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Morihiko Saito

Publications and source records attributed to Morihiko Saito.

At least 19 recordsLinked to original sources
arXiv2026

Defect of projective hypersurfaces with isolated singularities

Let $X$ be a hypersurface with isolated singularities defined by $f$ in ${\bf P^{n+1}}$ with $n>1$. The difference ${\rm def}(X):=h^{n+1}(X)-h^{n-1}(X)$ is called the defect of $X$ (for self-duality of the cohomology of $X$). It is known that its vanishing is closely related to ${\bf Q}$-factoriality of $X$ without assuming rational singularities when $n=3$. This number coincides with the dimension of the cokernel of the inclusion $H^{n-1}(X)\to{\rm IH}^{n-1}(X)$, the rank of the morphism from the vanishing cohomologies of $X$ to $H^{n+1}(X)$ for a one-parameter smoothing of $X$ with total space smooth, and also with the dimension of the unipotent monodromy part of the Milnor fiber cohomology of $f$ with degree $n$. In the case $X$ has only weighted homogeneous isolated singularities, the defect ${\rm def}(X)$ is then given by the $E_2$-term of the spectral sequence of the double complex with differentials ${\rm d}f\wedge$ and $\rm d$ by the $E_2$-degeneration of the pole order spectral sequence. It can be calculated explicitly using a computer even for analogues of the Hirzebruch quintic threefold with more than one hundred ordinary double points found by B.\ van Geemen and J.\ Werner in a compatible way with their computation. We give also an example with ${\rm def}(X)>0$ and $|{\rm Sing}\,X|=1$ in the non-projective cone case where $n=3$.

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arXiv2026

Factoriality of normal projective varieties

For a normal projective variety $X$, the $\mathbb Q$-factoriality defect $σ(X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove an improvement of a topological formula of S.G. Park and M. Popa asserting that $σ(X)\le h^{2n-2}(X)-h^2(X)$ by assuming only 1-semi-rational singularities instead of rational singularities, and the equality holds in the 2-semi-rational case. Here the singularities are called $j$-semi-rational if $R^kπ_*{\mathcal O}_{\widetilde{X}}=0$ for any $k\in[1,j]$ with $π:\widetilde{X}\to X$ a desingularization, $h^k(X):=\dim H^k(X,{\mathbb Q})$, and $n:=\dim X\ge2$. We also show (a slight generalization of) the assertion that $\mathbb Q$-factoriality implies factoriality if $X$ is a local complete intersection whose singular locus has at least codimension three. We then get a new proof for the projective case of Grothendieck's theorem asserting that $X$ is factorial if it is a local complete intersection whose singular locus has at least codimension four. We also show that a local complete intersection $X$ of dimension 3 having only isolated singularities is factorial if the (topological) defect ${\rm def}(X):=h^4(X)-h^2(X)$ vanishes, without any assumption on rational singularities.

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arXiv2026

Roots of Bernstein-Sato polynomials for projective hypersurfaces with ordinary double points

Let $X\subset{\mathbb P}^{n-1}$ be a hypersurface of degree $d\ge3$ with ordinary double points, where $n\ge3$. The roots of Bernstein-Sato polynomial of its defining polynomial $f$ are given up to sign by 1, $(n-1)/2$, and $j/d$ for $j\in{\mathbb Z}\cap[n,nd-n-p_f]$ with $p_f$ a positive integer. Here $p_f$ is bounded above by the minimal positive integer $q_s$ satisfying $\binom{q_s+n-1}{n-1}>s:=|{\rm Sing}\,X|$, and we can verify that $p_f$ coincides with $q_s$ in the case the singular points of $X$ are in ``general position". We show that this upper bound is sharp in the case $\binom{\lfloor d/2\rfloor+n-2}{n-1}\ge s$ or $\binom{d+n-3}{n-1}\ge sn$ by providing a homogeneous polynomial of degree $d$ such that the associated projective hypersurface has ordinary double points at given $s$ points in sufficiently general position and is nonsingular outside them (using a theorem of Alexander and Hirschowitz for the second case). It is conjectured that the above sharp bound under the first hypothesis can be extended naturally to the case where $X$ has only $A_2$-singularities instead of ordinary double points.

