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 \[ \mathrm{e}(\mathrm{Kum}^n)=n^3\sum_{d\mathrel{|} n}d \]
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