Positivity of divisors, generalized abundance and canonical bundle formula

Abstract

Much of the work in this dissertation is centered around the generalized abundance conjecture of Lazić and Peternell \cite{LP}. It is a generalization of the clasical abundance conjecture- arguably one of the most important problems in the whole of algebraic geometry. I now briefly summarize the contents of this thesis.\\

In the first chapter, we collect some classical facts about linear systems on complex projective varieties. Then we recall some standard facts from birational geometry- in particular singularities of pairs, the cone, contraction and basepoint free theorems for klt pairs and the main conjectures of the minimal model program. We then turn our attention to some of the tools we use: canonical bundle formulas and the Nakayama-Zariski decomposition \cite{Nak}. In the final section, we give a brief summary of the results of this thesis.\\

Chapter 2 is mostly about homogeneous vector bundles on flag varieties. It covers the contents of \cite{PC1}. I prove a global generation criteria for such bundles, namely that any such bundle which is nef is also globally generated. We then make a few results about the analogous question for bundles on abelian varieties.\\

Chapter 3 is about conjectures of Serrano and Campana, Chen and Peternell about (almost) strictly nef divisors. We also prove generalized abundance for a large class of uniruled varieties. This essentially covers the contents of \cite{PC}. Following methods of \cite{BCHM}, in the final section, we show that generalized abundance unconditionally holds for klt pairs with big boundary (see Theorem \ref{GA big}). This result is new to the thesis.\\

Chapter 4 is about an attempt to generalize a classical theorem of Kawamata \cite{Ka} to the setting of generalized klt pairs. In particular, we observe that Kawamata's theorem can be extended essentially to the setting of generalized abundance. The contents are essentially the same as the authors article \cite{PC2}. Corollary \ref{b-good} is new here and generalizes a previous result of \cite{PC2}.\\

In chapter 5, we give an inductive approach to generalized abundance using the nef dimension. This follows \cite{PC3}. In particular, we observe some new situations in which generalized abundance holds in arbitrary dimension. Theorem \ref{psef} proves a variation of the original inductive statement of \cite{PC3} and holds in the exact setting of generalized abundance.