Morihiko Saito

Morihiko Saito
Kyoto University | Kyodai · Research Institute for Mathematical Sciences

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Publications

Publications (119)
Article
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For a holomorphic function f on a complex manifold X, the Briançon-Skoda exponent eBS(f) is the smallest integer k with fk∈(∂f) (replacing X with a neighborhood of f-1(0)), where (∂f) denotes the Jacobian ideal of f. It is shown that eBS(f)≤dX(:=dimX) by Briançon-Skoda. We prove that eBS(f)≤[dX-2α~f]+1 with -α~f the maximal root of the reduced Bern...
Article
Full-text available
We use the Decomposition Theorem to derive several generalizations of the Clemens–Schmid sequence, relating asymptotic Hodge theory of a degeneration to the mixed Hodge theory of its singular fiber(s).
Preprint
For a holomorphic function $f$ on a complex manifold $X$, the Brian\c con-Skoda exponent $e^{\rm BS}(f)$ is the smallest integer $k$ with $f^k\in(\partial f)$ (replacing $X$ with a neighborhood of $f^{-1}(0)$), where $(\partial f)$ denotes the Jacobian ideal of $f$. It is shown that $e^{\rm BS}(f)\le d_X$ $(:=\dim X)$ by Brian\c con-Skoda. We prove...
Preprint
For a complex algebraic variety $X$, we introduce higher $p$-Du~Bois singularity by imposing isomorphisms between the sheaves of K\"ahler differential forms $\Omega_X^q$ and the shifted graded pieces of the Du~Bois complex $\underline{\Omega}_X^q$ for $q\le p$, extending natural isomorphisms on the smooth part of $X$. If $X$ is a reduced hypersurfa...
Article
Full-text available
We construct complex projective schemes with Lyubeznik numbers of their cones depending on the choices of projective embeddings. This answers a question of G. Lyubeznik in the characteristic 0 case. It contrasts with a theorem of W. Zhang in the positive characteristic case where the Frobenius endomorphism is used. Reducibility of schemes is essent...
Preprint
We study the vanishing cycle complex $\varphi_fA_X$ for a holomorphic function $f$ on a reduced complex analytic space $X$ with $A$ a Dedekind domain (for instance, a localization of the ring of integers of a cyclotomic field, where the monodromy eigenvalue decomposition may hold after a localization of $A$). Assuming the perversity of the shifted...
Article
Full-text available
We prove formulas for the localized Hirzebruch-Milnor class of a projective hypersurface in the case where the multiplicity of a generic hyperplane section is not 1. These formulas are necessary for the calculation of the localized Hirzebruch-Milnor class in the hyperplane arrangement case. To formulate them, we introduce the spectral Hirzebruch cl...
Article
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...
Preprint
We show that the frontier Hodge numbers $h^{p,q}$ (that is, for $pq(n-p)(n-q)=0$) do not change by passing to a desingularization of the singular fiber of a one-parameter degeneration of smooth projective varieties of dimension $n$ if the singular fiber is reduced and has only rational singularities. In this case the order of nilpotence of local mo...
Preprint
We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal $V$-filtration modulo the Jacobian ideal. We compare the Hodge ideal spectrum with the Steenbrink spectrum that can be described by using the microlocal $V$-filtration. As a consequence of a formula of Mustata...
Article
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...
Chapter
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...
Article
Full-text available
We present a relatively simple proof of the Thom-Sebastiani type theorem for underlying filtered $D$-modules in the constant Hodge module case where algebraic partial microlocalization is used. There is a well-known relation between multiplier ideals and $V$-filtrations of Kashiwara--Malgrange, and the above argument implies a Thom-Sebastiani type...
Article
Full-text available
We calculate the Bernstein-Sato polynomial (i.e. b-function) of a hyperplane arrangement with a reduced equation by using a generalization of Malgrange's formula together with a solution of Aomoto's conjecture due to Esnault, Schechtman, Viehweg. We show that the roots are greater than -2 and the multiplicity of -1 coincides with the (effective) di...
Article
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...
Article
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 gra...
Article
Full-text available
In this note we clarify some subtle points on the limit mixed Hodge structures and on the spectrum. These are more or less well-known to the specialists, but do not seem to be stated explicitly in the literature. However, as they do not seem to be obvious to the beginners, we consider them to be worth writing down explicitly. The general constructi...
Article
Full-text available
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 a hypersurface in projective space having only ordinary double points as singularities, if the projective space is even dimensional. In the odd dimensional case, we prove this conjec...
Article
We present a possibly simpler proof of the uniqueness of extensions of good sections for formal Brieskorn lattices. This uniqueness seems to be used in a recent paper of C. Li, S. Li, and K. Saito although it is not stated there explicitly. We also give some explanations of certain points which were not very clear in a previous paper on the structu...
Article
Full-text available
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...
Article
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...
Article
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 cor...
Article
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.
Article
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...
Article
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...
