Euler Characteristic of Sheaves and the Generalized Gauss-Bonnet Theorem
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Consider a sheaf $\mathscr{F}$ over a site $\mathscr{(C,J)}$, where $\mathscr{C}$ is a small category with a coverage $\mathscr{J}$. The Euler characteristic is defined as $\chi(\mathscr{F})=\sum\_{i\in\mathbf{Z}\_0^+}(-1)^i\dim\_\mathbb F(H^i(\mathscr{C},\mathscr{F}))$.


The Euler characteristic satisfies, on a compact orientable $2n$-dimensional Riemannian manifold without boundary, the generalized Gauss-Bonnet theorem:
$$\int\_M \mathrm{Pf}(\Omega)=(2\pi)^n\chi(M), \, $$
Here $\Omega$ is the curvature form of the Levi-Civita connection and the Pfaffian of $\Omega$ is $\mathrm{Pf}(\Omega)$.


The Question
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> 
> Is there an analog of the Gauss-Bonnet theorem in algebraic geometry? Specifically, is it possible to define the "curvature form $\mathscr{O}$ of a (Levi-Civita) connection on the quasi-coherent sheaf $\mathscr{F}$ over the site $\mathscr{(C,J)}$ ([this book, perhaps?](http://books.google.com/books?id=W3QuH7yGB0cC&pg=PA297&lpg=PA297&dq=curvature+of+sheaf&source=bl&ots=7UwYSPyYAA&sig=q5DMa7xygU4rH680k4ccWEMprIc&hl=en&sa=X&ei=ndmHU8-nBo6ZyASR-4DQDQ&ved=0CF8Q6AEwBQ#v=onepage&q=curvature%20of%20sheaf&f=false))" and the "sheaf Pfaffian $\mathscr{Pf}$" to obtain something like the Gauss-Bonnet, e.g. $\int\_\mathscr{F} \mathscr{Pf}(\mathscr{O})=(2\pi)^{\dim\mathscr{F}}\dim\_\mathbb FH^0(\mathscr{C},\mathscr{F})$, with $\dim\mathscr{F}$ the [(pure)](http://books.google.com/books?id=J5P5YzTfCXEC&pg=PA147&lpg=PA147&dq=dimension+of+a+sheaf&source=bl&ots=vXViaC2H62&sig=7FO2DE0i2BQydAJ9HfmOV6Oihw0&hl=en&sa=X&ei=HteHU-u8BZezyASQjoKYDQ&ved=0CJgBEOgBMA8#v=onepage&q=dimension%20of%20a%20sheaf&f=false) [dimension](https://mathoverflow.net/questions/88182/what-is-the-dimension-of-a-sheaf) of $\mathscr{F}$? (What is the "integral" supposed to be?)
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Thoughts
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**Note 1:** As for what the integration might be, [this answer](https://mathoverflow.net/questions/167577/expressing-the-lebesgue-integral-using-categories-the-difficulty-of-describing/167680#167680), which links to [this paper](http://arxiv.org/pdf/1311.3188v1.pdf) (I believe Theorem 3.2 of that paper) provides a possible answer. However, I'm not sure if that's the "integration" wanted in the conjectured Gauss-Bonnet for sheaves.


**Note 2:** For constructible sheaves on reductive groups, I found the following paper by [V. Kiritchenko](http://www.mccme.ru/%7Evalya/gauss.pdf). I am interested in a generalization of this to sheaves over sites.



The Gauss-Bonnet theorem for Sheaves
There is an answer in the case of constructible sheaves. The whole story can be found in Kashiwara and Shapira's book "Sheaves on Manifolds" Chapters VIII and IX.


I will try to summarize this rather long story. First, a prototype for a constructible sheaf is a locally constant sheaf supported on a *reasonable set*, e.g., semialgebraic, or subanalytic subset of some Euclidean space $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$. For example the constant sheaf with stalk $\bR$ defined on a compact reasonable set is constructible. The Euler characteristic of the cohomology of this sheaf is the Euler characteristic of that set.


