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Toru Ohmoto

Publications and source records attributed to Toru Ohmoto.

17 recordsLinked to original sources
arXiv2025

Thom polynomials for singularities of maps

This is a gentle introduction to a general theory of universal polynomials associated to classification of map-germs, called Thom polynomials. The theory was originated by Ren\'e Thom in the 1950s and has since been evolved in various aspects by many authors. In a nutshell, this is about intersection theory on certain moduli spaces, say `classifying spaces of mono/multi-singularities of maps', which provides consistent and deep insights into both classical and modern enumerative geometry with many potential applications.

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arXiv2024

Unstability problem of real analytic maps

As well-known, the $C^\infty$ stability of proper $C^\infty$ maps is characterized by the infinitesimal $C^\infty$ stability. In the present paper we study the counterpart in real analytic context. In particular, we show that the infinitesimal $C^\omega$ stability does not imply $C^\omega$ stability; for instance, a Whitney umbrella $\mathbb{R}^2 \to \mathbb{R}^3$ is not $C^\omega$ stable. A main tool for the proof is a relative version of Whitney's Analytic Approximation Theorem which is shown by using H. Cartan's Theorems A and B.

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arXiv2024

Universal polynomials for multi-singularity loci of maps

In the present paper, we prove the existence of universal polynomials which express multi-singularity loci classes of prescribed types for proper morphisms between smooth schemes over an algebraically closed field of characteristic zero -- we call them Thom polynomials for multi-singularity types of maps. It has been referred to as the Thom-Kazarian principle and unsolved for a long time. This result solidifies the foundation for a general enumerative theory of singularities of maps which is applicable to a broad range of problems in classical and modern algebraic geometry. In particular, it would contribute to a satisfactory answer to the rest of (an advanced form of) Hilbert's 15th problem and connect such classics to recent new interests in enumerations inspired by mathematical physics and other fields. A main feature of our proof is a striking use of algebro-geometric cohomology operations. Somewhat surprisingly, when trying to grasp a full perspective of classical enumerative geometry, we will inevitably encounter algebraic cobordism and motivic cohomology.

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arXiv2021

The dually flat structure for singular models

The dually flat structure introduced by Amari-Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, and such an excellent geometric structure is rarely defined on the entire space. To fix this trouble, in the present paper, we propose a novel generalization of the dually flat structure for a certain class of singular models from the viewpoint of Lagrange and Legendre singularity theory - we introduce a quasi-Hessian manifold endowed with a possibly degenerate metric and a particular symmetric cubic tensor, which exceeds the concept of statistical manifolds and is adapted to the theory of (weak) contrast functions. In particular, we establish Amari-Nagaoka's extended Pythagorean theorem and projection theorem in this general setup, and consequently, most of applications of these theorems are suitably justified even for such singular cases. This work is motivated by various interests with different backgrounds from Frobenius structure in mathematical physics to Deep Learning in data science.

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arXiv2018

Geometric algebra and singularities of ruled and developable surfaces

Any ruled surface in Euclidean 3-space is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra of type (0,3,1)). Combining this classical framework and Singularity Theory, we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invariants. In particular, using a theorem of G. Ishikawa, we show that local topological type of singular (non-cylindrical) developable surfaces is completely determined by vanishing order of the dual torsion, that generalizes an old result of D. Mond for tangent developables of non-singular space curves. Our approach would be useful for analysis on singularities arising in differential line geometry related with several applications such as robotics, vision theory and architectural geometry.

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arXiv2017

Binary differential equations at parabolic and umbilical points for $2$-parameter families of surfaces

We determine local topological types of binary differential equations of asymptotic curves at parabolic and flat umbilical points for generic $2$-parameter families of surfaces in $\mathbb P^3$ by comparing our projective classification of Monge forms and classification of general BDE obtained by Tari and Oliver. In particular, generic bifurcations of the parabolic curve are classified. The flecnodal curve is also examined by direct computations, and we present new bifurcation diagrams in typical examples.

