To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) complex geometry. If I understand correctly, complex geometry means many things to many different people. To some, it is an extension of algebraic geometry, while many others would immediately think of the Newlander-Nirenberg or Yau's proof of Calabi conjecture.
I am looking for some kind of research ``introduction'' to the analytic side of complex geometry. I want to develop a better appreciation of the pde theoretic tools that are relevant in complex geometry, the problems they can solve, and also some of the open problems in the field that are thought to be potentially amenable to analytic methods of attack. In other words, suppose an analyst wants to do research on complex geometry. What does he start by reading? Papers sometime down the line for sure, but initially they might be too specific/concentrated. I guess, if there were a textbook on complex geometry written by Yau, or Hormander, or Tao, amongst other people, that would be a starting point for me.
If my question is too broad/unfit for this site, I apologize. I realize that it is impossible that any single book/monograph/lecture note will cover all the analytic sides of complex geometry. But even a partial answer will be appreciated.
Lastly, just for example: if someone asked me what would be a good answer if someone asked the same question about real differential geometry? I would say I don't have a good answer, as the literature is just too huge. However, I would add that some of my favourites are Schoen/Yau's Lectures on Differential Geometry, Jost's Geometric Analysis, and perhaps Aubin's Nonlinear problems in Riemannian geometry. These books should certainly get one started.