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arXiv2026

Subtlety of oscillation indices of oscillatory integrals of real analytic functions

For a locally defined real analytic function $f$, we study the relation between the oscillation index of oscillatory integrals and the real log canonical threshold. The former is always negative, and its absolute value is greater than or equal to the latter. They coincide very often, but there are certain exceptional cases, and it is not very clear when the equality holds. In this note we give some sufficient conditions for the coincidence to hold or to fail. In the Newton-nondegenerate convenient homogeneous case, we show that the strict inequality holds if the number of variables $n$ is even and smaller than the degree $d$ of $f$ (or $f^{-1}(0)=\{0\}$), and the equality holds if $n$ is odd and $f^{-1}(0)=\{0\}$ (in particular, $d$ is even). The first assertion does not seem to be compatible with some standard formula in the literature, and there must be some error somewhere, although it does not seem easy to detect it inside this paper.

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arXiv2026

Some remarks on the generalized Hertling conjecture for Tjurina spectrum

We study the original version of the generalized Hertling conjecture on the variance of the Tjurina spectral numbers, which was proposed by Shi, Wang, and Zuo, and provide a sufficient condition for the original conjecture to fail, employing a theorem of Hertling in an essential way. We calculate certain examples using some codes in Singular.

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arXiv2026

Tjurina spectrum and graded symmetry of missing spectral numbers

For a hypersurface isolated singularity defined by a convergent power series $f$, the Steenbrink spectrum can be defined as the Poincaré polynomial of the graded quotients of the $V$-filtration on the Jacobian ring of $f$. The Tjurina subspectrum is defined by replacing the Jacobian ring with its quotient by the image of the multiplication by $f$. We prove that their difference (consisting of missing spectral numbers) has a canonical graded symmetry. This follows from the self-duality of the Jacobian ring, which is compatible with the action of $f$ as well as the $V$-filtration. It implies for instance that the number of missing spectral numbers which are smaller than $(n{+}1)/2$ (with $n$ the number of variables) is bounded by $[(μ{-}τ)/2]$. We can moreover improve the estimate of Briançon-Skoda exponent in the semisimple monodromy case.

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arXiv2026

Constant coefficient and intersection complex $L$-classes of projective varieties

For a projective variety $X$, we have the intersection complex $L$-classes $L_*(X)$ defined by Goresky-MacPerson using cohomotopy and also the constant coefficient $L$-class $L^c_*(X)$ defined by applying an $L$-class transformation (or $T_{1*}$) to a cubic hyperresolution of $X$. These coincide if $X$ is a $\mathbb Q$-homology manifold. We show that the two $L$-classes $L_*(X)$ and $L^c_*(X)$ differ if they do by replacing $X$ with an intersection of general hyperplane sections which has only $\mathbb Q$-homologically isolated singularities. Finding a good sufficient condition for the non-coincidence of $L_*(X)$ and $L^c_*(X)$ is thus reduced to the latter case, where a necessary and sufficient condition has been obtained in terms of the Hodge signatures of stalks of intersection complex in our previous paper. In the case of projective hypersurfaces having only isolated singularities, the difference between $L_*(X)$ and $L^c_*(X)$ is given by the Hodge signatures of the link cohomologies at singular points, and the Hodge signatures of the vanishing cohomologies give the difference between $L^c_*(X)$ and the virtual $L$-class of $X$, that is, the image by a retraction map of the $L$-class of a smooth deformation of $X$ in an ambient smooth projective variety $Y$ in the very ample case.

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arXiv2026

Strong monodromy conjecture for defining polynomials of projective hypersurfaces having only weighted homogeneous isolated singularities

Let $Z\subset{\bf P}^{n-1}$ be a hypersurface such that the associated reduced hypersurface $Z_{\rm red}$ has only weighted homogeneous isolated singularities. In the case $Z$ is a reduced curve or $Z_{\rm red}$ has only homogeneous isolated singularities with $n$ at least $4$, we show that the strong monodromy conjecture for a defining polynomial $f$ of $Z$ follows from arxiv:1609.04801v1 using in the reduced curve case a formula of Denef and Loeser for Newton-nondegenerate polynomials of three variables (which can be deduced in the applied case from the one for the two variable case) together with known results about the strong monodromy conjecture in the two variable case. Here an amazing cancellation occurs so that possible counterexamples fail. We also show the relation between the pole orders of topological zeta function and the root multiplicities of Bernstein-Sato polynomial in the case $Z$ has equimultiplicity and $Z_{\rm red}$ has only weighted homogeneous singularities with $n=3$ or $Z_{\rm red}$ has only homogeneous isolated singularities with $n>3$.

