3
$\begingroup$

Not sure if my question makes sense. Is there an area in complex geometry that is as analytic as possible? Actually what I wanted to ask is an area in complex geometry that is as non-algebraic as possible.

Surely I know some areas in complex geometry, for example Hodge theory, (hyper)Kahler manifolds, deformation of complex structures, canonical metrics, etc. But I only know the names, and I don't know, and frankly speaking, am afraid of, if working in any one of these areas or others would eventually need to learn a lot of tools from algebraic geometry. Personally I am not a big fan of algebraic geometry, but somehow would like to work in complex geometry if I could find an area I like.

So, you are very welcome to let me know an area in complex geometry you think is analytic. Then I will explore and take a brief look on related papers.

The books by Jean-Pierre Demailly and by Griffiths-Harris are towards more of the differential-geometric side of complex geometry rather than its algebraic side. But I am not sure if these books will tell (or guide) me a research area or direction. So any book or research monograph recommendation that leads to a research area in complex geometry that is as analytic as possible is also very welcome.

For my background, I know a little bit of real differential geometry. The posts I found in mathoverflow that are similar to my question are more like reference request, rather than a research topic request, so I hope it's ok to post my questions.

Improve this question
$\endgroup$
5
  • 2
    $\begingroup$ I think you should talk to potential advisors rather than random people on the internet. While I have not voted to close, I would not be surprised if this question ended up being closed (and I also think that questions like this never really get good answers here). I also think that it is unwise to insist that you are never going to learn any particular field. You go where the math leads you -- every time I have insisted that I disliked something, I have eventually been forced to learn it (and so far I have always learned that I actually liked it, once I got over my uninformed prejudices). $\endgroup$
    – Andy Putman
    Commented Oct 2, 2022 at 0:59
  • $\begingroup$ Mathematically, I want to point out that things that are classically considered analytic could be accessed via "algebraic" methods by recent works in condensed mathematics, cf. this MO post and Clausen–Scholze's lecture notes. $\endgroup$
    – Z. M
    Commented Oct 2, 2022 at 7:56
  • $\begingroup$ @AndyPutman I agree with you that questions like this may not get a good answer here. And not sure if there should be something like a wiki page so that every research area in math has a brief description (which would need the contributions of many researchers), so that researchers in other fields could have a rough picture. On the other hand, I cannot say I dislike algebraic geometry, I simply haven't learned it during my graduate study, and it seems I am kind of too old to learn it now (even the basic). Anyway, let see if there is anything which leads me to complex geometry. $\endgroup$
    – Ho Man-Ho
    Commented Oct 2, 2022 at 9:52
  • $\begingroup$ @Z.M Thanks, I will take a look. $\endgroup$
    – Ho Man-Ho
    Commented Oct 2, 2022 at 9:56
  • $\begingroup$ One possible answer to this is geometric analysis, which is quite an active subject indeed. This can be paraphrased as "partial differential equations on manifolds", and a substantial portion of it is intended to have applications to complex geometry. Hodge theory is a good historical example of this, and a more recent topic (that is definitely still an active area of research) is the study of Kähler–Einstein metrics. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 2, 2022 at 12:40

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .