TITLE: the central issues in complex geometry
QUESTION [5 upvotes]: I want to know about people in researching complex (maybe differential) geometry are careing about what currently ? For example ,$L^2$ estimate inspired by Lars Hormander is a very useful tool,and how does this theory be developed currently ? As myself , i like this method very much ,but i don't know which is the next important problem be solved by this method . How far will this method go ?  As well , just like the holomorphic morse inequalities , when it is proved by Demailly in 1985,twenty years passed , it seems that during thest twenty years there are no important results comes out in complex geometry ? I'm a beginner in complex geometry, i think that i can't scratch the direction of complex geometry ? I don't know people are careing about what in complex geometry ?Only doing some little questions or leave this field? So this is just the purpose of this question i asked , i want to communicate with all who are interested in this field.

REPLY [4 votes]: There also problems along the lines of proving L2 extension theorems for vector valued forms (there are such theorems already like the Ohsawa-Takegoshi extension theorem however, that applies to (n,1) forms. We want more general results). Besides, there is the problem of deformation invariance of plurigenera over Kahler manifolds (Siu proved it for the projective case).