Timeline for Hodge diamonds of complex threefolds
Current License: CC BY-SA 4.0
10 events
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| Oct 7, 2020 at 17:04 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 30, 2020 at 17:55 | comment | added | Michael Albanese | @R.vanDobbendeBruyn: The Hodge numbers of a non-Kähler surface $X$ satisfy $h^{1,0}(X) = h^{0,1}(X) - 1$, so that Hodge diamond cannot be realised. | |
| Sep 30, 2020 at 17:23 | comment | added | R. van Dobben de Bruyn | In addition, here is a post I wrote containing an example of the difficulty of the inverse Hodge problem: the Hodge diamond $$\begin{array}{ccccc}&&1&&\\&1&&1&\\0&&1&&0\\&1&&1&\\&&1&&\end{array}$$ can not be realised by an algebraic surface, but does not obviously violate symmetry, non-negativity, hard Lefschetz, etc. I don't immediately see whether this is also a counterexample in the non-Kähler case, but it could be a first thing to try. I expect that the situation for (simply connected) threefolds is no different. | |
| Sep 30, 2020 at 16:53 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 30, 2020 at 15:54 | comment | added | R. van Dobben de Bruyn | The existence of the Hopf surface and the techniques of this paper (specifically Cor. 3.8 ― will be 3.9 in final version) immediately imply that there are no universal linear relations between Hodge numbers of compact complex manifolds other than those induced by Serre duality. But that doesn't say much about the much harder "inverse Hodge" problem. | |
| Sep 30, 2020 at 13:43 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 30, 2020 at 13:13 | answer | added | Michael Albanese | timeline score: 10 | |
| Sep 30, 2020 at 11:05 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 30, 2020 at 10:52 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 30, 2020 at 10:42 | history | asked | user164740 | CC BY-SA 4.0 |