Higher Euler characteristics (possible generalizations)

Question

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?

Answer 1

We can associate to any $\mathbb{C}$-scheme $X$ in a canonical way a constructible function $\nu_{X}:X\rightarrow \mathbb{Z}$, which takes care of the singularities of the space $X$. This is proved in this paper Donaldson-Thomas type invariants via microlocal geometry. We can then define the weighted Euler characteristic of $X$ by $$ \chi(X,\nu_{X})=\sum_{n\in\mathbb{Z}}n\chi(\nu_{X}^{-1}(n)), $$ where $\chi$ is the topological Euler characteristic. The RHS is actually a finite sum and this is well-defined. The constructible function $\nu_{X}$ is quite mysterious and I don't think much is known about it. We know for example that $\nu_{X}(p)=(-1)^{\dim_{p}X}$ when $p\in X$ is a smooth point. So, when $X$ is smooth, we have $$ \chi(X,\nu_{X})=(-1)^{\dim X}\chi(X). $$ Another good situation is probably when $X$ can be written as the critical locus of some function. In this case we can use topological techniques (such as Milnor number) to compute the function $\nu_{X}$.