Morihiko Saito

Onb-function, spectrum and rational singularity

by duality [18], because R~Tr,CCy, = 0 for i > 0 by [6, 31] (this follows also from [13, 21, 23]) where rc is assumed projective. Here COy denotes the dualizing sheaf (i.e., the dualizing complex [18] shifted by the dimension to the right). The trace morphism (0.2) is injective, and its image is independent of the choice of resolution, because (0.2) is an isomorphism if Y is smooth. We will denote by CSy the image of (0.2). See (2.4.8) below. Now assume Y is a reduced divisor D on a complex manifold X of dimension n. Let f be a reduced defining equation of D on a neighborhood of a: C D. The b-function (i.e., Bernstein polynomial) of f is a monic polynomial bf(s) in s with rational coefficients, and is a generator of the ideal whose element b(s) satisfies the relation

Bernstein-Sato polynomials of arbitrary varieties

We introduce the notion of the Bernstein–Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible) using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein–Sato polynomial and the jumping coefficients of the multiplier ideals, and also a criterion for rational singularities in terms of the maximal root of the polynomial in the case of a reduced complete intersection. These are generalizations of the hypersurface case. We can calculate the polynomials explicitly in the case of monomial ideals.

Multiplier ideals, -filtration, and spectrum

For an effective divisor on a smooth algebraic variety or a complex manifold, we show that the associated multiplier ideals coincide essentially with the filtration induced by the filtration V constructed by B. Malgrange and M. Kashiwara. This implies another proof of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith and D. Varolin that any jumping coefficient in the interval (0,1] is a root of the Bernstein-Sato polynomial up to sign. We also give a refinement (using mixed Hodge modules) of the formula for the coefficients of the spectrum for exponents not greater than one or greater than the dimension of the variety minus one.

Multiplier ideals,

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.

b

The theory of mixed Hodge modules is applied to obtain results about the mixed Hodge structure on the intersection cohomology of a link of a subvariety in a complex algebraic variety. The main result, whose proof uses the purity of the intersection complex in terms of mixed Hodge modules, is a generalization of the semipurity theorem obtained by Gabber in the l-adic case. An application is made to the local topology of complex varieties.

-function, and spectrum of a hypersurface singularity

We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semipositivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semipositivity theorem for admissible variations of mixed Hodge structure in [FF] by using the theories of Cattani, Kaplan, Schmid, Steenbrink, and Zucker. This generalizes Kawamata’s classical result in the pure case. 2010 Mathematics Subject Classification: Primary 14D07; Secondary 32G20.

Mixed Hodge structures on the intersection cohomology of links

We study the algebraic Gauss-Manin system and the algebraic Brieskorn module associated to a polynomial mapping with isolated singularities. Since the algebraic Gauss-Manin system does not contain any information on the cohomology of singular fibers, we first construct a nonquasi-coherent sheaf which gives the cohomology of every fiber. Then we study the algebraic Brieskorn module, and show that its position in the algebraic Gauss-Manin system is determined by a natural map to quotients of local analytic Gauss-Manin systems, and its pole part by the vanishing cycles at infinity, comparing it with the Deligne extension. This implies for example a formula for the determinant of periods. In the two-dimensional case we can describe the global structure of the algebraic Gauss-Manin system rather explicitly.

Some Remarks on the Semipositivity Theorems

We give a combinatorial description of the roots of the Bernstein–Sato polynomial of a monomial ideal using the Newton polyhedron and some semigroups associated to the ideal.

Algebraic Gauss-Manin systems and Brieskorn modules

We show that for a polynomial map, the size of the Jordan blocks for the eigenvalue 1 of the monodromy at infinity is bounded by the multiplicity of the reduced divisor at infinity of a good compactification of a general fiber. The existence of such Jordan blocks is related to global invariant cycles of the graded pieces of the weight filtration. These imply some applications to period integrals. We also show that such a Jordan block of size greater than 1 for the graded pieces of the weight filtration is the restriction of a strictly larger Jordan block for the total cohomology group. If there are no singularities at infinity, we have a more precise statement on the monodromy.

Combinatorial Description of the Roots of the Bernstein–Sato Polynomials for Monomial Ideals

We introduce a class of cycles, called nondegenerate, strictly decomposable cycles, and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed Hodge structures does not vanish. This class contains certain cycles in the kernel of the Abel-Jacobi map. The construction gives a refinement of Noris argument in the case of a self-product of a curve. As an application, we show that a higher cycle which is not annihilated by the reduced higher Abel-Jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1-form.