math.AG — Algebraic Geometry

Algebraicity of the $\mathrm{G}_2$-invariant tensors on Yun's exceptional motives

Contributed by Benjamin Severn

Zhiwei Yun constructed, from a rigid Hecke eigensheaf for $\mathrm{G}_2$ on $\mathbb{P}^1$ minus three points, a family of rank-seven motives whose motivic Galois groups are dense in $\mathrm{G}_2$. We show that on this family the invariant cubic form is the class of an algebraic cycle: at every Hodge-generic parameter, every rational Hodge class on every tensor construction of the rank-seven piece is algebraic, and at every parameter the transported $\mathrm{G}_2$-invariant tensors are algebraic. The cycles are top components of Satake convolution fibres transported by Hecke correspondences. The projector that cuts out the rank-seven piece is made algebraic by a canonical weighted resolution of Yun's boundary divisor, built on the moduli stack of bundles from the stabiliser tori of its points, together with a characteristic-zero census of those points and a cycle-class comparison on varieties with quotient singularities. Yun announced that the cubic is motivated in Andr\'e's sense; the step from motivated to algebraic is new. This version, with its computational certificates and review bundle, is archived on Zenodo: https://doi.org/10.5281/zenodo.23478625