14 research outputs found
Algebraic computation of some intersection D-modules
Let be a complex analytic manifold, a locally
quasi-homogeneous free divisor, an integrable logarithmic connection with
respect to and the local system of the horizontal sections of on
. In this paper we give an algebraic description in terms of of the
regular holonomic D-module whose de Rham complex is the intersection complex
associated with . As an application, we perform some effective computations
in the case of quasi-homogeneous plane curves.Comment: 18 page
Maximal multihomogeneity of algebraic hypersurface singularities
From the degree zero part of logarithmic vector fields along an algebraic
hypersurface singularity we indentify the maximal multihomogeneity of a
defining equation in form of a maximal algebraic torus in the embedded
automorphism group. We show that all such maximal tori are conjugate and in
one-to-one correspondence to maxmimal tori in the degree zero jet of the
embedded automorphism group.
The result is motivated by Kyoji Saito's characterization of quasihomogeneity
for isolated hypersurface singularities and extends its formal version and a
result of Hauser and Mueller.Comment: 5 page
On `maximal' poles of zeta functions, roots of b-functions and monodromy Jordan blocks
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
Quasihomogeneity of isolated singularities and logarithmic cohomology
We characterize quasihomogeneity of isolated singularities by the injectivity
of the map induced by the first differential of the logarithmic differential
complex in the top local cohomology supported in the singular point.Comment: 5 page
Monodromy Jordan blocks, b-functions and poles of Zeta functions for germs of plane curves
We study the poles of several local zeta functions: the Igusa, topological and motivic zeta function associated to a germ of a holomorphic function in two variables. It was known that there is at most one double pole for (any of) these zeta functions which is then given by the log canonical threshold of the function at the singular point. If the germ is reduced Loeser showed that such a double pole always induces a monodromy eigenvalue with a Jordan block of size 2. Here we settle the non-reduced situation, describing precisely in which case such a Jordan block of maximal size 2 occurs. We also provide detailed information about the Bernstein-Sato polynomial in the relevant non-reduced situation, confirming a conjecture of Igusa, Denef and Loeser