Monge-Ampère
http://www.dm.unipi.it/~bertrand/amoeb-geotrop/node1.html
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Subsections
- Definition and first properties
- Connected components of the complement
- Spine
- Monge-Ampère measure
- Compactified Amoeba
- First application: Harnack curves
- Second application: dimers
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
in
, 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
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
(), 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
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
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
.
Given the Newton polytope , denote by the set of vertices of , and consider the ``moment map'':
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
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])