Deformations of Kähler manifolds where Hodge decomposition fails? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:40:06Z http://mathoverflow.net/feeds/question/42808 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42808/deformations-of-kahler-manifolds-where-hodge-decomposition-fails Deformations of Kähler manifolds where Hodge decomposition fails? Donu Arapura 2010-10-19T16:40:11Z 2010-10-20T15:45:24Z <p>This is partly inspired by answers to the question: <a href="http://mathoverflow.net/questions/42709/question-about-hodge-number/42735#42735" rel="nofollow">http://mathoverflow.net/questions/42709/question-about-hodge-number/42735#42735</a> . Is there a family of compact complex manifolds, where the general fibres are Kähler, but for which $E_1$ degeneration of the Hodge to de Rham spectral sequence fails at the special fibre? Or, even better, such that the special fibre has nonclosed holomorphic forms? </p> <p>I feel like I should know the answer, but somehow I don't. All the examples I know where the spectral sequence doesn't degenerate are nilmanifolds*, so they aren't even homotopic to Kähler manifolds by standard rational homotopy theoretic obstructions (e.g. they aren't formal). Also the famous Hironaka example [Ann. Math 1962] won't work either, because the special fibre is an algebraic variety, so the spectral sequence will degenerate (by an argument that can found in Deligne [Théorème de Lefschetz...]). Obviously, I haven't thought about this deeply enough, but perhaps someone else has**.</p> <p><strong>Footnotes</strong> </p> <p>*I was bit sloppy yesterday, since the examples I have in mind include solvmanifolds. However, there are still topological obstructions to these being Kähler due to Nori and myself.</p> <p>** From the answers, I gather that the work of Popovici suggests that there may be no counterexample.</p> http://mathoverflow.net/questions/42808/deformations-of-kahler-manifolds-where-hodge-decomposition-fails/42822#42822 Answer by Francesco Polizzi for Deformations of Kähler manifolds where Hodge decomposition fails? Francesco Polizzi 2010-10-19T19:34:43Z 2010-10-19T19:34:43Z <p>If any example exists, then the general fibre of the family cannot be projective. </p> <p>In fact, Dan Popovici ["Limits of projective manifolds under holomorphic deformations", arXiv.09102032] recently proved the following </p> <p><strong>Theorem.</strong> Let $\pi \colon \mathcal{X} \to \Delta$ be a complex analytic family of compact complex manifolds such that the fibre $X_t:=\pi^{-1}(t)$ is projective for all $t \neq 0$. Then $X_0:=\pi^{-1}(0)$ is Moishezon.</p> <p>Since Moishezon manifolds admit a projective algebraic modification, it follows that their Hodge-Frolicher spectral sequence degenerates at $E_1$. In particular, Hodge decomposition holds for $X_0$. Notice that in this case $X_0$ is Kähler if and only if it is projective. </p> http://mathoverflow.net/questions/42808/deformations-of-kahler-manifolds-where-hodge-decomposition-fails/42823#42823 Answer by Misha Verbitsky for Deformations of Kähler manifolds where Hodge decomposition fails? Misha Verbitsky 2010-10-19T19:38:21Z 2010-10-19T19:38:21Z <p>This is known, for projective (even Moishezon) manifolds as shown by Dan Popovici in his paper <a href="http://arxiv.org/abs/1003.3605" rel="nofollow">http://arxiv.org/abs/1003.3605</a> For general Kaehler manifold, this is conjectured. Popovici has proved that a property of "strong Gauduchon" is preserved in limits <a href="http://arxiv.org/abs/1009.5408" rel="nofollow">http://arxiv.org/abs/1009.5408</a> and (I think) there are no example of strong Gauduchon manifold without Hodge decomposition.</p>