Related papers: Deformed Calabi-Yau Completions

We shall develop a new deformation theory of geometric structures in terms of closed differential forms. This theory is a generalization of Kodaira -Spencer theory and further we obtain a criterion of unobstructed deformations. We apply…

By work of Davison and Meinhardt, the cohomological Hall algebra of a symmetric quiver with potential admits a geometrically defined filtration (the perverse filtration) whose associated graded is a supercommutative algebra. In the case of…

We study deformations of pairs (X,D), with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential graded Lie algebras theory and the…

Jake Goodman and Ulrich Kr\"ahmer have recently shown that a twisted Calabi-Yau algebra $A$ with modular automorphism $\sigma$ and dimension $d$ can be "untwisted," in the sense that the Ore extensions $A[X;\sigma]$ and $A[X^{\pm1};\sigma]$…

A graded quiver with superpotential is a quiver whose arrows are assigned degrees $c\in \{0, 1, \cdots, m\}$, for some integer $m \geq 0$, with relations generated by a superpotential of degree $m-1$. Ordinary quivers ($m=1)$ often describe…

The open string sector of the topological B-model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence generalizes the connection between CY $(m+2)$-folds and gauge theories on the worldvolume of…

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay…

Placing a set of branes at a Calabi-Yau singularity leads to an N=1 quiver gauge theory. We analyze F-term deformations of such gauge theories. A generic deformation can be obtained by making the Calabi-Yau non-commutative. We discuss…

The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries…

Let $G \leq \operatorname{SL}_{n+1}(\mathbb{C})$ act on $R = \mathbb{C}[X_1, \ldots, X_{n+1}]$ by change of variables. Then, the skew-group algebra $R \ast G$ is bimodule $(n+1)$-Calabi-Yau. Under certain circumstances, the algebra admits a…

We study moduli spaces of mirror non-compact Calabi-Yau threefolds enhanced with choices of differential forms. The differential forms are elements of the middle dimensional cohomology whose variation is described by a variation of mixed…

Given a quiver $Q$, a formal potential is called analytic if its coefficients are bounded by the terms of a geometric series. As shown by Toda, the potentials appearing in the deformation theory of complexes of coherent sheaves on complex…

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$.…

This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general…

We introduce a new notion of deformation of complex structure, which we use as an adaptation of Kodaira's theory of deformations, but that is better suited to the study of noncompact manifolds. We present several families of deformations…

Multiply-connected Calabi-Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli…

We prove that for $d \geq 2$, an algebraic $d$-Calabi-Yau triangulated category endowed with a $d$-cluster tilting subcategory is the stable category of a DG category which is perfectly $(d+1)$-Calabi-Yau and carries a non degenerate…

In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result…

We prove categorical systolic inequalities for the derived categories of 2-Calabi--Yau Ginzburg dg algebras associated to ADE quivers and explore their symplecto-geometric aspects.

We compare and generalise the various geometric constructions (due to Ringel, Lusztig, Schofield, Bozec, Davison...) of the unipotent generalised Kac-Moody algebra associated with an arbitrary quiver. These constructions are interconnected…