Monge-Ampère

http://www.dm.unipi.it/~bertrand/amoeb-geotrop/node1.html

Next: Tropical varieties Up: Tropical Geometry and Amoebas Previous: Tropical Geometry and Amoebas

Subsections

Amoebas of algebraic varieties

Definition and first properties

Take an algebraic variety in . Its amoeba is its image by the map

This name was first introduced by Gelfand, Kapranov and Zelevinsky in [GKZ94].

A first property of an amoeba is that it is closed.

Most of the properties we will mention concern amoebas of hypersurfaces, so that from now on, unless otherwise specified, we will consider a Laurent polynomial

, where the bold letters stand for -coordinate indeterminates (e.g ); is a finite subset of and means.

Let

be its zero set in . We study its amoeba . See an example of the picture of such an object in Figure 1.1

Figure 1.1: Example of an amoeba (taken from[Mik01])

Connected components of the complement

Theorem 1.1 Any connected component of is convex.

This is proved in [GKZ94]: it is because

is a domain of convergence of a certain Laurent series expansion of .

A useful function is the Ronkin function for the hypersurface: it is the function

defined by:

Theorem 1.2 (Ronkin) The Ronkin function is convex. It is affine on each connected component of

and strictly convex on .

See [PR00] for the study of the Ronkin function.

Actually, we will be able to see that it is affine on each connected component of

after the following propositions.

Proposition 1.3 The derivative of

with respect to is the real part ofProof. Write the coordinates in polar coordinates

. Then for fixed , and we haveDifferentiating with respect to

, we get

For in a connected component of , this is constant (since the homology class of the cycle in

remains unchanged) and was defined in [FPT00] to be the order of the component . They proved the following properties, all based on the residue formula (other proofs in [Rul01]):

Proposition 1.4 For in a component , is an integer.Proof. Consider for fixed

(), the integralBy the residue formula this is an integer (it counts the number of zeroes of the function

minus the numbers of poles, in the disk of boundary

), and since it depends continuously on the (), it is independent of them.

It is equal to

. Indeed,

Note that the fact that

is constant over any connected component of the complement implies that the partial derivatives of in each such connected component are constant, hence

is affine there!

Proposition 1.5 (Proposition 2.4 in [FPT00])

is a lattice point of the Newton polygon of (that is, the convex hull of the elements of for which .)Proof. The vector is in if and only if for any vector , .

Indeed, is in if and only if for any line passing through 0, its orthogonal projection on belongs to the projection of on (see Figure 1.2). By density we can assume that has a rational slope. The vector appearing here represents the slope of , and the scalar product can be seen as the projection on .

Figure 1.2: Condition for to belong to

Claim: is the number of zeroes (minus the order of the pole at the origin) of the one-variable Laurent polynomial inside the unit circle

(where is any point of , being the point where is computed).

But this polynomial has top degree equal to

. Hence we are done.

It remains to proof the claim. The numbers of zeroes (minus number of poles) of the function

in the disk is given by the usual formula

. We use a change of variable formula . The image of the circle

by this change of variable is a loop in , homologous to the sum where is the ``circle''

().

Hence

Topologically it has the following meaning:

is a -dimensional torus which does not intersect . Consider for each a loop of this torus (along which all the coordinates except

are constant), and let be a disk whose boundary is . Then is the intersection number of and

(see also [Mik00]).

Theorem 1.6 (Proposition 2.5 in [FPT00]) The map

sends two different connected components to two different points.

This implies that the number of connected components is finite, and less than or equal to the number of lattice points in .

Proof. Take two points and in , and let and . Let

such that for some positive . The claim in the preceding proof implies that and are the numbers of zeroes inside of the two polynomials and , where

and ; we choose such that i.e. they have the same argument. Hence

. Thus is the number of zeroes of inside the circle .

If , this means that has no zero in the ring , hence there is no point of the amoeba on the segment

(see Figure 1.3). This implies that and are in the same component.

Figure 1.3: The image of the ring by the map is sent to the segment

Spine

Define

where the range through the connected components of and

is the affine function whose restriction to coincides with .

Definition 1.7 ([PR00]) The spine of

is the corner locus of the function .

As we will see later, this is a tropical variety.

It is a deformation retract of the amoeba

(see [Rul01]). See Figure 1.4.

Figure 1.4: Amoeba and its spine (taken from[Mik01])

Remark: In [PR00] and [Rul01], the (non-obvious) relation between the coefficients of

and the coefficients of the ``tropical polynomial'' is studied (see later for the meaning of ``tropical polynomial'').

It is shown there that

where is the subset of of the for which there exist a connected component

of order of , and

for

It is also proved that, in the particular case where has no more than points and that no of these lie in an affine -dimensional subspace for ,

(remember that the are the coefficient of ).

Monge-Ampère measure

see [PR00], [Rul01]...

Compactified Amoeba

Given the Newton polytope , denote by the set of vertices of , and consider the ``moment map'':

In fact

is the restriction of the moment map where is the toric variety associated to .

The compactified amoeba

of is the closure of in .

See [Mik01].

First application: Harnack curves

Figure 1.5: Compactified amoeba of the amoeba of figure 1.1 (taken from[Mik01])

Definition 1.8 A curve of degree in is in maximal position with respect to the (generic) lines if is maximal (maximal number of ovals) There exist three disjoints arcs on one connected component such that .Theorem 1.9 (Mikhalkin)

, there exist only one maximal topological type (Harnack curve). If the number of generic lines is greater than , there is no such maximal topological type.

see [Ite03], [Mik00], [Mik01].

Second application: dimers

see [KO],[KOS].

Next: Tropical varieties Up: Tropical Geometry and Amoebas Previous: Tropical Geometry and Amoebas

Benoit BERTRAND 2003-12-19

Figure 1.6: The maximal topological type for (taken from[Mik01])

Page updated

Google Sites

Report abuse