\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip: the tropical Hodge conjecture
the tropical Hodge conjecture
Associated to \(X\) a tropical variety, we want to some geometry. We have a short exact sequence
\[ 0 \to \RR_X\to PA_X\to \Omega_X\to 0\]
where \(\Omega_X\) gives the tropical cotangent sheaf, whose sections are called tropical 1-forms. Another characterization is: given a fan \(\Sigma\subset \RR^n\), want to define \(F_k(\Sigma)\subset \bigwedge^k\ZZ^n\) the set of polyvectors generated by \(v_1\wedge \cdots \wedge v_k\), where \(v_1, \ldots, v_k\) belong to a single cone of \(\Sigma\). This associates to \(X\) a tropical variety \(F_k\) a constructible sheaf on \(X\) so that \(F_{k, x}=F_k(\Sigma(x))\), where \(\Sigma(x)\) is the tangent cone at \(x\). This defines for us a chain complex
\[C_\bullet(X, F_k)\]
which comes with a tropical \((p, q)\) homology which is the homology \(H_q(X, F_p)\). When \(X\) can be geometrically realized, this compute some limit of a mixed Hodge structure, and satisfies the Kähler/Hodge package. Finally, there exists a cycle class map
\begin{align*}
CH_k(\RR^n)\to H_{p}(\RR^n, F_p)&&
Z\mapsto \Sigma_\subset K_{cell} \weight(\Delta)\cdot v_\Delta \otimes \Delta
\end{align*}
where \(\Delta\) is the volume of the \(k\)-cell. The tropical Hodge conjecture is that this is an isomorphism.