Timeline for Negative holomorphic sectional curvature
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 6 at 20:38 | answer | added | AmorFati | timeline score: 1 | |
| Jan 12, 2011 at 17:46 | vote | accept | diverietti | ||
| Oct 21, 2010 at 21:08 | comment | added | diverietti | I guess you are right. If the metric is Kähler they coincide, otherwise not. SP let's suppose the metric is Kähler or that we are talking about holomorphic scalar curvature... | |
| Oct 21, 2010 at 12:55 | comment | added | Deane Yang | But is holomorphic scalar curvature necessarily equal to scalar curvature? It seems to me that scalar curvature has more terms in it. | |
| Oct 20, 2010 at 14:56 | comment | added | diverietti | That's exactly what I do. | |
| Oct 19, 2010 at 21:13 | comment | added | Deane Yang | I would define holomorphic scalar curvature to be $\sum_{i,j=1}^n R(e_i, \bar{e}_i, e_j, \bar{e}_j)$, where $e_1, \dots, e_n$ is a unitary basis of $T^{1,0}$. | |
| Oct 19, 2010 at 9:28 | comment | added | diverietti | Sorry, what's the holomorphic scalar curvature? Did you mean holomorphic sectional curvature? If yes, I don't think you are right: they are simply different objects... The first one is a tensor-like object the second one a function. | |
| Oct 19, 2010 at 0:47 | comment | added | Deane Yang | On third thought and based on some rather faint memories of when I actually knew this stuff, I believe that if the metric is Kahler, then the holomorphic scalar curvature is (a constant multiple of) the scalar curvature. So in that case negative holomorphic sectional curvature would appear to imply negative scalar curvature. But doesn't someone out there know the answer for sure? | |
| Oct 18, 2010 at 20:44 | answer | added | diverietti | timeline score: 9 | |
| Oct 17, 2010 at 14:27 | comment | added | Deane Yang | I would be interested in the proof, if this is true. It does require properties of the curvature tensor that are implied by the Kahler condition. | |
| Oct 15, 2010 at 16:46 | comment | added | diverietti | Anyway, I just need it in the Kähler case. | |
| Oct 15, 2010 at 16:46 | comment | added | diverietti | Actually, the answer is not in the pages you indicated. It is given as an exercise at the end of that chapter. I'll try to work it out and possibly post it here later. | |
| Oct 15, 2010 at 16:03 | comment | added | diverietti | Hey little brother, thank you very much! I go and check ! | |
| Oct 14, 2010 at 5:21 | comment | added | Gunnar Þór Magnússon | Oh! Hey brother! | |
| Oct 14, 2010 at 5:18 | comment | added | Gunnar Þór Magnússon | For the Kahler case you can see this on pages 177-178 of "Complex differential geometry" by Fangyang Zheng. For the non-Kahler case... do you like long calculations with curvature tensors in local coordinates? | |
| Oct 14, 2010 at 0:51 | comment | added | Deane Yang | On second thought I was careless. I do not see how to prove this and am not sure it is true. You can define something called holomorphic scalar curvature and that is negative. | |
| Oct 13, 2010 at 20:50 | comment | added | Deane Yang | This appears to me to be a straightforward consequence of the definitions of scalar and holomorphic sectional curvature and of a metric compatible with a complex structure. | |
| Oct 13, 2010 at 18:58 | history | asked | diverietti | CC BY-SA 2.5 |