In general, a constructible sheaf is really a complex of sheaves whose associated cohomology sheaves are locally constant.


We can form a $K$-theoretic group $K(\bR^n)$out of these constructible sheaves very much the way Grothendieck for coherent sheaves on algebraic manifolds. If $A, B,C$ are constructible complexes of sheaves, then we have a $K$-theoretic equality $B=A+C$ if there exists a short exact sequence of complexes $0\to A\to B\to C\to 0$ (It is a bit more complicated than this but I want to avoid talking about triangulated categories.) The Euler characteristic of the (hyper)cohomology of a complex $A$ induces a group morphism


$$\chi: K(\bR^n)\to \mathbb{Z}.$$ 


Kashiwara and Schapira have shown that the Abelian group $K(\bR^n)$ is isomorphic to two groups that have simpler descriptions.


The first group is the group $\newcommand{\eC}{\mathscr{C}}$ $\eC(\bR^n)$ of constructible functions. These are functions $f:\bR^n\to \mathbb{Z}$ with *finite* range such that, for any $t\in\bR$ $f^{-1}(t)$ is a subanalytic set. We obtain an Euler characteristic morphism


$$\chi:\eC(\bR^n)\to\mathbb{Z}, $$


defined by


$$\chi (f) =\sum\_{n\in\mathbb{Z}} n \chi\bigl(\;f^{-1}(n)\;\bigr). $$


The second group is the group $\newcommand{\eL}{\mathscr{L}}$ $\eL(\bR^n)$ of conical lagrangian cycles in $T^\*\bR^n$. These are cycles supported by lagrangian varieties in $T^\*\bR^n$ which are invariant under the rescaling along the fibers of the cotangent bundle.


Joseph Fu has given a very nice geometric description of an isomorphism $\eC(\bR^n)\to \eL(\bR^n)$. This isomorphism associates to each constructible $f$ function a lagrangian cycle called the conormal cycle $C^f$ of the constructible function $f$. For example, if $f$ is the indicator function of a compact, subanalytic submanifold $S\subset \bR^n$, then $C^f$ is the Lagrangian cycle defined by the conormal bundle of $S$ which is a subbundle of $T^\*\bR^n|\_S$.


The *normal cycle* $N^f$ of a constructible function $f$ is obtained by intersecting the conormal cycle $C^f$ with the unit sphere bundle of $T^\*\bR^n$. Thus, if $f$ is the indicator function of a compact, subanalytic submanifold $S\subset \bR^n$, then the normal cycle $N^f=N^S$ can be identified with the unit normal sphere bundle of the submanifold $S$. If $f$ is the characteristic function of a compact domain $D$ with smooth boundary $\newcommand{\pa}{\partial}$ $\pa D$ then the normal cycle is the graph of the Gauss map


$$ n:\pa D \to S^{n-1}\subset \bR^n,\;\;p\mapsto n(p), $$


where $n(p)$ denotes the outer unit normal vector to $\pa D$ at $p$.$\newcommand{\bZ}{\mathbb{Z}}$


The Euler characteristic morphism $\chi: \eL(\bR^n)\to\bZ$ was given an explicit description by Fu. More precisely there exists a universal form of degree $n-1$, $\omega\in \Omega^{n-1}(S^{n-1}\times \bR^n)$ such that, for any constructible function $f$, its Euler characteristic $\chi(f)$ is equal to the integral of $\omega$ over the normal cycle $N^f$ which is naturally a cycle in $S^{n-1}\times \bR^n$. When $f$ is the indicator function of a submanifold, the above result is none other that the usual Gauss-Bonnet. When $f$ is the indicator function of a compact affine simplicial complex in $\bR^n$, Fu's result leads to the computation of the Euler characteristic of the complex as an integral of a so called *curvature measure*. You can learn more about this at [Fu's homepage.](http://www.math.uga.edu/~fu/research/research.html)


