Holomorphy of Igusa's and topological zeta functions for homogeneous polynomials

Let F be a number field and f is an element of F [x(1),..., x(n)] \ F. To a ny completion K of F and any character of the group of units of the valuati on ring of K one associates Igusa's local zeta function Z(K) (k, f, s). The holomorphy conjecture states that for all except a finite number of comple tions K of F we have that if the order of does not divide the order of any eigenvalue of the local monodromy of f at any complex point of f(-1){0}, th en Z(K) (k, f, s) is holomorphic on C. The second author already showed tha t this conjecture is true for curves, i.e., for n=2. Here we look at the ca se of an homogeneous polynomial f, so we can consider {f=0} subset of or eq ual to Pn-1. Under the condition that chi (P-C(n-1)\{f=0})=0, we prove the holomorphy conjecture. Together with some results in the case when chi (P-C (n-1)\{f=0})=0, we can conclude that the holomorphy conjecture is true for an arbitrary homogeneous polynomial in three variables. We also prove the s o-called monodromy conjecture for a homogeneous polynomial f is an element of F [x(1), x(2), x(3)] with chi (P-C(2)\{f=0}) =0.