The toric geometry of some Niemeier lattices (Q1775233)

Let \(L\) be a lattice and \(R\subset L\) a root sublattice of maximal rank, and assume that \(R\) is an orthogonal sum of root lattices of type \(A_1\), \(A_2\) or \(D_4\). Since the root lattice \(R\) is similar to its dual, one can construct an associated complete singular toric variety \(X\), which has singularities of type \(\mathbb{C}/G\) (with \(G\simeq L/R\)). It is shown that the equivariant automorphisms of \(X\) are precisely the lattice automorphisms. A detailed description of the three cases where \(L\) is a Niemeier lattice (unimodular of dimension 24, cf. [\textit{H.-V. Niemeier}, J. Number Theory 5, 142--178 (1973; Zbl 0258.10009)]) is given.