#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?)" 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) dimension of $\mathscr{F}$? (What is the "integral" supposed to be?)

#Thoughts# Note 1: As for what the integration might be, this answer, which links to this paper (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. I am interested in a generalization of this to sheaves over sites.

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?)" 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) dimension of $\mathscr{F}$? (What is the "integral" supposed to be?)

Thoughts

Note 1: As for what the integration might be, this answer, which links to this paper (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. I am interested in a generalization of this to sheaves over sites.

#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?)" 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) dimension of $\mathscr{F}$? (What is the "integral" supposed to be?)

#Thoughts# Note 1: As for what the integration might be, this answer, which links to this paper (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. I am interested in a generalization of this to sheaves over sites.

#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?)" 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) dimension of $\mathscr{F}$? (What is the "integral" supposed to be?)

#Thoughts# Note 1: As for what the integration might be, this answer, which links to this paper (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. I am interested in a generalization of this to sheaves over sites.

#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 $\mathscr{(C,J)}$" 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) dimension of $\mathscr{F}$? (What is the "integral" supposed to be?)

#Thoughts# Note 1: As for what the integration might be, this answer, which links to this paper (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. I am interested in a generalization of this to sheaves over sites.

#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?)" 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) dimension of $\mathscr{F}$? (What is the "integral" supposed to be?)

#Thoughts# Note 1: As for what the integration might be, this answer, which links to this paper (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. I am interested in a generalization of this to sheaves over sites.