Searcharxiv⌕ Search

arXiv subjects

Nobuyoshi Takahashi

Publications and source records attributed to Nobuyoshi Takahashi.

13 recordsLinked to original sources
arXiv2026

Planar curve singularities having constant intersection multiplicity with a smooth boundary

We study deformations of curve singularities in a smooth surface having a constant local intersection multiplicity $w$ with a smooth boundary curve. In particular, we consider the codimension of the equisingular locus in the semiuniversal deformation space and prove results on the inclusion relations between ideals describing different kinds of deformations. A classification is given of singularities expected to appear in codimension $\leq 3$ in a general family.

math.AG↗
arXiv2025

Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes

Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $β$ a curve class on $X$ and $\mathfrak{d}=\sum w_iP_i$ an effective divisor on $D_{\mathrm{sm}}$. We consider the moduli space $\mathcal{M}_β(X, D, \mathfrak{d})$ of sheaves on $X$ which are direct images of rank-$1$ torsion-free sheaves on integral curves $C$ in $β$ such that $C|_D=\mathfrak{d}$, and show that each point of $\mathcal{M}_β(X, D, \mathfrak{d})$ is smooth over a point in the product of the Hilbert schemes of surface singularities of types $A_{w_i-1}$. Hence, $\mathcal{M}_β(X, D, \mathfrak{d})$ has symplectic singularities and admits a unique symplectic resolution.

math.AG↗
arXiv2025

Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras

Let $T$ be a Lie-Yamaguti algebra such that its standard enveloping Lie algebra $L(T)$ is semisimple and $[T, T, T]=T$. Then we give a description of representations of $T$ in terms of representations of $L(T)$ with certain additional data. Similarly, if $(T, σ)$ is an infinitesimal $s$-manifold such that $L(T)$ is semisimple, then any representation of $(T, σ)$ comes from a representation of $L(T)$.

math.RA↗
arXiv2021

Sheaves of maximal intersection and multiplicities of stable log maps

A great number of theoretical results are known about log Gromov-Witten invariants, but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov-Witten invariants. The first such calculation by Gross-Pandharipande-Siebert deals with multiple covers over rigid curves in the log Calabi-Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi-Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of "logarithmic" 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.

math.AG↗
arXiv2020

Log BPS numbers of log Calabi-Yau surfaces

Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb{A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

math.AG↗
arXiv2018

Local BPS Invariants: Enumerative Aspects and Wall-Crossing

We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincare polynomials of the moduli spaces for the curve classes $β$ having arithmetic genus at most 2. We formulate a conjecture that these Poincare polynomials are divisible by the Poincare polynomials of $((-K_S).β-1)$-dimensional projective space. This conjecture motivates upcoming work on log BPS numbers.

math.AG↗
arXiv2017

Quandles associated to Galois covers of arithmetic schemes

Let $X$ be a normal, separated and integral scheme of finite type over $\mathbb{Z}$ and $\mathcal{M}$ a set of closed points of $X$. To a Galois cover $\tilde{X}$ of $X$ unramified over $\mathcal{M}$, we associate a quandle whose underlying set consists of points of $\tilde{X}$ lying over $\mathcal{M}$. As the limit of such quandles over all étale Galois covers and all étale abelian covers, we define topological quandles $Q(X, \mathcal{M})$ and $Q^\mathrm{ab}(X, \mathcal{M})$, respectively. Then we study the problem of reconstruction. Let $K$ be $\mathbb{Q}$ or a quadratic field, $\mathcal{O}_K$ its ring of integers, $X=\mathrm{Spec} \mathcal{O}_K\setminus\{\mathfrak{p}\}$ the complement of a closed point such that $π_1(X)^\mathrm{ab}$ is infinite, and $\mathcal{M}$ a set of maximal ideals with density $1$. Using results from $p$-adic transcendental number theory, we show that $K$, $\mathfrak{p}$ and the projection $\mathcal{M}\to\mathrm{Spec} \mathbb{Z}$ can be recovered from the topological quandle $Q(X, \mathcal{M})$ or $Q^\mathrm{ab}(X, \mathcal{M})$.

math.NT↗
arXiv2017

On the multiplicity of reducible relative stable morphisms

Let $(Z, D)$ be a pair of a smooth surface and a smooth anti-canonical divisor. Denote by $\mathfrak{M}_β$ the moduli stack of genus $0$ relative stable morphisms of class $β$ with full tangency to the boundary. Let $C_1$ and $C_2$ be rational curves fully tangent to $D$ at the same point $P$ and assume that $C_1$ and $C_2$ are immersed and that $(C_1.C_2)_P=\min\{D.C_1, D.C_2\}$. Then we show that the contribution of $C_1\cup C_2$ to the virtual count of $\mathfrak{M}_{[C_1]+[C_2]}$ is $\min\{D.C_1, D.C_2\}$. As an example, we describe genus $0$ relative stable morphisms to $(\mathbb{P}^2, (\hbox{cubic}))$ of degree $4$ with full tangency, and examine how they contribute to the relative Gromov-Witten invariant.

math.AG↗
arXiv2013

Quandle varieties, generalized symmetric spaces and $φ$-spaces

We define a quandle variety as an irreducible algebraic variety $Q$ endowed with an algebraically defined quandle operation $\rhd$. It can also be seen as an analogue of a generalized affine symmetric space or a regular $s$-manifold in algebraic geometry. Assume that $Q$ is normal as an algebraic variety and that the action of the inner automorphism group has a dense orbit. Then we show that there is an algebraic group $G$ such that each orbit is isomorphic to the quandle $(G/H, \rhd_φ)$ associated to the group $G$, an automorphism $φ$ of $G$ and a subgroup $H$ of $G^φ$.

math.AG↗
arXiv2000

Mirror symmetry and C^\times

We show that counting functions of covers of $\mathbb{C}^\times$ are equal to sums of integrals associated to certain `Feynman' graphs. This is an analogue of the mirror symmetry for elliptic curves by Dijkgraaf.

math.AG↗
arXiv2000

Log mirror symmetry and local mirror symmetry

We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dxdy/xy over 2-chains whose boundaries lie on B_ϕwhere {B_ϕ} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for projective plane in a Calabi-Yau 3-fold by Chiang-Klemm-Yau-Zaslow.

math.AG↗