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arXiv2017

$C^1$-triangulations of semialgebraic sets

We show that every semialgebraic set admits a semialgebraic triangulation such that each closed simplex is $C^1$ differentiable. As an application, we give a straightforward definition of the integration $\int_X ω$ over a compact semialgebraic subset $X$ of a differential form $ω$ on an ambient algebraic manifold, that provides a significant simplification of the theory of semialgebraic singular chains and integrations. Our results hold over every (possibly non-archimedian) real closed field.

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arXiv2017

Thom polynomials in $\mathcal{A}$-classification I: counting singular projections of a surface

We study universal polynomials of characteristic classes associated to the $\mathcal{A}$-classification (i.e. up to right-left equivalence) of holomorphic map-germs $(\mathbb{C}^2,0) \to (\mathbb{C}^n, 0)$ $(n=2,3)$. That enables us to systematically treat with classical enumerative problems of lines of prescribed contact with a given projective surface in $3$ and $4$-spaces.

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arXiv2017

Classical formulae on projective surfaces and $3$-folds with ordinary singularities, revisited

As an application of universal polynomials for local and multi-singularities of maps, we revisit classical enumerative formulae of Salmon-Cayley-Zeuthen for projective surfaces and analogous formulae of Segre-(B.)Severi-Roth for projective $3$-folds. In particular, several examples of actual computation are given using universal polynomials for computing weighted Euler characteristics of singularity loci.

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arXiv2016

Projective classification of jets of surfaces in 3-space

We present a local classification of smooth projective surfaces in 3-space via projective transformations in accordance with singularity types of central projections up to codimension 4. We also discuss relations between our classification of Monge forms and bifurcations of parabolic curves and flecnodal curves.

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arXiv2014

Singularities and Characteristic Classes for Differentiable Maps

This is a note on my mini-course in the International Workshop on Real and Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here we introduce a new branch of the Thom polynomial theory for singularities of holomorphic maps, in which we replace counting singular points by computing weighted Euler characteristics. The main purpose is to apply this theory to the study on the vanishing topology of weighted homogeneous map-germs of finite A-codimension without any corank condition.

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arXiv2012

Characteristic classes of Hilbert schemes of points via symmetric products

We obtain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products. We apply the same methods for the calculation of generating series formulae for the Hirzebruch classes of the push-forwards of "virtual motives" of Hilbert schemes of a threefold. As corollaries, we obtain counterparts for the MacPherson (and Aluffi) Chern classes of Hilbert schemes of a smooth quasi-projective variety (resp. for threefolds). For a projective Calabi-Yau threefold, the latter yields a Chern class version of the dimension zero MNOP conjecture.

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arXiv2010

A note on Chern-Schwartz-MacPherson class

This is a note on MacPherson's Chern class for algebraic stacks, based on a previous paper of the author [arXiv:math/0407348]. We also discuss other additive characteristic classes in the same manner.

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arXiv2006

Generating Functions of Orbifold Chern Classes I : Symmetric Products

For a possibly singular complex variety $X$, generating functions of total "orbifold Chern homology classes" of symmetric products $S^nX$ are given. Those are very natural "Chern class versions" (in the sense of Schwartz-MacPherson) of known generating function formulae of (generalized) orbifold Euler characteristics of $S^nX$. In fact more generally we introduce the "class version" of the Dey-Wohlfahrt formula in classical group theory.

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arXiv2004

Equivariant Chern classes of singular algebraic varieties with group actions

We define the equivariant Chern-Schwartz-MacPherson class of a possibly singular algebraic variety with a group action over the complex number field (or a field of characteristic 0). In fact, we construct a natural transformation from the equivariant constructible function functor to the equivariant homology functor (in the sense of Totaro-Edidin-Graham), which may be regarded as MacPherson's transformation for (certain) quotient stacks. We discuss on other type Chern/Segre classes and give some applications generalizing orbifold Euler characteristics and Thom polynomials of singularities. The Verdier-Riemann-Roch formula takes a key role throughout.

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