MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

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?

share|cite|improve this question

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 $\nu_{X}$.

share|cite|improve this answer
Thanks, but I do not see how it helps to my question. This $\chi(X,\nu_X)$ is some other generalization, not what I meant – Dmitry Kerner Aug 1 '12 at 12:33
I don't know what you meant by "generalized Euler characteristic", but my example is defined for any $\mathbb{C}$-schemes and satisfies some stratification property (with a bit care). – Atsushi Kanazawa Nov 30 '12 at 0:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.