The main goal of this course will be to explain Hodge and Hodge-Lefschetz theory for smooth, projective varieties. These theories give extra structure on the cohomology groups H*(X,Z) for smooth, projective complex varieties X. If X is a (possibly singular) complex variety then there is a generalization of Hodge theory due to P. Deligne called mixed Hodge theory . If time permits I will explain this.
There is a substantial amount of background material before we can get to Hodge theory, starting with what a complex manifold is and how you define its cohomology. For this, I plan to follow Griffiths & Harris. We should be able to cover Chapters 0 and 1 thoroughly, and some of Chapters 2 and 3.
Over the course of the semester, I will post homework assignment on the web. I expect everyone to at least try to do the problems. I encourage collaboration, but I also expect you to write up solutions in your own words.