Positivity problems in higher dimensional geometry

We study a class of problems, in algebraic geometry, that can be roughly tied together under the title "positivity problems." That is, by exploiting various strong geometric properties of ample line bundles one is able to deduce information about algebraic varieties. We give four such examples:;First, we show that the fixed points, counted without multiplicity, of an endomorphism &phis; : A→ A with A an abelian variety can be effectively bounded based on the degree of &phis;. This gives a partial answer towards a conjecture of Shub and Sullivan along with a conjecture of Zhang.;Second, by using a positivity result about the ample cone of a smooth variety X defined over a number field with an endomorphism of degree greater than one on all curves, we show that the set of rational fixed points is finitely generated.;Third, we show that a conjecture of Hacon and McKernan is true in dimension three. That is, if X is a smooth projective variety of dimension 3 with non-negative Kodaira dimension, then there exists a universal constant r3 for which the pluricanonical maps, constructed by the linear series |rKX|, are birational to the Iitaka fibration whenever r > r 3.;Finally, we develop an analog for the Nakai Criterion for the big cone of a smooth projective variety. This gives a numerical criteria for testing when a divisor is big.