I am currently a first year PhD candidate and I need some help figuring out the top current research topics in Complex Algebraic and Differential Geometry (can be in both, or just one of the two).
I plan to ask my advisor next week, but I want to see if I can compile a list first and look up some topics on my own. Then next week I'll present some ideas first and see what he thinks.
I have research experience in algebraic coding theory (using L(D) and divisors), and Kahler metrics and sectional, ricci, and scalar curvature. I also have a good background in algebraic geometry (learning from Griffiths & Harris), Riemannian Geometry (from Gallot, and Lee), and Complex Geometry (Huybrechts and Demailley).
But the big question is what is the best and most popular topic in complex geometry to study and research in?
Thanks in advance.