  We show that the quantum geometry of the Fermi surface can be numerically
described by a 3-dimensional discrete quantum manifold. This approach not only
avoids singularities in the Fermi sea, but it also enables the precise
computation of the intrinsic Hall conductivity resolved in spin, as well as any
other local properties of the Fermi surface. The method assures numerical
accuracy when the Fermi level is arbitrarily close to singularities, and it
remains robust when Kramers degeneracy is protected by symmetry. The approach
is demonstrated by calculating the anomalous Hall and spin Hall conductivities
of a 2-band lattice model of a Weyl semimetal and a full-band ab-initio model
of zincblende GaAs.
