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Atsushi Takahashi

Publications and source records attributed to Atsushi Takahashi.

At least 19 recordsLinked to original sources
arXiv2026

The number of full exceptional collections modulo spherical twists for extended Dynkin quivers

This paper calculates the number of full exceptional collections modulo an action of a free abelian group of rank one for an abelian category of coherent sheaves on an orbifold projective line with a positive orbifold Euler characteristic, which is equivalent to the one of finite dimensional modules over an extended Dynkin quiver of ADE type by taking their derived categories. This is done by a recursive formula naturally generalizing the one for the Dynkin case by Deligne whose categorical interpretation is due to Obaid--Nauman--Shammakh--Fakieh--Ringel. Moreover, the number coincides with the degree of the Lyashko--Looijenga map of the Frobenius manifold for the orbifold projective line, which hints a consistency in some problems in Bridgeland's stability conditions and mirror symmetry.

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arXiv2025

Frobenius manifold for the nodal quiver

Starting from the Weierstrass elliptic function, we study the associated Frobenius structure, incorporating the perspective of derived categories, particularly that of homological mirror symmetry. Given a deformation of the Weierstrass elliptic function, we construct a primitive form normalized to be compatible with the period map for integral cycles, and obtain a Frobenius structure whose Frobenius potential is defined over the rational numbers. We also construct a Frobenius structure using elliptic Weyl group invariants (as opposed to Jacobi group invariants), and establish an isomorphism between these two Frobenius structures. We further examine the relationship between the degree of the Lyashko--Looijenga map modulo the modular group and the number of full exceptional collections up to the braid group action and translations, as well as the associated Gamma-integral structure.

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arXiv2023

Number of full exceptional collections modulo spherical twists for elliptic orbifolds

This paper calculates the number of full exceptional collections modulo an action of a group as the set generated by spherical twists for an abelian category of coherent sheaves on an orbifold projective line with a zero orbifold Euler characteristic. This is done by a recursive formula naturally generalizing the one for the Dynkin case by Deligne whose categorical interpretation is due to Obaid-Nauman-Shammakh-Fakieh-Ringel and an abelian category of coherent sheaves on an orbifold projective line with a positive orbifold Euler characteristic is due to Otani-Shiraishi-Takahashi.

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arXiv2021

Graded matrix factorizations of size two and reduction

We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C.T.C.Wall. We study this for the full strongly exceptional collections in the triangulated category of graded matrix factorizations constructed by H.Kajiura, K.Saito, and the second author.

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arXiv2021

Mirror symmetry for a cusp polynomial Landau-Ginzburg orbifold

For any triple of positive integers $A' = (a_1',a_2',a_3')$ and $c \in \mathbb{C}^*$, cusp polynomial $f_{A'} = x_1^{a_1'}+x_2^{a_2'}+x_3^{a_3'}-c^{-1}x_1x_2x_3$ is known to be mirror to Geigle-Lenzing orbifold projective line $\mathbb{P}^1_{a_1',a_2',a_3'}$. More precisely, with a suitable choice of a primitive form, Frobenius manifold of a cusp polynomial $f_{A'}$, turns out to be isomorphic to the Frobenius manifold of the Gromov-Witten theory of $\mathbb{P}^1_{a_1',a_2',a_3'}$. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any $G$ - a symmetry group of a cusp polynomial $f_{A'}$, we introduce the Frobenius manifold of a pair $(f_{A'},G)$ and show that it is isomorphic to the Frobenius manifold of the Gromov-Witten theory of Geigle-Lenzing weighted projective line $\mathbb{P}^1_{A,Λ}$, indexed by another set $A$ and $Λ$, distinct points on $\mathbb{C}\setminus\{0,1\}$. For some special values of $A'$ with the special choice of $G$ it happens that $\mathbb{P}^1_{A'} \cong \mathbb{P}^1_{A,Λ}$. Combining our mirror symmetry isomorphism for the pair $(A,Λ)$, together with the "usual" one for $A'$, we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta-function.

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arXiv2021

Serre dimension and stability conditions

We study relations between the Serre dimension defined as the growth of entropy of the Serre functor and the global dimension of Bridgeland stability conditions due to Ikeda-Qiu. A fundamental inequality between the Serre dimension and the infimum of the global dimensions is proved. Moreover, we characterize Gepner type stability conditions on fractional Calabi-Yau categories via the Serre dimension, and classify triangulated categories of the Serre dimension lower than one with a Gepner type stability condition.

