Iitaka Algebraic Geometry Pdf Download 2021

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Our primary tool for this will be the study of Iitaka fibration. Consider the class of Iitaka fibered spaces (or more specifically, log canonical pairs) which are special examples of algebraic fiber spaces. A generalization of the Iitaka fibrations, called also rational Iitaka fibrations, will be studied in order to provide a framework to deal with the Minimal Model Program. Iitaka fibrations give the best resource towards the classification of varieties with respect to the Minimal Model Program, and the interplay between Iitaka fibrations and birational geometry is an important aspect to address.

We begin with the study of the general theory of Iitaka fibered spaces, and then focus on elliptic threefolds. Finally, we will consider the study of the birational geometry of elliptic threefolds, more specifically, the minimal models of elliptic threefolds and the birational geometry of the minimal model of an elliptic threefold with section. Since elliptic threefolds are typically fibered by Iitaka fibrations (and even more interesting, elliptic fibrations), we will apply the general theory of Iitaka fibrations to elliptic threefolds, and also, as it will become more clear, to elliptic fourfolds with section.

In the first part, we will examine the class of Iitaka fibered spaces which are log canonical. We will prove that if a variety has an Iitaka fibration, then it is Iitaka fibered with respect to the canonical log pair, and thus, we will recover a version of the main theorem of Iitaka. We will also characterize the Iitaka fibered spaces with respect to the canonical log pair and characterize the structures that they have. We will also discuss the case of elliptic threefolds, showing that it is indeed Iitaka fibered with respect to their canonical log pair and that the the class of Iitaka fibered spaces is closed under birational morphism. We will also prove that if a variety has a log canonical quotient, then it is Iitaka fibered, and thus, we will recover the main theorem of Iitaka. In this part, we will also discuss the case of elliptic fourfolds with section, and show that it is indeed Iitaka fibered with respect to their canonical log pair.

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