Riemann–Roch polynomial
For a compact hyperkähler manifold $X$ of dimension $2n$, Huybrechts shows that the holomorphic Euler characteristic of a line bundle depends only on the Beauville–Bogomolov square $\operatorname{q}_X(\operatorname{c}_1(L))$, hence there is a polynomial $\operatorname{RR}_X(t)$ such that \[ \chi(X,L)=\operatorname{RR}_X\big(\operatorname{q}_X(\operatorname{c}_1(L))\big). \] See [Section 1, MR1664696].
[MR4554423] proves positivity of the coefficients of this polynomial in full generality. Explicit formulas for all currently known deformation types are collected in [MR4781884].
Known deformation types
- K3 and ${\mathrm K3}^{[n]}$-type: \[ \operatorname{RR}_{\mathrm{K3}^{[n]}}(t)=\binom{t/2+n+1}{n}. \]
- $\mathrm{Kum}_n$-type: \[ \operatorname{RR}_{\mathrm{Kum}_n}(t)=(n+1)\binom{t/2+n}{n}. \]
- $\mathrm{OG}_6$-type: same polynomial as $\mathrm{Kum}_3$-type, i.e. \[ \operatorname{RR}_{\mathrm{OG}_6}(t)=4\binom{t/2+3}{3}. \]
- $\mathrm{OG}_{10}$-type: same polynomial as ${\mathrm K3}^{[5]}$-type, i.e. \[ \operatorname{RR}_{\mathrm{OG}_{10}}(t)=\binom{t/2+6}{5}. \]
The per-type cards on the main page include the concrete $\operatorname{RR}_X(t)$ used for that type.
References
- MR1664696
- Huybrechts, D. (1999). Compact hyper-Kähler manifolds: basic results. Invent. Math., 135(1), 63–113. doi:10.1007/s002220050280 MR1664696
- MR4554423
- Jiang, C. (2023). Positivity of Riemann-Roch polynomials and Todd classes of hyperkähler manifolds. J. Algebraic Geom., 32(2), 239–269. MR4554423
- MR4781884
- Ríos Ortiz, Á. D. (2024). Riemann-Roch polynomials of the known hyperkähler manifolds. Bull. Soc. Math. France, 152(2), 169–184. MR4781884