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arXiv2025

Bernstein-Sato polynomials for projective hypersurfaces with weighted homogeneous isolated singularities

We present a quite efficient method to calculate the roots of Bernstein-Sato polynomial for a defining polynomial $f$ of a projective hypersurface $Z\subset{\mathbb P}^{n-1}$ of degree $d$ having only weighted homogeneous isolated singularities. We prove the $E_2$-degeneration of the pole order spectral sequence so that the computation of roots is reduced to the one of the Hilbert series of the Jacobian ring of $f$ except the special case where $f$ is annihilated by a nonzero vector field on ${\mathbb C}^n$ with linear function coefficients. In the three variable case with $d>4$ we may assume that this vector field is a linear combination of $x\partial_x, y\partial_y, z\partial_z$, where $f$ is called extremely degenerated; in particular, the latter case does not contain any essential indecomposable central hyperplane arrangement in ${\mathbb C}^3$. Combined with the self-duality of the Koszul complex and a theorem of Dimca and Popescu, it implies for $n=3$ with $d>4$ except the extremely degenerated case that $R_f=\frac{1}{d}({\mathbb Z}\cap[3,k'])\cup R_Z$. Here $R_f,R_Z$ are the roots of Bernstein-Sato polynomials of $f$ and $Z$ up to sign, and $k'=\max(2d-3,k_{\max}+3)$ with $k_{\max}$ the maximal degree of the ``torsion part" of the Jacobian ring, where the latter is known to be at most $2d-5$ in the hyperplane arrangement case.

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arXiv2025

Spectrum of cones of projective hypersurfaces with singularities isolated

Let $Z$ be a projective hypersurface such that its underlying reduced variety has only isolated singularities. In case its irreducible components have constant multiplicities, for instance if $\dim Z>1$, we show that the spectrum of its cone can be described by using the spectral numbers at singular points of the reduced hypersurface and the global degree. In the non-reduced plane curve case, assuming that the underlying reduced curve has only semi-weighted-homogeneous singularities, we express the spectrum of the cone in terms of the local weights and the weighted degrees and multiplicities of local irreducible components together with the degrees and multiplicities of global ones. These generalize a formula for reduced line arrangements. In the non-reduced ordinary (that is, semi-homogeneous) singularity case, the second formula is essentially equivalent to the one obtained by the third-named author.

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arXiv2024

Length of $D_Xf^{-α}$ in the isolated singularity case

Let $f$ be a convergent power series of $n$ variables having an isolated singularity at 0. For a rational number $α$, setting $(X,0)=({\mathbb C}^n,0)$, we show that the length of the ${\mathcal D}_X$-module ${\mathcal D}_Xf^{-α}$ is given by $\widetildeν_α+r_f\widetildeδ_α+1$. Here $r_f$ is the number of local irreducible components of $f^{-1}(0)$ (with $r_f=1$ for $n>2$), $\widetildeν_α$ is the dimension of the graded piece ${\rm Gr}_V^α$ of the $V$-filtration on the saturation of the Brieskorn lattice modulo the image of $N:=\partial_tt-α$ on ${\rm Gr}_V^α$ of the Gauss-Manin system, and $\widetildeδ_α:=1$ if $α\in{\mathbb Z}_{>0}$, and 0 otherwise. This theorem can be proved also by employing a generalization a recent formula of T. Bitoun in the integral exponent case. The theorem generalizes an assertion by T. Bitoun and T. Schedler in the weighted homogeneous case where the saturation coincides with the Brieskorn lattice and $N=0$. In the semi-weighted-homogeneous case, our theorem implies some sufficient conditions for their conjecture about the length of ${\mathcal D}_Xf^{-1}$ to hold or to fail.

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arXiv2024

Koszul complexes and spectra of projective hypersurfaces with isolated singularities

For a projective hypersurface $Z$ with isolated singularities, we generalize some well-known assertions in the nonsingular case due to Griffiths, Scherk, Steenbrink, Varchenko, and others about the relations between the Steenbrink spectrum, the Poincaré polynomial of the Jacobian ring, and the roots of Bernstein-Sato polynomial for a defining polynomial $f$ up to sign forgetting the multiplicities. We have to use the pole order spectrum and the alternating sum of the Poincaré series of certain subquotients of the Koszul cohomologies, and study the pole order spectral sequence. We show sufficient conditions for vanishing or non-vanishing of the differential $d_1$ of the spectral sequence, which are useful in many applications. We prove also symmetries of the dimensions of the subquotients of Koszul cohomologies, which are crucial for computing the roots of BS polynomials. We can deduce that the roots of BS polynomial whose absolute values are larger than $n-1-n/d$ are determined by the ``torsion part" of the Jacobian ring (modulo the roots of BS polynomial for $Z$) if all the singularities of $Z$ are weighted homogeneous. Here $d=°f$ and $n$ is the dimension of the ambient affine space.

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arXiv2024

Twisted logarithmic complexes of positively weighted homogeneous divisors

For a rank 1 local system on the complement of a reduced divisor on a complex manifold $X$, its cohomology is calculated by the twisted meromorphic de Rham complex. Assuming the divisor is everywhere positively weighted homogeneous, we study necessary or sufficient conditions for a quasi-isomorphism from its twisted logarithmic subcomplex, called the logarithmic comparison theorem (LCT), by using a stronger version in terms of the associated complex of $D_X$-modules. In case the connection is a pullback by a defining function $f$ of the divisor and the residue is $α$, we prove among others that if LCT holds, the annihilator of $f^{α-1}$ in $D_X$ is generated by first order differential operators and $α-1-j$ is not a root of the Bernstein-Sato polynomial for any positive integer $j$. The converse holds assuming either of the two conditions in case the associated complex of $D_X$-modules is acyclic except for the top degree. In the case where the local system is constant, the divisor is defined by a homogeneous polynomial, and the associated projective hypersurface has only weighted homogeneous isolated singularities, we show that LCT is equivalent to that $-1$ is the unique integral root of the Bernstein-Sato polynomial. We also give a simple proof of LCT in the hyperplane arrangement case under appropriate assumptions on residues, which is an immediate corollary of higher cohomology vanishing associated with Castelnuovo-Mumford regularity. Here the zero-extension case is also treated.

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arXiv2023

Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singularities

We extend the Hirzebruch-Milnor class of a hypersurface $X$ to the case where the normal bundle is nontrivial and $X$ cannot be defined by a global function, using the associated line bundle and the graded quotients of the monodromy filtration. The earlier definition requiring a global defining function of $X$ can be applied rarely to projective hypersurfaces with non-isolated singularities. Indeed, it is surprisingly difficult to get a one-parameter smoothing with total space smooth without destroying the singularities by blowing-ups (except certain quite special cases). As an application, assuming the singular locus is a projective variety, we show that the minimal exponent of a hypersurface can be captured by the spectral Hirzebruch-Milnor class, and higher du~Bois and rational singularities of a hypersurface are detectable by the unnormalized Hirzebruch-Milnor class. Here the unnormalized class can be replaced by the normalized one in the higher du~Bois case, but for the higher rational case, we must use also the decomposition of the Hirzebruch-Milnor class by the action of the semisimple part of the monodromy (which is equivalent to the spectral Hirzebruch-Milnor class). We cannot extend these arguments to the non-projective compact case by Hironaka's example.

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arXiv2023

Spectrum of non-degenerate functions with simplicial Newton polytopes

We show a precise proof of Steenbrink's formula for the spectrum of convenient Newton non-degenerate functions, and prove the symmetry of combinatorial polynomials in the simplicial case. Combined with the modified Steenbrink conjecture for spectral pairs (that is, weighted spectrum) which is recently proved in that case, this simplifies quite a lot of their calculations in such a case. We also introduce the $Γ$-spectrum of simplicial convenient non-degenerate functions as a first approximation of the spectrum, generalizing Arnold's picture in the 2 variable case. Analyzing their difference, we can find simple formulas for weighted spectrum in the 3 or 4 variable case. This is proved by using the symmetry of combinatorial polynomials, and fails in the non-simplicial case. Combining these with the Yomdin-Steenbrink formula for the spectrum, we can prove a formula for the spectrum of certain non-isolated surface singularities with simplicial non-degenerate Newton boundaries. As a byproduct of these arguments, we find an example where the Yomdin-Steenbrink formula for spectral pairs does not hold because of fusions of compact faces under projections.

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arXiv2023

Descent of nearby cycle formula for Newton non-degenerate functions

We prove a descent theorem of nearby cycle formula for Newton non-degenerate functions at the origin as well as its motivic version (without assuming the convenience condition). This is used in some papers without any proof although its proof is quite nontrivial because of the existence of coordinate hyperplanes which is completely neglected in the literature about the descent theorem. In the isolated singularity case, it implies some well-known formula for the number of Jordan blocks of the Milnor monodromy with the theoretically maximal size, using a standard estimate of weights. It also provides a proof of a modified version of the Steenbrink conjecture on spectral pairs for non-degenerate functions with simplicial Newton polytopes in the isolated singularity case (which is false in the non-simplicial case).

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