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Let $Y$ be a "nice" scheme. I am thinking projective varieties over an algebraically closed field, for now, but I am open to more general results.

In terms of singular homology (coefficients in $\mathbb{Z}$), one can define the Euler characteristic $\chi(Y)$.

My question is:

Can I express $\chi(Y)$ in terms of the Euler characteristic of certain coherent sheaves on $Y$, in terms of sheaf cohomology?

Most preferably, I would like $$\chi(Y)=\chi(Y,\mathcal{F})$$ for some particular sheaf $\mathcal{F}$.

I am sorry if this is really trivial or widely known, my searching and asking (in the real world) has led me nowhere so far.

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Jesko -- if you are willing to accept complexes of sheaves rather than just sheaves, then in characteristic 0 one may take the sum over all smooth strata of the Euler characteristics of the complexes of differential forms. – algori Jul 14 '11 at 8:42
Well, in a smooth flat family the fibres are diffeomorphic by Ehresmann theorem, so the topological Euler number is constant. But if the family is not smooth, it can definitely vary (think of a smooth plane cubic degenerating to a nodal one). – Francesco Polizzi Jul 14 '11 at 9:22
Over $\mathbb{C}$ one can use hodge theory to write $H^k(X,\mathbb{C})=\oplus_{p+q=k}H^q(X,\Omega_X^p)$, where $\Omega_X^p$ is the sheaf of $p$-forms on $X$. – Daniel Loughran Jul 14 '11 at 12:25
Also the top Chern class of the tangent sheaf gives the topological Euler characteristic (of a smooth, projective, complex variety). – Jason Starr Jul 14 '11 at 15:14
@Jesko: smooth strata means this: write $X = X_0 \cup X_1 \cup \cdots \cup X_n$ where $X_0$ is the non-singular locus of $X$, $X_1$ is the non-singular locus of $X \setminus X_0$, etc. – Steven Sam Jul 14 '11 at 18:03
up vote 4 down vote accepted

First, the answer/reference here might be exactly what you are looking for.

On the other hand, perhaps you just want to relate natural algebro-geometric structure to the classical Euler characteristic. Here is one way to do that:

A pure Hodge structure of weight $k$ is a finite dimensional complex vector space $V$ such that $V=\bigoplus_{k=p+q} H^{p,q}$ where $H^{q,p}=\overline{H^{p,q}}$. This gives rise to a descending filtration $F^{p}=\bigoplus_{s\ge p}H^{s,k-s}$. Define $\mathrm{Gr}^{p}_{F}(V)=F^{p}/ F^{p+1}=H^{p,k-p}$.

A mixed Hodge structure is a finite dimensional complex vector space $V$ with a real ascending weight filtration $\cdots \subset W_{k-1}\subset W_k \subset \cdots \subset V$ and a descending Hodge filtration $F$ such that $F$ induces a pure Hodge structure of weight $k$ on each $\mathrm{Gr}^{W}_{k}(V)=W_{k}/W_{k-1}$. Then define $H^{p,q}= \mathrm{Gr}^{p}_{F}\mathrm{Gr}^{W}_{p+q}(V)$ and $h^{p,q}(V) =\dim H^{p,q}$.

Let $Z$ be any quasi-projective algebraic variety. The cohomology groups with compact support $H^k_c(Z)$ are endowed with mixed Hodge structures by seminal work of Pierre Deligne.

The Hodge numbers of $Z$ are $h^{k,p,q}_{c}(Z)= h^{p,q}(H_{c}^k(Z))$, and the $E$-polynomial is defined as $$ E(Z; u,v)=\sum _{p,q,k} (-1)^{k}h^{k,p,q}_{c}(Z) u^{p}v^{q}. $$

From this, one gets the classical Euler characteristic $\chi(Z)=E(Z;1,1)$.

Note: if the counting function of $Z$ over finite fields is a polynomial in the order of the finite field, then $E(Z)$ is exactly the counting polynomial. From this point-of-view, in this case, the Euler characteristic is the number of $\mathbb{F}_1$-points.

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Indeed, the latter is what I want(ed)! Thanks for answering this old question. – Jesko Hüttenhain Apr 30 at 7:49

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