Related papers: Reduction of triangulated categories and Maximal M…
Modular invariance is known to constrain the spectrum of 2d conformal field theories. We investigate this constraint systematically, using the linear functional method to put new improved upper bounds on the lowest gap in the spectrum. We…
We show that the Calabi-Yau dimension of the stable module category of a selfinjective algebra of finite representation type is determined by the action of the Nakayama and suspension functors on objects. Our arguments are based on recent…
We study the cluster algebras arising from cluster tubes with rank bigger than $1$. Cluster tubes are $2-$Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid…
Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth…
Assume that $\D$ is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object $T$. We introduce the notion of relative cluster tilting objects, and $T[1]$-cluster tilting objects in $\D$, which are…
We develop the basic properties of $w$-simple-minded systems in $(-w)$-Calabi-Yau triangulated categories for $w \geq 1$. The main result is a reduction technique for negative Calabi-Yau triangulated categories. We show that the theory of…
We describe a reduction technique for stably 2-Calabi--Yau Frobenius extriangulated categories $\mathcal{F}$ with respect to a functorially finite rigid subcategory $\mathcal{X}$. The reduction of such a category is another category…
In this paper, we study boundedness questions for (simply-connected) smooth Calabi-Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology…
This is the extended version of the paper "Special Lagrangian conifolds, I: Moduli spaces", which discusses the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the…
This short note surveys the constructions of 3-Calabi--Yau triangulated categories with simple-minded collections due to Ginzburg and Kontsevich--Soibelman and the constructions of 2-Calabi--Yau triangulated categories with cluster-tilting…
We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution.…
For a self-orthogonal module $T$, the relation between the quotient triangulated category $D^b(A)/K^b({\rm add} T)$ and the stable category of the Frobenius category of $T$-Cohen-Macaulay modules is investigated. In particular, for a…
We study deformations of certain crepant resolutions of isolated rational Gorenstein singularities. After a general discussion of the deformation theory, we specialize to dimension $3$ and consider examples which are good (log) resolutions…
We construct a class of complete non-flat Calabi-Yau metrics on C^{N+1} for every N >= 3, which generalize the Taub-NUT metrics from C^2 and C^3 and whose tangent cone at infinity is R^N. The construction relies on the generalized…
We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and…
Our aim in this paper is to prove two results related to the three constructions of cluster categories: as orbit categories, as singularity categories and as cosingularity categories. In the first part of the paper, we prove the universal…
This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic…
We develop a general framework for $c$-vectors of 2-Calabi--Yau categories, which deals with cluster tilting subcategories that are not reachable from each other and contain infinitely many indecomposable objects. It does not rely on…
For a higher Nakayama algebra $A$ in the sense of Jasso-K\"{u}lshammer, we show that the singularity category of $A$ is triangulated equivalent to the stable module category of a self-injective higher Nakayama algebra. This generalizes a…
We introduce infinite discrete versions of the symmetric Nakayama representations by using techniques of persistence theory. After stabilising, we obtain a family triangulated categories which can be regarded as negative Calabi-Yau versions…