\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
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 \(f_*(\div(\phi))=P_1-P_2\).
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
\[\begin{tikzpicture}
\draw (-1.5,0) -- (1.5,0);
\draw (-1.5,0.5) -- (-0.5,0.5) -- (0.5,2) -- (1.5,2);
\node at (2.5,0) {\(R\)};
\draw (-0.5,0.5) -- (-0.5,-0.5) (0.5,2) -- (0.5,-0.5);
\draw[dashed] (-1.5,-0.5) -- (1.5,-0.5);
\draw[dashed] (-1.5,3) -- (1.5,3);
\end{tikzpicture}
\]