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Let $X$ be a proper smooth algebraic space over $\mathbb C$ (which amounts, due to Artin, to giving a Moishezon space: a compact complex manifold whose dimension equals the transcendence degree of its field of meromorphic functions). I'd like to know if the classical Hodge theory holds on the cohomology of $X$ (e.g. degeneration of the Hodge vs de Rham spectral sequence and Hodge symmetry) and, if it does, could someone give a reference.

I will sketch a "proof" in the following. There are details to check. The case for proper smooth algebraic varieties over $\mathbb C$ is proved by Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales (Pub. math. de IHES, tome 35, 1968). Some key ingredients in Deligne's proof are: Chow's lemma, Hironaka's resolution of singularities and Serre duality. One has Chow's lemma for algebraic spaces (Knutson, Algebraic spaces, chp IV, 3.1), so there exists a projective birational morphism from a projective smooth algebraic variety to $X.$

In the beginning we used two terminologies (algebraic space and Moishezon space) to remind myself that there is a GAGA issue when defining $H^q(X,\Omega^p)$ (of course we assume $X$ is not Kähler, so there's no *-operator and we don't use $(p,q)$-forms in defining $H^{q,p};$ we use $H^p(X,\Omega^q)$ as its definition). In his thesis, B. Toen proved (among many other things) GAGA for proper DM-stacks; he refered (SGA1, XII) on p.176 for GAGA for proper algebraic spaces, but I couldn't find the statement for algebraic spaces there. The proof in SGA1 uses Chow's lemma to reduce to the projective case which then follows from Serre's result, and one can probably use Chow's lemma for algebraic spaces to generalize the proof to algebraic spaces. I'm not sure if this is what Toen was thinking, since Knutson's book is not in the references in Toen's thesis. (I'll be grateful if someone can provide a reference where this is written down.)

For Serre duality, I learned from wiki that it works for holomorphic vector bundles on compact complex manifolds (not necessarily Moishezon), which suffices for our argument. Again, can someone give a reference for this version of Serre duality?

This is the sketched proof for the degeneration of Hodge vers de Rham and the Hodge symmetry, on proper smooth $\mathbb C$-algebraic spaces. The degeneration part (but not the Hodge symmetry part) can also be proved using reduction mod p and Deligne-Illusie.

I'll appreciate if someone can point out a mistake or give a reference.

Edit: This seems to be well-known to the experts. See Deformations of Kähler manifolds where Hodge decomposition fails?

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It seems likely that Toen meant that the method of SGA1 carries over nearly verbatim to prove GAGA for proper algebraic spaces (which is true). In that sense, it's a reasonable reference for the result (though the reader may appreciate being told that this is why it is being referenced there). Just read the proof and check for yourself. –  BCnrd Dec 19 '10 at 1:09
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Serre in his paper on duality see Commentari Mathematici Helvetici vol 29 1955 proved serre duality for complex manifolds.. –  Mohan Ramachandran Dec 20 '10 at 20:32
    
continued; as long as the cohomology is hausdorff as a topological vector space. –  Mohan Ramachandran Dec 20 '10 at 20:34

1 Answer 1

up vote 5 down vote accepted

If the Hodge to de Rham sequence degenerates for a smooth compact complex manifold $M$, it degenerates for any smooth compact manifold bimeromorphic to $M$, see theorem 5.22 of Deligne, Griffiths, Morgan, Sullivan, Real homotopy theory of K\"ahler manifolds (theorem 5.22 is about the $dd^c$ lemma, but this is equivalent as shown earlier in the same paper). Moishezon spaces are bimeromorphically projective, hence bimeromorphically K\"ahler.

Re the Serre duality for arbitrary smooth compact complex manifolds: I don't know a reference off hand, but if memory serves it can be found somewhere in Chapter 1 of Peters-Steenbrink, Mixed Hodge structures.

Again, if memory serves, in Chapter 2 of the same book the authors sketch a proof that the Hodge symmetry holds for the cohomology of any bimeromorphically K\"ahler manifold.

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