Classes of tropical linear spaces

Hirzebruch引入了一个光滑射影复簇$X$的函数，称为\{$\chi_y$-genus}，\[\chi_y（X）= \sum_{p，q}（-1）^{p+q}\dim H^{p，q}（X）\，y^p\]其中$H^r（X，\mathbb C）=\bigoplus H^{p，r-p}（X）$是X$上的Hodge结构; y的不同特化提供了几个重要的欧拉特征型不变量。[Gross，1705.05719]将$\chi_y$-亏格扩展到代数环面的子变种$X$，它们在[Tevelev，math/0412329]的意义上是\n {sch\“on}，并描述了如何从~$X$的热带化计算它。我们将研究超平面排列补的$\chi_y$亏格作为拟阵不变量的性质，并试图确定这是否是赋值的。

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.