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arXiv2021

Strange duality between the quadrangle complete intersection singularities

There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and C.T.C.Wall. We derive this duality from the mirror symmetry, the Berglund-Hübsch transposition of invertible polynomials, and our previous results about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.

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arXiv2021

Gamma integral structure for an invertible polynomial of chain type

The notion of the Gamma integral structure for the quantum cohomology of an algebraic variety was introduced by Iritani, Katzarkov-Kontsevich-Pantev. In this paper, we define the Gamma integral structure for an invertible polynomial of chain type. Based on the $Γ$-conjecture by Iritani, we prove that the Gamma integral structure is identified with the natural integral structure for the Berglund-Hübsch transposed polynomial by the mirror isomorphism.

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arXiv2020

A Frobenius manifold for $\ell$-Kronecker quiver

We construct a Frobenius structure whose intersection form coincides with the generalized Cartan matrix of the $\ell$-Kronecker quiver $K_{\ell}$ and underlying complex manifold is isomorphic to the space of stability conditions for the bounded derived category of finitely generated modules over the path algebra $\mathbb{C} K_{\ell}$.

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arXiv2019

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik--Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles. The purpose of this paper is to prove the algebraic counterpart of the Orlik--Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund--Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov--Smirnov, whose Euler matrix are calculated in terms of the "zeta function" of the inverse of the polarization. As a corollary, it turns out that the homological mirror symmetry holds at the level of lattices, namely, the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the intersection form (with a suitable sign).

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arXiv2018

Lattices for Landau-Ginzburg orbifolds

We consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries. Berglund, Hübsch, and Henningson proposed a duality between such pairs giving rise to mirror symmetry. We define an orbifoldized signature for such a pair using the orbifoldized elliptic genus. In the case of three variables and based on the homological mirror symmetry picture, we introduce two integral lattices, a transcendental and an algebraic one. We show that these lattices have the same rank and that the signature of the transcendental one is the orbifoldized signature. Finally, we give some evidence that these lattices are interchanged under the duality of pairs.

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arXiv2018

Hochschild cohomology and orbifold Jacobian algebras associated to invertible polynomials

Let $f$ be an invertible polynomial and $G$ a group of diagonal symmetries of $f$. This note shows that the orbifold Jacobian algebra $\mathrm{Jac}(f,G)$ of $(f,G)$ defined by the authors and Elisabeth Werner in arXiv:1608.08962 is isomorphic as a $\mathbb{ZZ}/2\mathbb{ZZ}$-graded algebra to the Hochschild cohomology $\mathsf{HH}^*(\mathrm{MF}_G(f))$ of the dg-category $\mathrm{MF}_G(f)$ of $G$-equivariant matrix factorizations of $f$ by calculating the product formula of $\mathsf{HH}^*(\mathrm{MF}_G(f))$ given by Shklyarov in arXiv:1708.06030. We also discuss the relation of our previous results to the categorical equivalence.

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arXiv2017

A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines

Entropy of categorical dynamics is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov-Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

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arXiv2017

On the categorical entropy and the topological entropy

To an exact endofunctor of a triangulated category with a split-generator, the notion of entropy is given by Dimitrov-Haiden-Katzarkov-Kontsevich, which is a (possibly negative infinite) real-valued function of a real variable. In this paper, we propose a conjecture which naturally generalizes the theorem by Gromov-Yomdin, and show that the categorical entropy of a surjective endomorphism of a smooth projective variety is equal to its topological entropy. Moreover, we compute the entropy of autoequivalences of the derived category in the case of the ample canonical or anti-canonical sheaf.

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arXiv2017

Orbifold Jacobian algebras for exceptional unimodal singularities

This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.

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arXiv2016

Orbifold Jacobian algebras for invertible polynomials

An important invariant of a polynomial $f$ is its Jacobian algebra defined by its partial derivatives. Let $f$ be invariant with respect to the action of a finite group of diagonal symmetries $G$. We axiomatically define an orbifold Jacobian $\mathbb{Z}/2\mathbb{Z}$-graded algebra for the pair $(f,G)$ and show its existence and uniqueness in the case, when $f$ is an invertible polynomial. In case when $f$ defines an ADE singularity, we illustrate its geometric meaning.

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arXiv2016

On simply-laced generalized root systems

We show the uniqueness and existence of the Euler form for a simply-laced generalized root system. This enables us to show that the Coxeter element for a simply-laced generalized root system is admissible in the sense of R.~W.~Carter. As an application, the isomorphism classes of simply-laced generalized root systems with positive definite Cartan forms are classified by Cartar's admissible diagrams associated to their Coxeter elements.

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