On numerically trivial automorphisms of threefolds of general type

In this paper, we prove that the group Aut(Q)(X) of numerically trivial automorphisms are uniformly bounded for smooth projective three folds X of general type which either satisfy q(X) >= 3 or have a Gorenstein minimal model. If X is furthermore of maximal Albanese dimension, then |Aut(Q)(X)| <= 4, and equality can be achieved by an unbounded family of three folds previously constructed by the third author. Along the way we prove a Noether type inequality for log canonical pairs of general type with the coefficients of the boundary divisor from a given subset C subset of (0,1] such that C boolean OR {1} attains the minimum.