The Noether inequality for smooth minimal 3-folds
  Let X be a smooth projective minimal 3-fold of general type. We prove the sharp inequality K^3_X >= (2 /3)(2p_g(X) - 5), an analogue of the classical Noether inequality for algebraic surfaces of general type 
Fabrizio Catanese (Universitaet Bayreuth), Meng Chen (Fudan   University), De-Qi Zhang (National University of Singapore)