**Question**: Is there a smooth rational variety $X$ of even complex dimension $2n$ such that the intersection form on $H^{2n}(X,\mathbb{Z})$ is the Leech lattice?

My motivation is mainly curiosity combined with the fact that the other well-known symmetric, positive definite forms appear in this way, i.e. the well-known lattices $\Gamma_{4k}$ (of which $\Gamma_{8}$ is the famous $E_{8}$ lattice) appear as the intersection bilinear form of an complete intersection of two quadrics in $\mathbb{CP}^{4k-2}$. Due to work of Libgober and Wood ([On the topological structure of even dimensional complete intersections, Trans.Amer.Math.Soc. **267** (1981) 637-660][1]) the Leech lattice does not appear as the intersection form of any smooth complete intersection.

If such an example is not known I would also be curious to know if it is true without the rationality assumption or if there is an example containing the Leech lattice as a unimodular summand of the intersection form, etc.


  [1]: https://www.jstor.org/stable/1998676