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Morihiko Saito

Publications and source records attributed to Morihiko Saito.

At least 73 records · Page 4Linked to original sources
arXiv2013

Twistor deformation of rank one local systems

We determine the twistor deformation of rank one local systems on compact Kaehler manifolds which correspond to smooth twistor modules of rank one in the sense of C. Sabbah. Our proof is rather elementary, and uses a natural description of the moduli space of rank one local systems together with the canonical morphism to the Picard variety. The corresponding assertion for smooth twistor modules of rank one follows from the theory of C. Simpson and has been known to the specialists, according to T. Mochizuki.

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arXiv2013

Ambiguity of certain Chow-Kuenneth projectors

We show that the ambiguity of Murre's Chow-Kuenneth projector for degree 1 has certain good properties, assuming only that it factors through a Chow motive of a smooth irreducible curve. This is compatible with a picture obtained by using Beilinson's conjectural mixed motives, and implies for instance the independence of the Chow motive defined by the middle Chow-Kuenneth projector for surfaces.

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arXiv2013

Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at infinity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor product with a certain sheaf of formal microlocal differential operators instead of the formal completion.

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arXiv2013

On the definition of mixed Hodge modules

We give some details of a simpler definition of mixed Hodge modules which has been announced in some papers. Compared with earlier arguments, this new definition is simplified by using Beilinson's maximal extension together with stability by subquotients systematically.

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arXiv2013

Some remarks on the semi-positivity theorems

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 semi-positivity 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 semi-positivity 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.

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arXiv2013

Hirzebruch-Milnor classes of complete intersections

We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusinski and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.

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arXiv2013

Number of Jordan blocks of the maximal size for local monodromies

We prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In case the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.

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arXiv2012

Some remarks on limit mixed Hodge structure and spectrum

We give some remarks on limit mixed Hodge structure and spectrum. These are more or less well-known to the specialists, and do not seem to be stated explicitly in the literature. However, they do not seem to be completely trivial to the beginners, and may be worth writing down explicitly.

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arXiv2011

On the local zeta functions and the b-functions of certain hyperplane arrangements

Conjectures of J. Igusa for p-adic local zeta functions and of J. Denef and F. Loeser for topological local zeta functions assert that (the real part of) the poles of these local zeta functions are roots of the Bernstein-Sato polynomials (i.e. the b-functions). We prove these conjectures for certain hyperplane arrangements, including the case of reduced hyperplane arrangements in three-dimensional affine space.

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arXiv2010

Relative Chow-Kuenneth decompositions for morphisms of threefolds

We show that any nonconstant morphism of a threefold admits a relative Chow-Kuenneth decomposition. As a corollary we get sufficient conditions for threefolds to admit an absolute Chow-Kuenneth decomposition. In case the image of the morphism is a surface, this implies another proof of a theorem on the absolute Chow-Kuenneth decomposition for threefolds satisfying a certain condition, which was obtained by the first author with P. L. del Angel. In case the image is a curve, this improves in the threefold case a theorem obtained by the second author where the singularity of the morphism was assumed isolated and the condition on the general fiber was stronger.

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arXiv2010

Symmetric products of mixed Hodge modules

Generalizing a theorem of Macdonald, we show a formula for the mixed Hodge structure on the cohomology of the symmetric products of bounded complexes of mixed Hodge modules by showing the existence of the canonical action of the symmetric group on the multiple external self-products of complexes of mixed Hodge modules. We also generalize a theorem of Hirzebruch and Zagier on the signature of the symmetric products of manifolds to the case of the symmetric products of symmetric parings on bounded complexes with constructible cohomology sheaves where the pairing is not assumed to be non-degenerate.

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arXiv2010

A variant of Neron models over curves

We study a variant of the Neron models over curves which is recently found by the second named author in a more general situation using the theory of Hodge modules. We show that its identity component is a certain open subset of an iterated blow-up along smooth centers of the Zucker extension of the family of intermediate Jacobians and that the total space is a complex Lie group over the base curve and is Hausdorff as a topological space. In the unipotent monodromy case, the image of the map to the Clemens extension coincides with the Neron model defined by Green, Griffiths and Kerr. In the case of families of Abelian varieties over curves, it coincides with the Clemens extension, and hence with the classical Néron model in the algebraic case (even in the non-unipotent monodromy case).

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arXiv2010

Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations

We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the singular fiber are both rational homology manifolds. Also we introduce quasi-semistable degenerations and calculate the limit mixed Hodge structure by constructing the weight spectral sequence. As a corollary we show non-triviality of the space of vanishing cycles of the Lefschetz pencil associated with a tensor product of any two very ample line bundles except for the case of even-dimensional projective space where two has to be replaced by three.

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arXiv2010

First Milnor cohomology of hyperplane arrangements

We show a combinatorial formula for a lower bound of the dimension of the non-unipotent monodromy part of the first Milnor cohomology of a hyperplane arrangement satisfying some combinatorial conditions. This gives exactly its dimension if a stronger combinatorial condition is satisfied. We also prove a non-combinatorial formula for the dimension of the non-unipotent part of the first Milnor cohomology, which apparently depends on the position of the singular points. The latter generalizes a formula previously obtained by the second named author.

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arXiv2009

Jumping coefficients and spectrum of a hyperplane arrangement

In an earlier version of this paper written by the second named author, we showed that the jumping coefficients of a hyperplane arrangement depend only on the combinatorial data of the arrangement as conjectured by Mustata. For this we proved a similar assertion on the spectrum. After this first proof was written, the first named author found a more conceptual proof using the Hirzebruch-Riemann-Roch theorem where the assertion on the jumping numbers was proved without reducing to that for the spectrum. In this paper we improve these methods and show that the jumping numbers and the spectrum are calculable in low dimensions without using a computer. In the reduced case we show that these depend only on fewer combinatorial data, and give completely explicit combinatorial formulas for the jumping coefficients and (part of) the spectrum in the case the ambient dimension is 3 or 4. We also give an analogue of Mustata's formula for the spectrum.

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arXiv2009

Hausdorff property of the Neron models of Green, Griffiths and Kerr

We prove the Hausdorff property of the Neron modle of the family of intermediate Jacobians which is recently defined by Green, Griffiths and Kerr assuming that the divisor at infinity is smooth. Using their result, this implies in this case the analyticity of the closure of the zero locus of an admissible normal function. The last assertion is also obtained by Brosnan and Pearlstein generalizing their method in the curve case.

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