  The holomorphy conjecture states roughly that Igusa's zeta function
associated to a hypersurface and a character is holomorphic on $\mathbb{C}$
whenever the order of the character does not divide the order of any eigenvalue
of the local monodromy of the hypersurface. In this article we prove the
holomorphy conjecture for surface singularities which are nondegenerate over
$\mathbb{C}$ with respect to their Newton polyhedron. In order to provide
relevant eigenvalues of monodromy, we first show a relation between the
normalized volume (which appears in the formula of Varchenko for the zeta
function of monodromy) of faces in a simplex in arbitrary dimension. We then
study some specific character sums that show up when dealing with false poles.
In contrast with the context of the trivial character, we here need to show
fakeness of certain poles in addition to the candidate poles contributed by
$B_1$-facets.
