Hyperkaehler.info

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 \[ \mathrm{e}(\mathrm{OG}_{10})=176904 \]

References

MR1032930

Göttsche, L. (1990). The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann., 286(1–3), 193–207. doi:10.1007/BF01453572 MR1032930

MR1219901

Göttsche, L., & Soergel, W. (1993). Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann., 296(2), 235–245. doi:10.1007/BF01445104 MR1219901

mozgovoy-phd

Mozgovoy, S. (2006). The Euler number of O’Grady’s ten-dimensional symplectic manifold [PhD thesis]. Universität Mainz.

MR2282256

Rapagnetta, A. (2007). Topological invariants of O’Grady’s six dimensional irreducible symplectic variety. Math. Z., 256(1), 1–34. doi:10.1007/s00209-006-0022-2 MR2282256