# Euler Characteristic of Sheaves and the Generalized Gauss-Bonnet Theorem #
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 #
>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?)

# Thoughts #
**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/~valya/gauss.pdf). I am interested in a generalization of this to sheaves over sites.