arXiv subjects

Susumu Tanabé

Publications and source records attributed to Susumu Tanabé.

14 recordsLinked to original sources

arXiv2025

Andr{é} and Simone Weil: Mathematics, social activism and Indian culture

This is an essay on the relation of Andr{é} and Simone Weil with Indian culture and Sanskrit literature, especially the Bhagavad G{ī}t{ā}, a Hindu scripture which they knew well, which they quoted extensively, and which guided them in making important life decisions. In addressing this question, we will also talk about the life paths of the two Weils, and more specifically about certain aspects that relate to their deep convictions.

math.HO↗

arXiv2024

Note sur le lieu discriminant d'une hypersurface de bi-degré $(m, n).$

We present a new topological method to study the discriminantal loci of an algebraic variety defined in a product of projective spaces. Our approach relies on an efficient use of groupoid to describe the monodromy. As an example, we treat here the discriminantal loci of an hypersurface of bi-degree $(m, n).$

math.AG↗

arXiv2021

Asymptotic critical value set and Newton polyhedron

We present an effective method to investigate the asymptotic critical value set of a polynomial map. For this purpose we propose a method to construct rational curves with reduced number of terms present in its parametric representation. In this way we show that the asymptotic critical value set contains the critical value of an polynomial associated to so called bad face of the Newton polyhedron. Our main technical tool is the toric geometry that has been introduced into this question by A.Némethi and A.Zaharia

math.AG↗

arXiv2017

On monodromy representation of period integrals associated to an algebraic curve with bi-degree (2,2)

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve $\cal Y$ with bi-degree (2,2) in a product of projective lines ${\Bbb P}^{1} \times {\Bbb P}^{1}$. We calculate two differenent monodromy representations of period integrals for the affine variety ${\cal X}^{(2,2)}$ obtained by the dual polyhedron mirror variety construction from $\cal Y$. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals that gives us a proper subgroup of the monodromy group. It turns out both representations admit a Hermitian quadratic invariant form that is given by a Gram matrix of a split generator of the derived category of coherent sheaves on on $\cal Y$ with respect to the Euler form.

math.AG↗

arXiv2016

Maximally reducible monodromy of bivariate hypergeometric systems

We investigate branching of solutions to holonomic bivariate hypergeometric systems of Horn's type. Special attention is paid to the invariant subspace of Puiseux polynomial solutions. We mainly study Horn systems defined by simplicial configurations and Horn systems whose Ore-Sato polygons are either zonotopes or Minkowski sums of a triangle and segments proportional to its sides. We prove a necessary and sufficient condition for the monodromy representation to be maximally reducible, that is, for the space of holomorphic solutions to split into the direct sum of one-dimensional invariant subspaces.

math.CV↗

arXiv2015

Towards effective detection of the bifurcation locus of real polynomial maps

We answer to a problem raised by recent work of Jelonek and Kurdyka: how can one detect by rational arcs the bifurcation locus of a polynomial map $\bR^n\to\bR^p$ in case $p>1$. We describe an effective estimation of the "nontrivial" part of the bifurcation locus.

math.AG↗

arXiv2006

Logarithmic vector fields and multiplication table

This is a review article on the Gauss-Manin system associated to the complete intersection singularities of projection. We show how the logarithmic vector fields appear as coefficients to the Gauss-Manin system. We examine further how the multiplication table on the Jacobian quotient module calculates the logarithmic vector fields tangent to the discriminant and the bifurcation set . As applications, we establish signature formulae for Euler characteristics of real hypersurfaces and real complete intersections by means of these fields.

math.AG↗

arXiv2005

On Horn-Kapranov uniformisation of the discriminantal loci

In this note we give a rational uniformisation equation of the discriminant loci associated to a non-degenerate affine complete intersection variety. To show this formula we establish a relation of the fibre-integral with the hypergeometric function of Horn and that of Gel'fand-Kapranov-Zelevinski.

math.AG↗

arXiv2004

Transformée de Mellin des intégrales- fibres associées à l'intersection complète non-dégénérée

Mellin transform of fibre integral is calculated for certain classes of non-degenerate affine complete intersections. The lattice structure of the poles of the Mellin transform is clarified by means of the mixed Hodge structure of the affine variety in question. One establishes the relation of the fibre integral with the hypergeometric function of Horn and that of Gel'fand-Kapranov-Zelevinski. As an application we prove an hypothesis proposed by Berglund, Candelas et altri on the mirror symmetry.

math.AG↗

arXiv2004

Hodge structure of fibre integrals associated to the afine hypersurface in a torus

We calculate the fibre integrals of the hypersurface in a torus in the form of their Mellin transforms. Especially, our method works efficiently for an affine hypersurface defined by a so called simpliciable polynomial. The relations between poles of Mellin transforms of fibre integrals, the mixed Hodge structure of the cohomology of the hypersurface, the hypergeometric differential equation and the Euler characteristic of fibres are clarified.

math.AG↗

arXiv2004

Combinatorial aspects of the mixed Hodge structure

This is a review article on the combinatorial aspects of the mixed Hodge structure of a Milnor fibre of the isolated hypersurface singularity. We give a purely combinatorial method to compute spectral pairs of the singularity under the assumption of simplicial Newton boundary and non-degeneracy of the germ.

math.AG↗

arXiv2004

Invariant of the hypergeometric group associated to the quantum cohomology of the projective space

We present a simple method to calculate the Stokes matrix for the quantum cohomology of the projective spaces ${CP}^{k-1}$ in terms of certain hypergeometric group. We present also an algebraic variety whose fibre integrals are solutions to the given hypergeometric equation.

math.AG↗

arXiv2001

Transformée de Mellin des intégrales-fibres associées aux singularités isolées d'intersection comlète quasihomogène

The Mellin transform of fibre integral is calculated for certain isolated singularities of quasihomogeneous complete intersections (especially the unimodal singualrities of the list by Giusti and Wall). We show the property of symmetry between spectra of Gauss-Manin and shed light on the lattice structure of poles of the Mellin transform by means of topological data of the singularity. As an application, Hodge numbers of the Milnor fibre are expressed by counting the spectra of Gauss-Manin.

math.AG↗

arXiv2000

Hyperbolic Cauchy problem and Leray's residue formula

We give an algebraic description of wave fronts that appear in strictly hyperbolic Cauchy problem. Concrete form of definig function of wave front issued from initial algebraic variety is obtained by the aid of the Gauss-Manin systems satisfied by Gel'fand -Leray integrals.

math.AP↗