Factor of automorphy

Symbol-free definition
Let $$G$$ be a group acting on a complex manifold $$X$$. Then, $$G$$ also acts on the vector spaces of holomorphic functions from $$X$$ to $$\Z,\C,\C^*$$, as vector space actions.

Each of these gives a linear representation of $$G$$ over $$\C$$.

A factor of automorphy is defined as a 1-cocycle for the action of $$G$$ on the space of holomorphic functions to $$\C^*$$. Two factors of automorphy are termed equivalent if they they give the same element in the cohomology group, or in other words, if they differ multiplicatively by a coboundary.

Thus, the equivalence classes of factors ofautomorphy can be identified with the first cohomology group for the action on the space of holomorphic function to $$\C^*$$.

Automorphic functions
Associated with any factor of automorphy, we can construct a space of automorphic functions. An automorphic function $$f$$ for a factor of automorphy is a function from $$X$$ to $$\C$$ satisfying:

$$f(g^{-1}.x) = \gamma_g(x)f(x)$$

Note that if $$f$$ were everywhere a nonzero function, then that would make $$\gamma$$ into a coboundary. Thus, in some sense, it is the presence of zeroes of $$f$$ that prevents $$\gamma$$ from being a coboundary.

The automorphic functions for a given factor of automorphy form a vector space. Further, the product of automorphic functions corresponding to two factors of automorphy is an automorphic function corresponding to their product. Thus, we can associate to the entire cohomology group a ring comprising those functions that are finite linear combinations of automorphic functions corresponding to various factors of automorphic functions.

Chern class
Given any factor of automorphy, we can define an associated Chern class. The Chern class is the same for equivalent factors of automorphy.