I will explain how a refinement of the Green-Lazarsfeld Generic 
Vanishing Theorem, general Fourier-Mukai transform theory, and the Syzygy 
Theorem of Evans-Griffith, can be used to prove a higher dimensional 
analogue of the classical Castelnuovo-de Franchis inequality for surfaces. 
I will then use this inequality to provide some bounds on the irregularity of special classes of projective varieties. Joint work with Giuseppe 
Pareschi.