HOLOMORPHIC PRINCIPAL BUNDLES WITH AN ELLIPTIC CURVE AS THE STRUCTURE GROUP

Let Lambda subset of C be the Z-module generated by 1 and root-1 . tau, where tau is a positive real number. Let Z := C/Lambda be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X. Let theta(1) and theta(2) be two C-infinity real closed two-forms on X such that the Hodge type (0, 2) component of the form theta(1) + tau root-1 . theta(2) vanishes, and the elements in H-2 (X, C) represented by theta(1) and theta(2) are contained in the image of H-2 (X, Z). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle E-Z over X, we construct a pair of closed differential forms on X of the above type.