Timeline for Thom conjecture in CP3
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 15, 2021 at 20:12 | history | edited | Danny Ruberman | CC BY-SA 4.0 |
Updated to reflect recent preprint answering the question.
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| Jan 23, 2015 at 22:14 | comment | added | Danny Ruberman | The degree 4 hypersurface in $CP^3$ has minimal $b_2$ in its homology class so a Thom conjecture holds in this case. The signature of an arbitrary smooth representative of a homology class in $H_4(CP^3)$ is determined by the degree, and is given by the same formula as for the hypersurface of that degree. If $d$ is even, then it is necessarily spin. By Donaldson's theorem C, the degree 4 hypersurface has the minimal $b_2$ among spin manifolds with signature $-16$. For higher degrees, there is a gap between $b_2$ of the hypersurface and the best lower bounds for $b_2$ with given signature. | |
| Jan 22, 2015 at 12:15 | comment | added | Marko Slapar | The references are very helpful. Freedman's work does provide counterexamples in higher (even?) dimensional complex projective spaces. It does seem at first glance that not much is known for CP3. Am I correct in thinking that the "simplest" submanifolds must also be taut? | |
| Jan 21, 2015 at 20:36 | vote | accept | Marko Slapar | ||
| Jan 20, 2015 at 15:00 | history | answered | Danny Ruberman | CC BY-SA 3.0 |