Classes of tropical linear spaces

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Hirzebruch introduced a function of a smooth projective complex variety $X$ known as the \emph{$\chi_y$-genus},\[\chi_y(X) = \sum_{p,q}(-1)^{p+q}\dim H^{p,q}(X)\, y^p\] where $H^r(X,\mathbb C)=\bigoplus H^{p,r-p}(X)$ is the Hodge structure on~$X$; different specialisations of $y$ provide several important Euler-characteristic-type invariants.[Gross, 1705.05719] extends the $\chi_y$-genus to subvarieties $X$ of algebraic tori which are \emph{sch\"on} in the sense of [Tevelev, math/0412329], and describes how to compute it from the tropicalisation of~$X$. We will be studying the properties of the $\chi_y$ genus of a hyperplane arrangement complement as a matroid invariant and trying to determine if this is valuative.