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 ai 1 ,...,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{ai 1 ,...,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
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