Euler characteristic
The topological Euler characteristic $\mathrm{e}(X)$ is the alternating sum of the Betti numbers of a manifold. It is also the top Chern number, so for a $2n$-dimensional hyperkähler manifold $X$ we are interested in $\int\mathrm{c}_{2n}(X)$.
- K3[n]-type
- The Euler characteristics are the coefficients of the expansion of $1/\eta(q)^{24}$, see also OEIS:A006922. This is proven in [Corollary 2.10(b), MR1032930]. Written out (see Theorem 0.1 of op. cit) it reads \[ \sum_{n=0}\mathrm{e}(\mathrm{K3}^{[n]})t^n=\prod_{m=1}^{+\infty}(1-t^m)^{-24} \]
- Kumn-type
- The Euler characteristics are the coefficients of the expansion of $\frac{1}{6912}(6\mathrm{E}_2^2\mathrm{E}_4 - 8\mathrm{E}_2\mathrm{E}_6 + 3\mathrm{E}_4^2 - \mathrm{E}_2^4)$, where $\mathrm{E}_2,\mathrm{E}_4,\mathrm{E}_6$ are the Eisenstein series of weights 2, 4, and 6, see also OEIS:A282211. In a closed formula (see [Corollary 1, MR1219901]) it reads \[ \mathrm{e}(\mathrm{Kum}^{n})=(n+1)^3\sum_{d\mathrel{|} n+1}d \]
- OG6
- In [Theorem 2.2.3, MR2282256] it is shown that \[ \mathrm{e}(\mathrm{OG}_6)=1920 \]
- OG10
- In [mozgovoy-phd] it is shown that \[ \mathrm{e}(\mathrm{OG}_{10})=176904 \]
| dimension | K3 | K3[n]-type | Kumn-type | OG6 | OG10 |
|---|---|---|---|---|---|
| OEIS:A006922 | OEIS:A282211 | ||||
| 2 | 24 | ||||
| 4 | 324 | 108 | |||
| 6 | 3200 | 448 | 1920 | ||
| 8 | 25650 | 750 | |||
| 10 | 176256 | 2592 | 176904 | ||
| 12 | 1073720 | 2744 | |||
| 14 | 5930496 | 7680 | |||
| 16 | 30178575 | 9477 | |||
| 18 | 143184000 | 18000 | |||
| 20 | 639249300 | 15972 |
References
- MR1032930
- Göttsche, Lothar. "The Betti numbers of the Hilbert scheme of points on a smooth projective surface." In: Math. Ann. 286 (1990), pp. 193–207. doi:10.1007/BF01453572
- MR1219901
- Göttsche, Lothar and Soergel, Wolfgang. "Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces." In: Math. Ann. 296 (1993), pp. 235–245. doi:10.1007/BF01445104
- mozgovoy-phd
- Mozgovoy, Sergey. "The Euler number of O'Grady's ten-dimensional symplectic manifold." PhD thesis, Universität Mainz (2006)
- MR2282256
- Rapagnetta, Antonio. "Topological invariants of O'Grady's six dimensional irreducible symplectic variety." In: Math. Z. 256 (2007), pp. 1–34. doi:10.1007/s00209-006-0022-2