# Hyperkaehler.info

the geography of compact irreducible holomorphic symplectic (or hyperkähler) varieties

### 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