CHAPTER 1

The tropics

We start with the simplest of the three worlds, the tropical world. Tropical

geometry is a kind of piecewise linear combinatorial geometry which arises when

one starts to think about algebraic geometry over the so-called tropical semi-ring.

This chapter will give a rather shallow introduction to the subject. We will start

with the definition of the tropical semi-ring and some elementary algebraic geometry

over the tropical semi-ring. We move on to the notion of parameterized tropical

curve, which features in Mikhalkin’s curve counting results. Next, we introduce the

type of tropical objects which arise in the Gross-Siebert program: aﬃne manifolds

with singularities. These arise naturally if one wants to think about curve counting

in Calabi-Yau manifolds. We end with a duality between such objects given by the

Legendre transform.

1.1. Tropical hypersurfaces

We begin with the tropical semi-ring,

Rtrop

= (R, ⊕, ).

Here R is the set of real numbers, but with addition and multiplication defined by

a ⊕ b := min(a, b)

a b := a + b.

Of course there is no additive inverse. This semi-ring became known as the tropical

semi-ring in honour of the Brazilian mathematician Imre Simon. The word tropical

has now spread rapidly.

We would like to do algebraic geometry over the tropical semi-ring instead of

over a field. Of course, since there is no additive identity in this semi-ring, it is not

immediately obvious what the zero-locus of a polynomial should be. The correct,

or rather, useful, intepretation is as follows. Let

Rtrop[x1,

. . . , xn]

denote the space of functions f :

Rn

→ R given by tropical polynomials

f(x1,...,xn) =

(i1,...,in)∈S

ai1,...,in x11

i

· · ·

xnni

where S ⊆

Zn

is a finite index set. Here all operations are in

Rtrop,

so this is really

the function

f(x1,...,xn) = min{ai1,...,in +

n

k=1

ikxk | (i1,...,in) ∈ S}

This is a piecewise linear function, and the tropical hypersurface defined by f,

V (f) ⊆

Rn,

as a set, is the locus where f is not linear.

3

http://dx.doi.org/10.1090/cbms/114/02