On the problem of deformation invariance of plurigenera
In this dissertation, we study the problem of the deformation invariance of plurigenera of algebraic varieties using techniques from the Mori and Iitaka Programs. In characteristic zero, we study the problem for smooth families of non-negative Kodaira dimension. In this case, we reformulate a famous theorem of Siu in terms of a condition on the central fiber of certain special models of the relative Iitaka fibration of X over T. Under some additional assumptions, this gives algebraic proofs of deformation invariance of plurigenera, generalizing results of Nakayama and Kawamata. In positive and mixed characteristic, we construct examples of families of smooth surfaces where all sufficiently divisible plurigenera fail to be constant, answering a quesion of Katsura and Ueno. In particular, invariance of plurigenera does not follow from the MMP and Abundance Conjecture. Lastly, we show that invariance of all sufficiently divisible plurigenera holds for families of quasi-elliptic surfaces and certain families of log Calabi-Yau fibrations of small relative dimension.