Abelian differentials on singular varieties and variations on a theorem of Lie-Griffiths

We compare various notions of meromorphic and holomorphic differential forms on (singular) analytic varieties. In particular we find that every meromorphic form gives rise to a canonical principal value current with support in the variety. Demanding that this current be <(partial derivative)over bar>-closed we obtain a useful analytic description of the dualizing sheaf. We then go on to generalize the Lie-Griffiths theorem: On one hand it is shown that a rational trace is enough to ensure algebraicity of the data. On the other hand we prove that zero trace implies that the form is holomorphic in the sense of currents. In fact, we prove a more general, local version of the theorem.