The correspondence between algebraic and analytic geometry is thought-provoking. The GAGA makes this precise to some extent. But there is more of this analogy. As I checked an article on analytic space, most of the notions there seems to have an algebraic analogue. I had been wondering why people studied analytic geometry separately to the extent it has been. It would be nice to know about some theorems in analytic geometry that have no analogue in algebraic geometry, and this would justify the study of this subject separately. Here I mean theorems that do need to use the full power of analytic geometry; not things like Hodge decomposition for which Kähler is sufficient.
Non-algebraic theorems in analytic geometry
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0I don't understand your comment re: Hodge decomposition. What is the "full power" of analytic geometry? – 2010-08-10
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0Hodge decomposition is true over Kahler manifolds. You do not need a complex analytic structure. – 2010-08-10
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0Hodge decomposition, as I know it, is true for compact Kähler manifolds. Unless I am mistaken, Kähler manifolds by definition have an integrable complex structure. Are you saying that Hodge decomposition also holds for almost Kähler manifolds? (By almost Kähler I mean Kähler minus the condition that the almost complex structure be integrable.) – 2010-08-10
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0Oops, I screwed up badly. Instead of saying "complex analytic structure", I should have said, "structure of an analytic set", ie that it is a zero set of some power series, instead of polynomials in the case of algebraic sets. – 2010-08-10
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0Dear LLN BBK, Most people would think that complex analytic geometry is about spaces which are *locally* zero loci of power-series (not globally), and hence would include any complex manifold (including Kahler ones) as part of the subject. – 2010-08-10
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0@Matt E: Sorry for the mistake. I seem to be saying all sorts of wrong things one after the other. Note that I said "analytic set", and not "analytic space". Of course analytic spaces are which look locally like analytic sets. But if some theorem is true for complex manifolds, it need not hold for all analytic sets. So a theorem for kaehler manifolds needs more data than for analytic sets. The question still remains, what are some theorems which are true precisely for analytic spaces, without strengthening to algebraic spaces, or to complex manifolds? – 2010-08-10
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2Sorry, I still don't get it: every complex manifold is an analytic space in the sense you linked to above. – 2010-08-11