Bernstein-sato Polynomial versus Cohomology of the Milnor Fiber for Generic Arrangements
نویسنده
چکیده
In this note we determine the Bernstein-Sato polynomial bQ(s) of a generic central arrangement Q = ∏k i=1 Hi of hyperplanes. We establish a connection between the roots of bQ(s) and the degrees of the generators for the top cohomology of the corresponding Milnor fiber. This connection holds for all homogeneous polynomials. We also introduce certain subschemes of the arrangement determined by the roots of bQ(s).
منابع مشابه
Bernstein-sato Polynomial versus Cohomology of the Milnor Fiber for Generic Hyperplane Arrangements
Let Q ∈ C[x1, . . . , xn] be a homogeneous polynomial of degree k > 0. We establish a connection between the Bernstein-Sato polynomial bQ(s) and the degrees of the generators for the top cohomology of the associated Milnor fiber. In particular, the integer uQ = max{i ∈ Z : bQ(−(i+n)/k) = 0} bounds the top degree (as differential form) of the elements in H DR (Q(1), C). The link is provided by t...
متن کاملThe Monodromy Conjecture for Hyperplane Arrangements
The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πic) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every such c is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane...
متن کاملMULTIPLIER IDEALS, b-FUNCTION, AND SPECTRUM OF A HYPERSURFACE SINGULARITY
We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain...
متن کاملBernstein-sato Polynomials of Hyperplane Arrangements
Using a generalization of Malgrange’s formula and a solution of Aomoto’s conjecture due to Esnault, Schechtman and Viehweg, we calculate the Bernstein-Sato polynomial (i.e. b-function) of a hyperplane arrangement with a reduced equation, and show that its roots are greater than−2 and the multiplicity of −1 coincides with the (effective) dimension. As a corollary we get a new proof of Walther’s ...
متن کاملOn Milnor Fibrations of Arrangements
We use covering space theory and homology with local coefficients to study the Milnor fiber of a homogeneous polynomial. These techniques are applied in the context of hyperplane arrangements, yielding an explicit algorithm for computing the Betti numbers of the Milnor fiber of an arbitrary real central arrangement in C3, as well as the dimensions of the eigenspaces of the algebraic monodromy. ...
متن کامل