English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

For a germ $f$ on a complex manifold $X$, we introduce a complex derived fromthe Liouville form acting on logarithmic differential forms, and give anexactness criterion. We use this Liouville complex to connect properties of the$D$-module generated by $f^s$ to homological data of the Jacobian ideal;specifically we show that for a large class of germs the annihilator of $f^s$is generated by derivations. Through local cohomology, we connect thecohomology of the Milnor fiber to the Jacobian module via logarithmicdifferentials.  In particular, we consider (not necessarily reduced) hyperplane arrangements:we prove a conjecture of Terao on the annihilator of $1/f$; we confirm in manycases a corresponding conjecture on the annihilator of $f^s$ but we disprove itin general; we show that the Bernstein--Sato polynomial of an arrangement isnot determined by its intersection lattice; we prove that arrangements forwhich the annihilator of $f^s$ is generated by derivations fulfill the StrongMonodromy Conjecture, and that this includes as very special cases allarrangements of Coxeter and of crystallographic type, and allmulti-arrangements in dimension 3.

The Jacobian module, the Milnor fiber, and the $D$-module generated by $f^s$