An algebraic proof of Iitaka‚s conjecture C2, 1

Abstract. We give a proof of Iitaka‚s conjecture C2, 1 using only elementary methods from algebraic geometry. The main point we show is that, given a non-isotrivial and relatively minimal model of a family $ f : X \rightarrow B $, where X is a surface and B is a curve, both smooth and projective, the direct image of the relatively canonical sheaf of differentials has strictly positive degree.