Pluricanonical systems for 3-folds and 4-folds of general type

A celebrated result by Hacon-McKernan and Takayama asserts that there exist integers n(d) and m(d) such that for every complex projective variety X of general type and dimension d, the n-th plurigenus P(n) is greater than 0 if n>n(d) and the m-th pluricanonical map f(m) is birational if m>m(d). Effective values for n(d), m(d) are not known, except for d=1,2,3.
It is reasonable to expect that supposing the volume of X sufficiently large we can obtain better results: in this talk we will give explicit bounds for d=3, improving a result by Todorov. Moreover, always in the case of large volume, we will characterize 3-folds for which f(4) is not birational, thus showing that the bounds are in some sense optimal. We will also give explicit bounds for 4-folds of general type and we will show some generalizations to higher dimensional varieties.