Article
Full-text available
Using the graded duality of the cohomology of the Koszul complexes defined by the partial derivatives of homogeneous polynomials with one-dimensional singular loci, we get some formulas for their Poincare series generalizing a well-known result in the isolated singularity case. We also show some relations with the spectrum in the sense of Steenbrin...
Article
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...
Article
We show the graded duality of the cohomology groups of the Koszul complexes defined by the partial derivatives of homogeneous polynomials with one-dimensional singular loci, generalizing a well-known result in the isolated singularity case. The top cohomology of the Koszul complex is not necessarily Cohen-Macaulay, but is an extension of Cohen-Maca...
Article
Full-text available
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.
Article
Full-text available
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 H...
Article
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...
Article
Full-text available
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...
Article
Full-text available
We give three new proofs of a theorem of C. Sabbah asserting that the weight filtration of the limit mixed Hodge structure at infinity of cohomologically tame polynomials coincides with the monodromy filtration up to a certain shift depending on the unipotent or non-unipotent monodromy part.
Article
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...
Article
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 re...
Article
We explain some recent developments in the theory of Neron models for families of Jacobians associated to variations of Hodge structures of weight -1. Comment: 8 pages
Article
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 three...
Article
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 tot...
Article
We study the map associating the cohomology class of an admissible normal function on the product of punctured disks, and give some sufficient conditions for the surjectivity of the map. We also construct some examples such that the map is not surjective.
Article
Full-text available
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 o...
Article
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 mor...
Article
Full-text available
Let $X$ be an irreducible complex analytic space with $j:U\into X$ an immersion of a smooth Zariski open subset, and let $\bV$ be a variation of Hodge structure of weight $n$ over $U$. Assume $X$ is compact K\"ahler. Then provided the local monodromy operators at infinity are quasi-unipotent, $IH^k(X, \bV)$ is known to carry a pure Hodge structure...
Article
Full-text available
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...
Article
We construct a relative Chow–Künneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle...
Article
As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge conjecture is inductively equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomolog...
Article
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...
Article
We introduce real log canonical threshold and real jumping numbers for real algebraic functions. A real jumping number is a root of the $b$-function up to a sign if its difference with the minimal one is less than 1. The real log canonical threshold, which is the minimal real jumping number, coincides up to a sign with the maximal pole of the distr...
Article
Full-text available
We introduce a spectrum for arbitrary varieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for the coefficients of integral exponents. We show a relation to the graded pieces of the multiplier ideals by usi...
Article
We give a survey on b-function, spectrum, and multiplier ideals together with certain interesting relations among them including the case of arbitrary subvarieties.
Article
Full-text available
We show that the Hodge and pole order filtrations are globally different for sufficiently general singular projective hypersurfaces in case the degree is 3 or 4 assuming the dimension of the projective space is at least 5 or 3 respectively. We then study an algebraic formula for the global Hodge filtration in the ordinary double point case conjectu...
Article
We give a formalism of mixed sheaves on varieties over a subfield of the complex number field.
Article
We give an introduction to a theory of b-functions, i.e. Bernstein-Sato polynomials. After reviewing some facts from D-modules, we introduce b-functions including the one for arbitrary ideals of the structure sheaf. We explain the relation with singularities, multiplier ideals, etc., and calculate the b-functions of monomial ideals and also of hype...
Article
We show that the etale cohomology (with compact supports) of an algebraic variety $X$ over an algebraically closed field has the canonical weight filtration $W$, and prove that the middle weight part of the cohomology with compact supports of $X$ is a subspace of the intersection cohomology of a compactification $X'$ of X, or equivalently, the midd...
Article
We show that a smooth projective variety admits a Chow-Kunneth decomposition if the cohomology has level at most one except for the middle degree. This can be extended to the relative case in a weak sense if the morphism has only isolated singularities, the base space is 1-dimensional, and the generic fiber satisfies the above condition.
Article
Using the theory of mixed perverse sheaves, we extend arguments on the Hodge conjecture initiated by Lefschetz and Griffiths to the case of the Tate conjecture, and show that the Tate conjecture for divisors is closely related to the de Rham conjecture for nonproper varieties, finiteness of the Tate-Shafarevich groups, and also to some conjectures...
Article
We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes...
Article
We describe the roots of the Bernstein-Sato polynomial of a monomial ideal using reduction mod p and invariants of singularities in positive chracteristic. We give in this setting a positive answer to a problem of Takagi, Watanabe and the second author, concerning the dependence on the characteristic for these invariants of singularities.
Article
We give explicit formulas for the Hodge filtration on mixed Hodge modules associated with certain hypersurfaces.
Article
For a subvariety of a smooth projective variety, consider the family of smooth hypersurfaces of sufficiently large degree containing it, and take the quotient of the middle cohomology of the hypersurfaces by the cohomology of the ambient variety and also by the cycle classes of the irreducible components of the subvariety. Using Deligne's semisimpl...
Article
Full-text available
We generalize Griffiths' theorem on the Hodge filtration of the primitive cohomology of a smooth projective hypersurface, using the local Bernstein-Sato polynomials, the V-filtration of Kashiwara and Malgrange along the hypersurface and the Brieskorn module of the global defining equation of the hypersurface.
