Multiplier 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 relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the case of locally conical divisors along a stratification, generalizing a formula of Mustata in the case of hyperplane arrangements. We also give another proof of a formula of U. Walther on the b-function of a generic hyperplane arrangement, including the multiplicity of -1.

[1]  P. Deligne,et al.  Equations differentielles à points singuliers reguliers , 1970 .

[2]  P. Deligne.  Le formalisme des cycles évanescents , 1973 .

[3]  R. Randell.  Milnor fibrations of lattice-isotopic arrangements , 1997 .

[4]  M. Saito,et al.  Mixed Hodge modules , 1990 .

[5]  H. Esnault,et al.  Cohomology of local systems on the complement of hyperplanes , 1992 .

[6]  J. H. M. Steenbrink,et al.  Mixed Hodge Structure on the Vanishing Cohomology , 1977 .

[7]  A generalization of Griffiths's theorem on rational integrals , 2005, math/0501253.

[8]  Mircea Mustata.  Multiplier ideals of hyperplane arrangements , 2004 .

[9]  F. Pham,et al.  Singularités des systèmes différentiels de Gauss-Manin , 1995 .

[10]  M. Saito,et al.  Modules de Hodge Polarisables , 1988 .

[11]  Robert Lazarsfeld,et al.  Jumping coefficients of multiplier ideals , 2003, math/0303002.

[12]  P. Orlik,et al.  The Milnor fiber of a generic arrangement , 1993 .

[13]  jason howald.  Multiplier ideals of monomial ideals , 2000 .

[14]  J. Demailly.  L2-Methods and Effective Results in Algebraic Geometry , 1995 .

[15]  János Kollár,et al.  Singularities of Pairs , 1996, alg-geom/9601026.

[16]  B. Malgrange.  Intégrales asymptotiques et monodromie , 1974 .

[17]  M. Vaquié.  Irrégularité des revêtements cycliques des surfaces projectives non singulières , 1992 .

[18]  Daniel C. Cohen,et al.  On Milnor Fibrations of Arrangements , 1995 .

[19]  A. Varchenko.  Zeta-function of monodromy and Newton's diagram , 1976 .

[20]  A. G. Kouchnirenko.  Polyèdres de Newton et nombres de Milnor , 1976 .

[21]  Masaki Kashiwara,et al.  Vanishing cycle sheaves and holonomic systems of differential equations , 1983 .

[22]  Egbert Brieskorn,et al.  Die monodromie der isolierten singularitäten von hyperflächen , 1970 .

[23]  H. Esnault.  Fibre de Milnor d'un cône sur une courbe plane singulière , 1982 .

[24]  A N Varčenko,et al.  ASYMPTOTIC HODGE STRUCTURE IN THE VANISHING COHOMOLOGY , 1982 .

[25]  B. Malgrange,et al.  Le polynome de Bernstein d’une singularite isolee , 1975 .

[26]  G. Laumon,et al.  A Series of Modern Surveys in Mathematics , 2000 .

[27]  H. Terao,et al.  Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors , 1994 .

[28]  J. Steenbrink.  Intersection form for quasi-homogeneous singularities , 1977 .

[29]  Masaki Kashiwara,et al.  B-functions and holonomic systems , 1976 .

[30]  A. Dimca,et al.  Some consequences of perversity of vanishing cycles , 2003, math/0308078.

[31]  A. Nadel.  Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature , 1990 .

[32]  Multiplier Ideals of Sufficiently General Polynomials , 2003, math/0303203.

[33]  Robert Lazarsfeld,et al.  Positivity in algebraic geometry , 2004 .

[34]  J. H. M. Steenbrink,et al.  On the mixed Hodge structure on the cohomology of the Milnor fibre , 1985 .

[35]  M. Saito,et al.  Exponents and Newton polyhedra of isolated hypersurface singularities , 1988 .