(This is just rewriting my recent edit as it’s own post). Basically I’ve combined some proofs from the past year and a half into one write-up. (algebraic) KLT Deformation Invariance of Plurigenera for Kodaira dimension >= dim(X) – 3: First I remark that klt abundance should already hold in that case using some previous results (Hacon / Lai ?), then I use either adjoint ideals or Hodge Theory to derive the desired theorem on plurigenera which so far is only known to hold using the techniques of complex analysis. Needs some editing / proofreading (April 6, 2015)
edit: The above link somehow went down for a bit, but seems to be working again!
(July 9 ) Below is my original attempt for all non-negative kodaira dimensions (from last summer, although recently cleaned up) which my advisor doesn’t like but has so far failed to convince me is incorrect: