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Morihiko Saito

Publications and source records attributed to Morihiko Saito.

At least 55 records · Page 3Linked to original sources
arXiv2019

D-modules generated by rational powers of holomorphic functions

We prove some sufficient conditions in order that a root of the Bernstein-Sato polynomial contributes to a difference between certain D-modules generated by rational powers of a holomorphic function; for instance, this holds in the case of isolated singularities with semisimple Milnor monodromies. We then construct an example where a root does not contribute to a difference. This also solves an old open problem about the relation between the Milnor monodromy and the exponential of the residue of the Gauss-Manin connection on the saturation of the Brieskorn lattice. This shows that the structure of Brieskorn lattices can be more complicated than one might imagine.

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arXiv2019

Degeneration of pole order spectral sequences for hyperplane arrangements of 4 variables

For essential reduced hyperplane arrangements of 4 variables, we show that the pole order spectral sequence degenerates almost at $E_2$, and completely at $E_3$, generalizing the 3 variable case where the complete $E_2$-degeneration is known. These degenerations are useful to determine the roots of Bernstein-Sato polynomials supported at the origin. For the proof we improve an estimate of the Castelnuovo-Mumford regularity of logarithmic vector fields which was studied by H. Derksen and J. Sidman.

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arXiv2018

Rank one local systems on complements of hyperplanes and Aomoto complexes

We show that the cohomology of a rank 1 local system on the complement of a projective hyperplane arrangement can be calculated by the Aomoto complex in certain cases even if the condition on the sum of the residues of connection due to Esnault et al is not satisfied. For this we have to study the localization of Hodge-logarithmic differential forms which are defined by using an embedded resolution of singularities. As an application we can compute certain monodromy eigenspaces of the first Milnor cohomology group of the defining polynomial of the reflection hyperplane arrangement of type $G_{31}$ without using a computer.

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arXiv2018

Weight zero part of the first cohomology of complex algebraic varieties

We show that the weight 0 part of the first cohomology of a complex algebraic variety $X$ is a topological invariant, and give an explicit description of its dimension using a topological construction of the normalization of $X$, where $X$ can be reducible, but must be equidimensional. The first assertion is known in the $X$ compact case by A. Weber, where intersection cohomology is used. Note that the weight 1 or 2 part of the first cohomology is not a topological (or even analytic) invariant in the non-compact case by Serre's example.

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arXiv2018

Thom-Sebastiani theorems for filtered D-modules and for multiplier ideals

We give a proof of the Thom-Sebastiani type theorem for holonomic filtered $D$-modules satisfying certain good conditions (including Hodge modules) by using algebraic partial microlocalization. By a well-known relation between multiplier ideals and $V$-filtrations of Kashiwara and Malgrange, the argument in the proof implies also a Thom-Sebastiani type theorem for multiplier ideals, which cannot be deduced from a already known proof of the Thom-Sebastiani theorem for mixed Hodge modules (since the latter gives only the information of graded pieces of multiplier ideals). We also sketch a more elementary proof of the Thom-Sebastiani type theorem for multiplier ideals (as communicated to us by M.~Mustaţǎ), which seems to be known to specialists, although it does not seem to be stated explicitly in the literature.

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arXiv2018

On the structure of Brieskorn lattices, II

We give a simple proof of the uniqueness of extensions of good sections for formal Brieskorn lattices, which can be used in a paper of C. Li, S. Li, and K. Saito for the proof of convergence in the non-quasihomogeneous polynomial case. Our proof uses an exponential operator argument as in their paper, although we do not use polyvector fields nor smooth differential forms. We also present an apparently simpler algorithm for an inductive calculation of the coefficients of primitive forms in the Brieskorn-Pham polynomial case. In a previous paper on the structure of Brieskorn lattices, there were some points which were not yet very clear, and we give some explanations about these, e.g. on the existence and the uniqueness of primitive forms associated with good sections, where we present some rather interesting examples. In Appendix we prove the uniqueness up to a nonzero constant multiple of the higher residue pairings in some formal setting which is different from the one in the main theorem. This is questioned by D. Shklyarov.

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arXiv2017

Deformations of abstract Brieskorn lattices

We study certain deformations of abstract Brieskorn lattices in fixed abstract Gauss-Manin systems, and show that the ambiguity of expressions of deformations coming from automorphisms of base spaces is essentially the same as the one coming from the choice of opposite filtrations, and hence is finite dimensional, although the freedom of parameters in the expressions of deformations is infinite dimensional. As a consequence, we can prove the non-existence of a versal deformation of the Fourier transform of this abstract Brieskorn lattice with expected dimension. This shows that the generation condition is quite essential for the existence of versal deformations with expected dimensions in the absolute case.

