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) 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.

Question: Is there a smooth rational variety $X$ of complex dimension $4n$, $n \in \mathbb{N}$; such that the intersection form on $H^{4n}(X,\mathbb{Z})$ 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) 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.

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) appears 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" Transactions of the American Mathematical Society (1981)) 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.

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) 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.