A new geometric approach to problems in birational geometry
  A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them which induces these isometries? In this work a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence. 
Chen-Yu Chi, Shing-Tung Yau