in the associated group cohomology. It is typical to write down the group as the lattice defining , hence contains the isomorphism classes of line bundles on . First chern class of line bundles on complex tori From the exponential exact sequence the connecting morphism is the first Chern class map, sending an isomorphism class of a line bundle to its associated first Chern class. It turns out there is an isomorphism between and the module of alternating forms on the lattice , . Therefore, can be considered as an alternating -valued 2-form on . If has factor of automorphy then the alternating form can be expressed as for and . Example For a normalized period matrix expanded using the standard basis of we have the column vectors defining the lattice . Then, any alternating form on is of the form where a number of compatibility conditions must be satisfied. Sections of line bundles and theta functions For a line bundle given by a factor of automorphy , so and , there is an associated sheaf of sections where with open. Then, evaluated on global sections, this is the set of holomorphic functions such that which are exactly the theta functions on the plane. Conversely, this process can be done backwards where the automorphic factor in the theta function is in fact the factor of automorphy defining a line bundle on a complex torus. Hermitian forms and the Appell-Humbert theorem For the alternating -valued 2-form associated to the line bundle , it can be extended to be -valued. Then, it turns out any -valued alternating form satisfying the following conditions for any is the extension of some first Chern class of a line bundle . Moreover, there is an associated Hermitian form satisfying for any . Neron-Severi group For a complex torus we can define the Neron-Serveri group as the group of Hermitian forms on with Equivalently, it is the image of the homomorphism from the first Chern class. We can also identify it with the group of alternating real-valued alternating forms on such that . Example of a Hermitian form on an elliptic curve For an elliptic curve given by the lattice where we can find the integral form by looking at a generic alternating matrix and finding the correct compatibility conditions for it to behave as expected. If we use the standard basis of as a real vector