Article
Full-text available
We study the Brieskorn modules associated to a germ of a holomorphic function with non-isolated singularities and show that the Brieskorn module has naturally the structure of a module over the ring of microdifferential operators of non-positive degree, and that the kernel of the morphism to the Gauss–Manin system coincides with the torsion part fo...
Article
We introduce the notion of 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 betw...
Article
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 is...
Article
Full-text available
For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate it explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to...
Article
We reformulate a conjecture of Deligne on 1-motives by using the integral weight filtration of Gillet and Soulé on cohomology, and prove it. This implies the original conjecture up to isogeny. If the degree of cohomology is at most two, we can prove the conjecture for the Hodge realization without isogeny, and even for 1-motives with torsion. Let X...
Article
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 co...
Article
For a smooth complex projective variety $ X $ defined over a number field, we have filtrations on the Chow groups depending on the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the...
Article
Let X be a smooth quasi-projective variety over the algebraic closure of the rational number field. We show that the cycle map of the higher Chow group to Deligne cohomology is injective and the higher Hodge cycles are generated by the image of the cycle map as conjectured by Beilinson and Jannsen, if the cycle map to Deligne cohomology is injectiv...
Article
Consider an external product of a higher cycle and a usual cycle which is algebraically equivalent to zero. Assume there exists an algebraically closed subfield k such that the higher cycle and its ambient variety are defined over k, but the image of the usual cycle by the Abel-Jacobi map is not. Then we prove that the external product is nonzero i...
Article
We show that Mildenhall's theorem implies that the indecomposable higher Chow group of a self-product of an elliptic curve over the complex number field is infinite dimensional, if the elliptic curve is modular and defined over rational numbers. For the moment we cannot prove even the nontriviality of the indecomposable higher Chow group by a compl...
Article
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 construct...
Article
We reformulate a conjecture of Deligne on 1-motives by using the integral weight filtration of Gillet and Soul\'e on cohomology, and prove it. This implies the original conjecture up to isogeny. If the degree of cohomology is at most two, we can prove the conjecture for the Hodge realization without isogeny, and even for 1-motives with torsion. Com...
Article
this paper, we call (k 1 ; n 1 ); : : : ; (k g ; n g ) the Puiseux pairs of (V; 0) with respect to the coordinates x; y. It is known that the Puiseux pairs are independent of the coordinates as long as the condition k 1 ? n 1 is satisfied (cf. for example [1], [15]). This follows also from (2.6) below. We will assume always this condition, unless t...
Article
Let f: X → S be a projective morphism of complex manifolds with relative dimension n such that S is an open disk and f is semistable (i.e. X0: = f −1 (0) is a divisor with normal crossings). Then J. Steenbrink [18] constructed a limit mixed Hodge structure by using a resolution of the nearby cycle sheaf. This limit mixed Hodge structure coincides w...
Article
We give a formalism of arithmetic mixed sheaves including the case of arithmetic mixed Hodge structures, and show the nonvanishing of certain higher extension groups, and also the nontriviality of the second Abel-Jacobi map for zero cycles on a smooth proper complex variety of any dimension under the existence of a nontrivial global two-form. For t...
Article
. We show that the size of the Jordan blocks with eigenvalue one of the monodromy at infinity is estimated in terms of the weights of the cohomology of the total space and a general fiber. Let f : X ! S be a morphism of complex algebraic varieties with relative dimension n. Assume S is a smooth curve. Let U be a dense open subvariety of S such that...
Article
Full-text available
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. T...
Article
les. S. Bloch [3] conjectured that the vanishing of the transcendental cycles (i.e. that of p g ) would be equivalent to: (0.3) The Albanese map is injective. Note that the noninjectivity of the Albanese map in the case p g 6= 0 is a theorem of D. Mumford [6], and Bloch's conjecture is proved at least if X is not of general type [4]. See also [1],...
Article
Using the theory of mixed Hodge Modules, we introduce the notion of mixed Hodge complex on an algebraic variety, and establish the relation between the filtered complex of Du Bois and the corresponding complex of mixed Hodge Modules. Some application to the Du Bois singularity is given.
Article
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Using local cohomology and algebraic -Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, P. Monsky, A. Adolphson and S. Sperber.
Article
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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 non quasi-coherent sheaf which gives the cohomology of every fiber. Then we st...
Article
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We introduce the notion of filtered perversity of a filtered differential complex on a complex analytic manifold $X$, without any assumptions of coherence, with the purpose of studying the connection between the pure Hodge modules and the \lt-complexes. We show that if a filtered differential complex $(\cM^\bullet,F_\bullet)$ is filtered perverse t...
Article
F or aµ-constant deformation of holomorphic functions with isolated singularities, we define a period mapping by associating the Brieskorn module to each point, where we use the Gauss-Manin system to define the parallel translation. We show that it induces a finite morphism of the µ-constant stratum of a miniversal deformation. This means that the...

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Project
Study the cohomology of singular complex projective hypersurfaces and complete intersections.