# Complex analytic vs algebraic geometry

This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next.

It looks to me, that complex-analytic geometry has lost its relative positions since 50's, especially if we compare it to scheme theory. Are there internal mathematical reasons for why that happened?

As we all know, lot's of techniques which were later adapted by algebraic geometrers were originally developed in the complex analytic setting (sheaves, local algebra machinery, etc.). In the 50's, Serre, Cartan, Grothendieck and others seemed to have been developing scheme theory somewhat in parallel with complex-analytic spaces (results on coherent sheaves, base change and cohomology theorems, etc.). But already in the 60's it seems like complex analytic geometry started to lag behind - Grothendieck developed Hilbert scheme in 61, and it took already 5 years for Douady to build a complex-analytic analogue. Then in the 80's came Fulton's monograph on intersection theory, and as far as I can tell intersection theory for complex analytic spaces is still far from the polished form of Fulton. Finally, there is a lot of foundational algebraic work on stacks, and it seems that the amount of publications on complex-analytic stacks is far smaller.

And in general, my impression is that the amount of people doing foundational work in complex analytic geometry is minuscule compared to algebraic geometry (in the recent years I've only come across works of Mauro Porta about the derived complex-analytic set-up and analytic stacks), compared to how active it was before 1990's (there was a nice summary of results by Grauert and Remmert here going back to 1996, showing how active the field was).

Edit: 1. As others have pointed out in the comments, I should have been more specific about what I meant by complex-analytic geometry, since areas like Kahler geometry are very active right now. I basically meant that fundamental side of complex analytic geometry, which shared machinery with algebraic geometry in the 50-60's.

1. The point of asking this question at all is (besides curiosity) to learn whether there are some well-understood limitations to the subject.
• There are Quot spaces in the complex analytic setting. This is a consequence of Douady spaces. – Jason Starr Dec 15 '18 at 19:59
• Isn’t Kahler geometry a large and active area, a part of complex geometry but outside of algebraic geometry? – Sam Hopkins Dec 16 '18 at 0:03
• What possible purpose can it serve for you to ask whether "complex-analytic geometry became a much less active field"? If there is a beautiful problem in complex analysis that you would like to solve, I recommend that you learn about that and work on that regardless of questions like this one. – Jason Starr Dec 16 '18 at 1:33
• I would like to be clearer about whether "complex geometry" is meant to include Kaehler geometry (which is intensely active and in white heat around Siu and Demailly and many others) or do we mean the story of Grauert, Remmert, Stein, Cartan, Douady and others of Stein spaces and the construction of a kind of SCV analogue of algebraic geometry for complex spaces? I see this second direction as relatively cool at the moment, and I would agree that I don't know why. – Ben McKay Dec 16 '18 at 15:17
• @BenMcKay, yes I mainly meant this second direction. – Bananeen Dec 16 '18 at 15:19

Though I am not an expert on this I think that the shift toward algebraic geometry is not entirely sociological. Consider the following statement which is true in both the category of schemes and analytic spaces :

The push-foward of a coherent sheaf by a proper map is coherent.

In algebraic geometry, this statement is rather a routine-like statement, once you have the tools crafted by Grothendieck. In the complex-analytic setting, this is a hard theorem (due to Grauert and Remmert) and no simple proof of it is known.

Another result one could be interested is Mori's bend-and-break lemma. It is probably one of the most important tool in modern birationnal geometry (and was celebrated as one of the most important results in algebraic geometry in the late 70's early 80's). The original proof goes through caracteristic $$p$$ techniques (namely the use of the Froebenius morphism to amplify a given vector bundle).

Siu and others have claimed to have a purely analytical proof of the bend-and-break lemma. But as far as I can tell (I am not an expert, but I know a bit of birational geometry) their analytical proofs are very hard to follow.

In my opinion, many people in the 70's and 80's left analytical geometry to embrace algebraic geometry because so many powerful and beautiful results had simple and crystalline (in the non-Grothendickian sense) proofs in the algebraic category while their analytic counterparts looked, at the time, either out of reach or extremely hard to prove.

A very down-to-earth baby example at the undergraduate level is the following:

A regular function on the affine line which has a non-isolated zero vanishes everywhere.

In the algebraic category, there is a one-line proof using Euclidean division. In the analytic setting (that is for analytic functions over $$\mathbb{C}$$), you have to work a little to prove this.

Note however that the Empire may be striking back. Indeed, outside of Siu's and Demailly's circles, which are, in my opinion, not very active anymore, there is a new approach to the minimal model program using the Ricci-flow and the techniques Perelman introduced to prove the Poincaré conjecture. In a word, the new idea is to start with a (smooth) variety whose canonical bundle is not nef, run the Ricci flow on it and hope that it will converge to a minimal model. So, at least as far as birational geometry is concerned, it seems that analysis is back!

• I think perhaps the OP is merely confused. As anybody can freely read, Douady proved representability of Quot in the analytic setting. Here is a link to Douady's paper: numdam.org/item/AIF_1966__16_1_1_0 It might be unwise to engage in argumentative speculations if, in fact, the OP is simply confused. – Jason Starr Dec 15 '18 at 23:42
• @JasonStarr Did you read my answer or did you post your comment for the sake of it? – Libli Dec 15 '18 at 23:44
• @Libli. I read what you wrote. I think that the OP, or anybody, would do better to read beautiful mathematics (about which the OP is apparently confused) rather than play pointless word games about which field of mathematics is more active. – Jason Starr Dec 16 '18 at 1:22
• @JasonStarr I think all reasonable mathematicians agree that doing mathematics is more interesting than playing "power games" in academia. However, I think OP's question is more than a pointless word game. To frame it more precisely, we may ask "Why did the number of professional algebraic geometers explode in 1960's while the number of analytic geometers didn't?". The premise of this question is a statistical fact and there might be a legitimate sociological or mathematical reason behind it. – geometer Dec 16 '18 at 9:42
• I would like to see statistics. I do not accept that "the premise of this question is a statistical fact". I suspect the basis of this question is confusion on the part of the OP, and the basis of further discussion is projection of our own biases on a question that, finally, is vacuous. – Jason Starr Dec 16 '18 at 9:59

I'm not sure this question is a good fit form mathoverlow, but here are a few thoughts. I'll probably delete this answer in a while.

1. Let me elaborate my comment concerning sociology a bit further. A (sub)field (not necessarily in mathematics) becomes "hot" when exciting new techniques are introduced that resolve old problems, and create new directions. But at some point it seems to "cool", at least when observed from a distance, when some of these have played out. This seems to have happened with several complex variables in the period of the late 1940's- 1960's with the introduction, and subsequent development, of coherence and other sheaf theoretic ideas. After that I have the sense, from my colleagues who work in the area, that the sorts of the questions that people in SCV think about are further away from the interests of algebraic geometers, and so are perhaps less visible to us.
2. If by "complex analytic geometry", one includes areas such as complex differential geometry etc. then the subject is very much alive and very active, with many interactions with algebraic geometry.