Norm of the Riemann curvature tensor
Hitchin and Sawon computed in [MR1813238] the $\mathcal{L}^2$ norm of the Riemann curvature tensor $R$ of a hyperkähler manifold $X$ of (real) dimension $4k$: \begin{equation} \frac{1}{(192\pi^2k)^k}\frac{||R||^{2k}}{(\operatorname{vol} X)^{k-1}}=\int_X\mathrm{td}_X^{1/2} \end{equation}
| dimension | K3 | K3[n]-type | Kumn-type | OG6 | OG10 |
|---|---|---|---|---|---|
| $\displaystyle\frac{(n+3)^n}{4^n n!}$ | $\displaystyle\frac{(n+1)^{n+1}}{4^n n!}$ | ||||
| 2 | 1 | ||||
| 4 | 25/32
$\approx0.78125$ | 27/32
$\approx0.84375$ | |||
| 6 | 9/16
$\approx0.56250$ | 2/3
$\approx0.66667$ | 2/3
$\approx0.66667$ | ||
| 8 | 2401/6144
$\approx0.39079$ | 3125/6144
$\approx0.50863$ | |||
| 10 | 4/15
$\approx0.26667$ | 243/640
$\approx0.37969$ | 4/15
$\approx0.26667$ | ||
| 12 | 59049/327680
$\approx0.18020$ | 823543/2949120
$\approx0.27925$ | |||
| 14 | 15625/129024
$\approx0.12110$ | 64/315
$\approx0.20317$ | |||
| 16 | 214358881/2642411520
$\approx0.08112$ | 43046721/293601280
$\approx0.14662$ | |||
| 18 | 243/4480
$\approx0.05424$ | 1953125/18579456
$\approx0.10512$ | |||
| 20 | 137858491849/3805072588800
$\approx0.03623$ | 285311670611/3805072588800
$\approx0.07498$ |
References
- MR1813238
- Hitchin, Nigel and Sawon, Justin. "Curvature and characteristic numbers of hyper-Kähler manifolds." In: Duke Math. J. 106 (2001), pp. 599–615. doi:10.1215/S0012-7094-01-10637-6