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This is partly inspired by answers to the question: Question about Hodge numberQuestion about Hodge number . 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?

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**.

Footnotes

*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.

** From the answers, I gather that the work of Popovici suggests that there may be no counterexample.

Donu Arapura
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This is partly inspired by answers to the question: Question about Hodge number . 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?

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 nilmanifoldsnilmanifolds*, 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 hashas**.

Footnotes

*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.

** From the answers, I gather that the work of Popovici suggests that there may be no counterexample.

Donu Arapura
• 31k
• 2
• 82
• 143

Deformations of Kähler manifolds where Hodge decomposition fails?

This is partly inspired by answers to the question: Question about Hodge number . 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?

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.