\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip: a sketch of compactification in tropical geometry
a sketch of compactification in tropical geometry
Note that we have some ideas of compactification already: after all we can define tropical polynomials on \(\RR^n\), and \(\TT^n\) is a compactification of \(\RR^n\).
The space \(\TT^n\) has a decomposition into pieces \(\RR_I\), which we call the torus orbits. The key notion that we will need to study is how many ``-'infinity'''s does a point \(x\) belong to.
definition 0.0.1
Given \(x\in \TT^n\), we define the sedentarity of \(x\) to be the number of \(-\infty\)'s belonging to \(q\) in its coordinate description.
We say that \(\sigma\in \TT^n\) is a polyhedron if it is the closure of a polyhedron in one of the torus orbits of \(\TT^n\).
From these compact pieces, we can define a tropical variety: a locally open subset of polyhedra satisfying a balancing condition.