structure on X(B) and the symplectic structure on X̂(B).

In particular, we would like to see the tropical curves on B to correspond to J-holomorphic curves on X̌(B). However, we do not know whether this is the case or not.

Example 9. Consider the case where B is a circle

B = ℝ/τ_{2}ℤ, τ_{2} > 0

X(B) would then be a torus

X(B) = ℂ/ < 1, iτ_{2} >

Example 10. We generalize this to the case where

B = ℝ^{n}/Γ (Γ : a lattice)

X(B) would then be a complex torus of dimension n.

Example 11. We construct B by removing six points from the boundary of a tetrahedron. Let Ξ = Conv{(−1, −1, −1), (3, −1, −1), (−1, 3, −1), (−1, −1, 3)} ⊂ ℝ^{3} Then

B = ∂Ξ − {P_{1}, ⋯, P_{6}}. [figure: ·; ·; P_{1}; ·; P_{2}; ·; ·; P_{3}; ·]

The open sets are given by {U_{σ}, U_{v}} where σ is a 2-face of the tetrahedron and v is a vertex. The set U_{σ} is the interior of the 2-face σ. We can think of it an open subset of 𝔸^{2}. This gives an affine structure on U_{σ}. Explicitly, we have

σ : a 2 − face of Ξ,
Ψ_{σ} : Int(σ) ↪ 𝔸^{2} ⊂ ℝ^{3}

For each vertex v of Ξ, let U_{v} be a suitably chosen open neighborhood of v such that it does not include any of the points P_{i}. We can map U_{v} to an open set in ℝ^{2} by taking the quotient

ψ_{v} : U_{v} ↪ ℝ^{3} ↠ ℝ^{3}/ℝv
ψ_{v} ∘ ψ_{σ}^{−1} ∈ Aff(Z^{2})

This affine structure then gives us a X(B) that can be compactified by adding degenerate tori over P_{1}, ⋯, P_{6}. The resulting manifold is homeomorphic to a K3 surface as a symplectic manifold. However, there are obstructions in extending the complex structure to the compatification and X̄(B) cannot be made biholomorphic to a K-3 surface.

2.3 Tropical line bundles

Let B be a tropical affine manifold. The structure sheaf of B is then

O_{B}(U) = tropical functions on U

= convex piecewise linear functions with integral gradient

and we define the Picard by

P ic(X) = H^{1}(X, O_{X}^{∗})

In analogy with the Picard group for holomorphic line bundle.

O_{B}^{∗} = sheaf of affine linear functions
P ic(B) = H^{1}(B, O_{B}^{∗})

A line bundle L ∈ H^{1}(B, O_{B}^{∗}) is represented by (U_{i} ∩ U_{j}, α_{i j}), where {U_{i}} is an open cover of B and α_{i j} is a section of O^{∗} on U_{i} ∩ U_{j}. A section of this line bundle β ∈ L(U) is then given by β_{i} ∈ O_{B}(U ∩ U_{i}) such that β_{i} − β_{j} = α_{i j} on U_{i} ∩ U_{j}.

Example 12. Let B = R/Z. We have the following short exact sequence

0 ⟶ ℝ ⟶O_{B}^{∗} ⟶ ℤ ⟶ 0

This gives rise to the following long exact sequence of the homology groups

0 ⟶ ℝ ⟶ ℝ ⟶ ℤ ⟶ H^{1}(B, ℝ) ⟶ H^{1}(B, O_{B}^{∗}) ⟶ H^{1}(B, ℤ) ⟶ 0
[figure: ≅; 0; ||; ℝ]

We therefore have the following short exact sequence, which is basically the same as the one for holomorphic line bundle, except that the ‘Jacobian’ has now become a 1-sphere rather than a torus

0 ⟶ ℝ/ℤ ⟶ H^{1}(B, O_{B}^{∗}) ⟶ ℤ ⟶ 0
[figure: Jacobian of B]

We thus see that each tropical line bundle is classified by an integer invariant, (degree) and the space of equivalence classes of tropical line bundles with a given degree d is a circle.

References

[1] M. Forsberg, M. Passare and A. Tsikh (2000). Laurent determinants and arrangements of hyperplane amoebas. Advances in Math. 151, 45–70.

[2] I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky (1994). Discriminants, resultants, and multidimensional determinants. Mathematics: Theory and Applications. Birkhäuser Bosten, Inc., Boston, MA.

[3] M. Kontsevich and Y. Soibelman (2001). Homological mirror symmetry and torus fibrations. Symplectic geometry and mirror symmetry. World Sci. Publishing, River Edge, NJ, 203–263.

[4] G. Mikhalkin (2001). Amoebas of algebraic varieties. arXiv:math. AG/0108225.

[5] G. Mikhalkin (2003). Enumerative tropical algebraic geometry in ℝ^{2}. arXiv:math.AG/0312530.

[6] M. Passare and H. Rullgård (2000). Amoebas, Monge-Ampére measures, and triangulations of the Newton polytope. Preprint, Stockholm University.

[7] A. Strominger, S. T. Yau and E. Zaslow (1996). Mirror symmetry is Tduality, Nuclear Phys. B, 479, 243–259.