DescriptionThis thesis presents some of my works on projective geometry. The first part of this thesis provides some background based on the joint work with Li, Sturm, and Wang. The second part of this thesis will give a lower bound of the paired alpha invariant for polarized manifolds (X, L) and a fixed divisor D, which generalizes the results in Berman. The third part provides a simple proof of the equivalence relation between two different criteria on the existence of constant scalar curvature Kahler (cscK) metrics on toric manifolds.