Hodge theory and $G_4$ fluxes in weighted projective spaces: Galois action
We extend the explicit study of $G_4$-fluxes and general Hodge cycles from the ordinary Fermat sextic fourfold to tame Fermat-type hypersurfaces in weighted projective space. The main new feature in the weighted setting is that the Galois action on the cyclotomic period data need not preserve the $(2,2)$-subspace. As a consequence, the rational reconstruction of an integral self-dual class can involve additional middle-cohomology components, increasing the norm of the corresponding flux. We work at maximally symmetric Fermat points, where the period matrices and symmetry-invariant Hodge loci can be computed explicitly. Using Movasati's description of periods, cyclotomic period matrices, and Hermite/Smith normal form reductions, we construct the relevant integral lattices of symmetric self-dual classes in middle cohomology. This gives a controlled test of whether symmetric general Hodge cycles can satisfy the M2-brane tadpole bound. Our main conclusion is empirical. In the degree 12 example in $\mathbb{P}_{1,1,1,1,4,4}$, and in the degree 8 weighted example used as a comparison, the shortest computed symmetric general Hodge cycles overshoot the tadpole bound. In the degree 36 example in $\mathbb{P}_{1,1,1,9,12,12}$, which has $h^{1,1}=11$, the most general example we have, the tadpole conjecture is indeed verified at the maximally symmetric locus, although the computations get difficult and computationally expensive. These computations suggest that, in weighted Fermat examples, the relevant notion of a ``symmetric flux'' must take into account not only automorphisms of the variety but also the Galois action on the period field. Non-uniform Galois orbits provide a natural arithmetic mechanism by which symmetric Hodge classes can acquire large tadpole charge.