The main objects of study in this paper are the poles of several local zeta
functions: the Igusa, topological and motivic zeta function associated to a
polynomial or (germ of) holomorphic function in n variables. We are interested
in poles of maximal possible order n. In all known cases (curves,
non-degenerate polynomials) there is at most one pole of maximal order n which
is then given by the log canonical threshold of the function at the
corresponding singular point.
For an isolated singular point we prove that if the log canonical threshold
yields a pole of order n of the corresponding (local) zeta function, then it
induces a root of the Bernstein-Sato polynomial of the given function of
multiplicity n (proving one of the cases of the strongest form of a conjecture
of Igusa-Denef-Loeser). For an arbitrary singular point we show under the same
assumption that the monodromy eigenvalue induced by the pole has a Jordan block
of size n on the (perverse) complex of nearby cycles.Comment: 8 pages, to be published in Journal of Topolog
