Let $X$ be projective and Gorenstein (over $\mathbb{C}$), of dimension $n$, then $\chi(\mathcal{O}_X)=(-1)^n\chi(\omega_X)$. Hence a "generalization": $\chi(\omega^{\otimes k}_X)$. 

I'd like something of this sort for the topological Euler characteristic. For example, suppose $X$ is smooth, so $\chi(X)=c_n(T_X)$. We could consider $c_n(T^{\otimes k}_X)$. More generally, let $\lambda$ be a Young tableau (symmetrization pattern), then we can consider $c_n(T^{[\lambda]}_X)$. In a similar way, starting from $\chi(X)=\sum(-1)^{p+q}h^p(\Omega^q_X)$ one could suggest $\sum(-1)^{p+q}h^p((\Omega^q_X)^{[\lambda]})$

I'd like the generalized Euler characteristic to be still defined on a broad class of topological spaces. (Or at least for any quasi-projective variety.) So, the suggestions above only give a motivating idea. Also, I'd like the generalized E.char. to be additive (at least for algebraic stratifications).

Is there something known in this direction?