Half twists of Hodge structures of CM-type

Abstract

To a Hodge structure $V$ of weight $k$ with CM by a field $K$ we associate Hodge structures $V_{-n/2}$ of weight $k+n$ for $n$ positive and, under certain circumstances, also for $n$ negative. We show that these 'half twists' come up naturally in the KugaSatake varieties of weight two Hodge structures with CM by an imaginary quadratic field.