Some Results on the Degeneracy of Entire Curves and Integral Points in the Complements of Divisors



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In this dissertation, we first discuss some of the important results in Nevanlinna Theory and Diophantine Approximation Theory. Next, a result by the author and Min Ru \cite{LR} is presented. In chapter 3, we extend the Second Main Theorem to the case of holomorphic curves into algebraic varieties intersecting numerically equivalent ample divisors. In chapter 4, we improve Ru's defect relation (see \cite{ru15}) and the height inequality (see \cite{ru}) in the case when X is a normal projective surface and Dj, 1≤jq, are big and asymptotically free divisors without irreducible common components on X. Lastly, the author and Gordon Heier approach the hyperbolic problem by projections from Pn+2 to Pn.



Navenlinna Theory, Diophantine approximation, Holomorphic curves