Generalized Weil’s reciprocity law and multiplicativity theorems

Abstract:

Fix a one-dimensional group variety $G$ with Euler–characteristic $\chi (G)=0$, and a quasi–projective variety $Y$, both defined over $\mathbf {C}$. For any $f\in Hom(Y,G)$ and constructible sheaf $\mathcal {F}$ on $Y$, we construct an invariant $c_{\mathcal {F}}(f)\in G$, which provides substantial information about the topology of the fiber–structure of $f$ and the structure of $\mathcal {F}$ along the fibers of $f$. Moreover, $c_{\mathcal {F}}:Hom(Y,G)\to G$ is a group homomorphism.