We will discuss the proof of Kodaira's embedding theorem which gives us an answer whether a complex manifold can be embedded into a projective space. The statement of the theorem is as follows. A complex manifold $X$ is projective if and only if $X$ admits an integral Kähler class. The key algebraic input of this proof is using Kodaira's vanishing theorem and the blowup to mimic the proof of projectivity when $X$ is a curve. Time abiding, I will give one interesting application of this theorem which says that the torus $\Pic^{0}(X)$ of a projective manifold $X$ is again projective.

I will try to avoid talking too much about analysis in detail, but will show some hints of analytic techniques. Speaker(s): Hyunsuk Kim (UM)