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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.

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    $\begingroup$ Donaldson's work uses such algebraic geometry in an important way: roughly (and inaccurately) speaking, his formulation of K-stability is as an ``asymptotic'' GIT condition on Hilbert schemes (in the sense of Mumford). His point of view on the Yau-Tian-Donaldson conjecture is to view the Kahler-Einstein (more generally constant scalar curvature) condition as a moment map (also due to Fujiki); then an infinite dimensional version of the classical Kempf-Ness Theorem predicts that such metrics exist provided a algebro-geometric stability condition holds. $\endgroup$
    – Ruadhaí Dervan
    Apr 26, 2016 at 22:01
  • $\begingroup$ Of course, the work of many others (including especially Sun and Tian who you mention) is closely related to these ideas. To emphasise, I mean moduli theory and scheme theory play important roles in K-stability. $\endgroup$
    – Ruadhaí Dervan
    Apr 26, 2016 at 22:11
  • $\begingroup$ This is known as Fujiki-Donaldson picture :) $\endgroup$
    – user21574
    Apr 29, 2016 at 0:50
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    $\begingroup$ Many (most? all?) of the problems considered in compact complex differential geometry come from algebraic geometry, so you can't lose knowing about that stuff. That said, the machines of differential geometry run best on manifolds (or at least on things that are "mostly" manifolds), so complex geometers tend not to touch schemes, stacks, motives or derived categories so much. (There are those who do; google Kahler complex spaces.) $\endgroup$
    – Gunnar Þór Magnússon
    May 3, 2016 at 1:54
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    $\begingroup$ As an example of the kinds of complications that come up when you run up against the limitations of complex differential geometry, I really like Demailly's description of the proof of hard Lefschetz with multiplier ideal sheaves in Chapter 16 of [1] below. There the semi-positive (i.e., natural diffeo-geometric condition) case is almost trivial, but the naturally algebro-geometric condition demands pages upon pages of hard work. [1]: www-fourier.ujf-grenoble.fr/~demailly/manuscripts/… $\endgroup$
    – Gunnar Þór Magnússon
    May 3, 2016 at 1:58

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