Serre's GAGA

Our goal is to introduce Serre’s GAGA theorem and some of its applications. Serre’s GAGA establishes a bridge between algebraic geometry and analytic geometry. For a proper algebraic variety, the category of algebraic coherent sheaves is equivalent to the category of analytic coherent sheaves on its analytification. Moreover, the cohomology groups of a coherent sheaf agree with those of its analytification. I will also review the necessary preliminaries, including the definitions of algebraic varieties and coherent sheaves, as well as the cohomology theory of coherent sheaves.