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arXiv2017

Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points

Let Y be a hypersurface in projective space having only ordinary double points as singularities. We prove a variant of a conjecture of L. Wotzlaw on an algebraic description of the graded quotients of the Hodge filtration on the top cohomology of the complement of Y except for certain degrees of the graded quotients, as well as its extension to the Milnor cohomology of a defining polynomial of Y for degrees a little bit lower than the middle. These partially generalize theorems of Griffiths and Steenbrink in the Y smooth case, and enable us to determine the structure of the pole order spectral sequence. We then get quite simple formulas for the Steenbrink and pole order spectra in this case, which cannot be extended even to the simple singularity case easily.

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arXiv2017

Spectral Hirzebruch-Milnor classes of singular hypersurfaces

We introduce spectral Hirzebruch-Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch-Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hyperplane section is not 1. These formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases. Here we use the Thom-Sebastiani theorem for the underlying filtered $D$-modules of vanishing cycles, from which we can deduce the Thom-Sebastiani type theorem for spectral Hirzebruch-Milnor classes. For the Chern classes after specializing to $y=-1$, we can give a relatively simple formula for the localized Milnor classes, which implies a new formula for the Euler numbers of projective hypersurfaces, using iterated hyperplane sections. Applications to log canonical thresholds and Du Bois singularities are also explained; for instance, the latter can be detected by using Hirzebruch-Milnor classes in the projective hypersurface case.

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arXiv2017

Hodge ideals and microlocal V-filtration

We show that the Hodge ideals in the sense of Mustata and Popa are quite closely related to the induced microlocal V-filtration on the structure sheaf, defined by using the microlocalization of the V-filtration of Kashiwara and Malgrange. More precisely the former coincide, module the ideal of the divisor, with the part of the latter indexed by positive integers, although they are different without modulo the ideal in general. This coincidence implies that the $j$-log-canonicity in their sense is determined by the microlocal log-canonical threshold of the divisor, which coincides with the maximal root of the reduced (or microlocal) Bernstein-Sato polynomial up to a sign.

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arXiv2016

Mixed Hodge modules and mixed twistor modules

We explain some fundamental differences between the theories of mixed Hodge modules and mixed twistor modules (including the difference in weight system on the nearby cycle functor) which do not seem to be clarified explicitly in the literature.

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arXiv2016

Bernstein-Sato polynomials of hyperplane arrangements

We show that the Bernstein-Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange's formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than -2 and the multiplicity of the root -1 is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein-Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein-Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.

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arXiv2016

Hirzebruch-Milnor classes and Steenbrink spectra of certain projective hypersurfaces

We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.

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arXiv2015

Hilbert series of graded Milnor algebras and roots of Bernstein-Sato polynomials

We show that there is a pair of homogeneous polynomials such that the sets of roots of their Bernstein-Sato polynomials which are strictly supported at the origin are different although the sets of roots which are not strictly supported at the origin are the same and moreover their graded Milnor algebras have the same Hilbert series. This shows that the roots of the Bernstein-Sato polynomials strictly supported at the origin cannot be determined uniquely by the Hilbert series of the Milnor algebras. This is contrary to certain hyperplane arrangement cases. It also implies that a nonzero torsion element with pure degree in the Milnor algebra does not necessarily contribute to a root of the Bernstein-Sato polynomial in an expected way. This example is found by using Macaulay2 and RISA/ASIR.

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arXiv2015

Fields of definition of Hodge loci

We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of finite transcendence degree if X is defined over k and the component contains a k-rational point. We also prove a similar assertion for the Hodge locus inside the Hodge bundle if the Hodge bundle together with the connection is defined over k. This is closely related with the theory of absolute Hodge classes. The proof uses the spread of the Hodge locus, and is quite similar to the case of the zero locus of an admissible normal function.

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arXiv2014

Normal functions and spread of zero locus

If there is a topologically locally constant family of smooth algebraic varieties together with an admissible normal function on the total space, then the latter is constant on any fiber if this holds on some fiber. Combined with spreading out, it implies for instance that an irreducible component of the zero locus of an admissible normal function is defined over k if it has a k-rational point where k is an algebraically closed subfield of the complex number field with finite transcendence degree. This generalizes a result of F. Charles that was shown in case the normal function is associated with an algebraic cycle defined over k.

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arXiv2014

Graded duality for filtered D-modules

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the structure sheaf. We also prove that the local cohomology sheaves can be calculated by using the higher direct images of the twists of the associated sheaf complex on the projective cotangent bundle. These are closely related to local duality, essentially due to Grothendieck.

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