The Jacobian module, the Milnor fiber, and the D-module generated by \(f^s\)

详细信息    查看全文

• 作者：Uli Walther
• 关键词：Mathematics Subject Classification14F10 ; 14N20 ; 13D45 ; 32S22 ; 58A10 ; 14F40 ; 14J17 ; 32C38
• 刊名：Inventiones mathematicae
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：207
• 期：3
• 页码：1239-1287
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1297
• 卷排序：207

文摘

For a germ f on a complex manifold X, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness 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 the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials. In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1 / f; we confirm in many cases a corresponding conjecture on the annihilator of \(f^s\) but we disprove it in general; we show that the Bernstein–Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of \(f^s\) is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.