\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: rational equivalence in tropical geometry

rational equivalence in tropical geometry

SECTIONS
Download .tex

definition 0.0.1

Let \(Z\subset Q\) be a subcycle (bounded) rationally equivalent to \(0\) if there exists a cycle \(Y, \dim(Y)=\dim(Z)\) and a morphism \(Y\to Q\) and a (bounded) rational function \(\phi\in \mathcal K^*_Y\) such that \[f_*(\deg(\phi)\cdot Y)=Z.\] If you're familiar with rational equivalence in algebraic geometry, the main difference is that we ask the cycle \(Y\) to be a subvariety of \(X\). Originally, the definition used that \(Y\) was a subset of \(X\), but unfortunately this definition is not compatible with pushforward. The belief is that the definition of tropical rational function is too rigid (so our definition for tropical rational equivalence) is a bit more flexible.

definition 0.0.2

Let \(X\) be a tropical variety. We define the tropical chow groups \(A^{(b)}_\bullet(X)=Z^{(b)}_\bullet(X)/\sim^{(b)}\) Let \(Q\subset Q\times \RR\). Let \(\phi_p\) be the pullback of \(\max(x, p')\). Then \(F_p=\mathrm{Div}(\phi\cdot F)\subset Q\times \{Q\}\cong Q\) Given \(P_1, 2\in Q\) we say that the are \(\sim^\RR\) if there exists a cycle \(F\) and points \(q_1, q_2\in \RR\) so that \(P_i=F_{q_i}\).

proposition 0.0.3

A cycle \(P_1\sim^b 0\) if and only if \(Z_1\sim^\RR 0\) Suppose that \(P_1\sim^\RR P_2\). Then there exists \(F\subset Q\times \RR\) with \(F_{q_i}=P_i\). Take \(R=F\), and define \(\phi:=\pi^*\psi\). We see that \((\pi_Q)_* (\pi^*(\psi^*\psi)\cdot F)=P_1-P_2\). NOw suppose instead that \(P_1\sim^b P_2\). Then there exists \(R\) and \(\phi\in \mathcal K_R\) such that \(f_*(\mathrm{Div}(\phi))=P_1-P_2\). Tke \(F:=(f\times \id)_* (\text{graph}(\phi))\). Cruicially, the function \(\phi\) is bounded so that the intersection with a sufficiently large slice is empty
figure 0.0.4:sufficiently large slices have empty intersection
Observe that if \(R\subset P\) has \(R\sim^b 0\) and \(g: P\to Q\) is a map of tropical varities that \(g_*Z\sim^b0\). However, consider the morphism this gives an example of \(Z\sim 0\) but \(f_*Z\not\sim 0\).
figure 0.0.5: