Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? Complex algebraic and analytic geometry with connections to differential geometry and analysis. The topics which are researched by Jean-Pierre Demailly, Simon Donaldson, Gang Tian, Song Sun and Christian Schnell (that is, Kahler geometry, Kahler-Einstein metrics, K-stability, mixed Hodge modules etc. ).
There is an intersection of an impressive number of mathematical fields there, so it looks fascinating even though I'm not a differential/complex geometer myself.
But how many ideas and constructions of post-Grothendieck algebraic geometry do these people use in their everyday research? Homological algebra? Derived categories? Different cohomology theories? Schemes? Stacks? Motives? That's what I'm interested in knowing.