Effective boundedness results in algebraic and analytic geometry

Abstract

The goal of this thesis is to study several aspects arising from the notion of boundedness of algebraic varieties.

In the first part of the thesis we will focus on the problem of stability of pluricanonical maps. It basically says that the maps associated to the canonical divisor of a variety of positive Kodaira dimension stabilize uniformly. We give a positive answer to this problem in the special case the Iitaka fibration has maximal variation and its image is not uniruled. Moreover, we provide effective bounds.

It turns out that in order to extend these results when the image of the Iitaka fibration is uniruled, one needs to study the pseudo-effective threshold. The problem reduces to proving that in the set of pseudo-effective thresholds, 1 is not an accumulation point from below. Motivated by this problem, we study in details the accumulation points of this set. In particular, we prove Fujita's log spectrum conjecture. Roughly speaking, it says that in the set of pseudo-effective thresholds the accumulation points can be reached only from above.

In the second part of this work we work out similar properties of the nef threshold. For example, we characterize the pairs (X,D) such that K_{X}+D is a limit of ample divisors of the form K_{X}+&alpha D. This result turns out to have applications in the study of Kähler-Einstein metrics with with cone-edge singularities, described by the cone-angle 2&pi(1-&alpha) for &alpha in (0, 1). We show that in the negative scalar curvature case, if such Kähler-Einstein metrics exist for all small cone-angles then they exist for every &alpha in ((n+1)/(n+2), 1), where n is the dimension. We also give a characterization of the pairs that admit negatively curved cone-edge Kähler-Einstein metrics with cone-angle close to 2&pi. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed. We characterize the obstruction using boundedness of Fano varieties.

These methods developed for Kähler-Einstein metrics can be used to investigate the geometry of cusped complex hyperbolic manifolds. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We give effective bounds on the number of complex hyperbolic manifolds with given upper bounds on the volume. Moreover, we estimate the number of ends of such manifolds

in terms of their volume.