Timeline for Is the minimal Chern number of a toric manifold at least 2?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 23, 2019 at 23:36 | comment | added | Nick L | See my answer below. | |
| Jun 23, 2019 at 23:36 | answer | added | Nick L | timeline score: 2 | |
| Jun 23, 2019 at 9:20 | comment | added | BrianT | Nick L, thanks a lot. Could you extend your answer with a bit more details ? | |
| Jun 23, 2019 at 9:19 | comment | added | BrianT | Thank you Will Sawin, sorry for the mistake, of course it is $n+1$ for $\mathbb{P}^n(\mathbb{C})$... | |
| Jun 23, 2019 at 3:13 | comment | added | Nick L | This also fails in complex dimension $2$. Any exceptional $E=\mathbb{P}^{1}$ with $E^{2} = -1$, has $c_{1}.E = 1$ by the adjunction formula. So take any non-minimal toric surface, for example $\mathbb{P}^{2}$ blown up in a point. So basically almost all toric surfaces have minimal Chern number equal to 1. | |
| Jun 23, 2019 at 2:12 | comment | added | Will Sawin | The minimal Chern number of $\mathbb P^n (\mathbb C) $ is $n+1$ because the first Chern class is $n+1$ times the hyperplane class (Euler sequence). | |
| Jun 22, 2019 at 21:46 | comment | added | BrianT | Thank you for your comment. I don't really understand what positivity means on $A$, which is a homology class. By definition, $N_M$ is positive, since it is a minimum over positive integers. Your example seems wrong: the minimal Chern number of $\mathbb{P}^n(\mathbb{C})$ is $2$, for any $n>0$. Therefore $N_{ \mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^2(\mathbb{C})}$ equals $2$ as well. | |
| Jun 22, 2019 at 20:50 | comment | added | Will Sawin | Is there no positivity condition on $A$? If not then this is just equivalent to taking the gcd of $\langle c, A \rangle$ over a basis of values of $A$. It seems easy to have this to be $1$ in the toric case, for instance for $\mathbb P^1(\mathbb C) \times \mathbb P^2(\mathbb C)$. | |
| Jun 22, 2019 at 15:46 | history | asked | BrianT | CC BY-SA 4.0 |