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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.

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You may consider asking this question in mathoverflow. But even so, the question is too board. –  John Mar 14 at 2:39
What about the existence of Kahler-Einstein metrics on for positive Chern class? –  Alex Youcis Mar 14 at 6:18
@AlexYoucis: Will that be a good candidate? The big conjecture in the field has just been proven. –  John Mar 14 at 6:29
@John I have no idea what would be a good candidate--that was just one example I knew of :) –  Alex Youcis Mar 14 at 7:31
I really think that this type of question is best discussed with an expert in the areas that interest you. In particular, I don't think you want lots of opinions from people on the internet. –  Sam Lisi Mar 15 at 22:12

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