Basic definitions

1. What is an abstract tropical variety (without embedding, as ringed spaces, rigid analytical space, non-archimedean field given by a covering of charts) ?
   What is a good family of local models? Gluing maps.
2. What is an abstract idempotent variety?

Detailed discussion (moderated and contributed by Margaret Symington; see also the figure of the white board of that session):

In the subsequent discussion of the basic definition of tropical varieties, the workshop participants expressed a desire to lay out a couple of definitions, sorting out the names of different objects arising in the tropical realm. Here is what was proposed (in the lower right hand section of the white board):

Given an ideal $I$ in $K^*\subset K=\overline{\mathbb{C}(t)}$ (the Puiseux series), consider two maps,

$${\rm val}: K^*\rightarrow \mathbb{R}^*$$

and

$$({\rm val,\ phase}): K^*\rightarrow \mathbb{C}^*$$

where the valuation map `` val'' takes the value of the smallest exponent in a Puiseux series and `` phase'' takes the argument of the coefficient of the term with the smallest exponent. Then

• a tropical variety is the image of ${\rm val}$,
• a complex tropical variety is the image of $({\rm val,\ phase})$, and
• a real tropical variety is the subset of a complex tropical variety on which the phase is real-valued.

In the $p$-adic setting (lower center of the white board) one has analogous objects of interest: the image of the valuation map in $\mathbb{Q}$ and the image of the valuation and the phase in $\mathbb{Q} \times \overline{F^*_p}$ where $\overline{F^*_p}$ is the closure of the set of multiplicative generators of the algebraic closure of $F_p$.

More generally (in characteristic zero), let $I$ be an ideal in the Laurent polynomial ring $K[z_1^{\pm 1}, \ldots, z_n^{\pm 1}]$ and let $V(I)$ be its affine variety $V(I)\subset (K^*)^n$. Then the corresponding tropical and complex tropical varieties are the images of $\rm val$ and ${\rm (val,\ phase)}$ in $(\mathbb{Q}^*)^n$ and $(\mathbb{C}^*)^n$. The fiber of the projection from such a complex tropical variety to the corresponding tropical variety is a torus, while the fiber of the projection from the real tropical variety is a subset of $\mathbb{Z}_2^n$.

Now view an ideal $I$ in $K[z_1^{\pm 1}, \ldots, z_n^{\pm 1}]$ as a family of ideals $I_t$ in $\mathbb{C}[z_1^{\pm 1}, \ldots, z_n^{\pm 1}]$. As $t\rightarrow 0$ the complex algebraic varieties $V(I_t)$ converge in the Hausdorff limit to the complex tropical variety for $I$. In general the complex tropical variety is not homeomorphic to $V(I_t)$ for $t\ne 0$. For curves, if the complex tropical variety is smooth, then it is homeomorphic to $V(I_t)$ for small $t$.

Question: When is the complex tropical variety homeomorphic or homotopic to the algebraic variety $V(I_t)$ for small $t$?

Question: Can one define a Hodge theory for complex tropical varieties that is consistent with the limit of the classical Hodge theory?

Moreover, the following questions and needs were stated:

1. Patchworking theorem: Is there a refinement of tropical geometry that will give information about a single (classical) hypersurface?

2. Need to define maps, line bundles, vector bundles

Back to the main index for Amoebas and tropical geometry.