This book is a compilation of the author''s thesis work. The book studies a class of problems that can be roughly tied together under the title "positivity problems." The logically splits into four parts: fixed points of endomorphisms of abelian varieties, finite generation of the rational points of endomorphisms of degree greater than one on all curves, universal constants of pluricaonical maps in dimension 3, and a Nakai type criterion for the big cone of a smooth projective variety.

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