On meromorphic functions defined by a differential system of order $1$

Abstract

Abstract. -Given a germ h of holomorphic function on (C n , 0), we study the condition: "the ideal Ann D 1/h is generated by operators of order 1". We obtain here full characterizations in the particular cases of Koszul-free germs and unreduced germs of plane curves. Moreover, we prove that this condition holds for a special type of hyperplane arrangements. These results allow us to link this condition to the comparison of de Rham complexes associated with h.

Linear free divisors and the global logarithmic comparison theorem

“…Several authors have further investigated the range of validity of LCT, and established interesting links with the theory of D-modules, in particular in [4], [6], [11], [27], and [28].…”

LOGARITHMIC COMPARISON THEOREM AND D-MODULES: AN OVERVIEW

“…On the other hand, does A(1/h) imply H(h)? This is true for isolated singularities [30], Koszul-free germs, and suspensions of unreduced plane curve z N + g(x 1 , x 2 ) (see [31]); this question is still open. Finally, the condition A(1/h) does not imply A(h) in general.…”

“…We give here some results about the meaning of the condition A(1/h) (see [31] Reciprocally, what does remain true? We recall that the condition A(1/h) always implies B(h).…”

“…Finally, the condition A(1/h) does not imply A(h) in general. Indeed, Calderón's example h = x 1 x 2 (x 1 + x 2 )(x 1 + x 2 x 3 ) satisfies LCT(D), A(1/h), B(h), H(h) and not A(h) (see [4], [5], [6], [10], [31]). Meanwhile, condition A(h) is not unrealistic, since we have the following characterization of A(1/h) for Koszul-free germs: Moreover, condition A(h) is satisfied when h defines an isolated singularity (see below).…”

Quasihomogeneity of isolated hypersurface singularities and logarithmic cohomology

“…• (log D) computes the cohomology of U and one says that the logarithmic comparison theorem, or briefly LCT, holds for D. The characterization of LCT is an open problem and the subject of active current research [Tor04]. Mainly the two extremal cases of isolated singularities and free divisors have been studied.…”