scholarly journals On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

2021 ◽  
Author(s):  
Wahei Hara ◽  
Keyword(s):  
Finite Length ◽  
Moduli Spaces ◽  
Homogeneous Spaces ◽  
Alternative Proof ◽  
Crepant Resolution ◽  
Derived Equivalence ◽  
Orthogonal Decompositions ◽  
Crepant Resolutions

The Abuaf-Ueda flop is a 7-dimensional flop related to G<sub>2</sub> homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

scholarly journals On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties

2020 ◽  
pp. 2050034
Author(s):  
Salvatore Floccari ◽  
Lie Fu ◽  
Ziyu Zhang
Keyword(s):  
Recent Result ◽  
Moduli Spaces ◽  
Betti Number ◽  
Defect Group ◽  
Crepant Resolution ◽  
Tate Conjecture ◽  
Chow Motive ◽  
Crepant Resolutions ◽  
The Difference ◽  
Cubic Fourfold

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O’Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of (semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kähler varieties of O’Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford–Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kähler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford–Tate conjecture 7.3 holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.

scholarly journals Crepant resolutions and open strings

2019 ◽  
Vol 2019 (755) ◽  
pp. 191-245 ◽  
Author(s):  
Andrea Brini ◽  
Renzo Cavalieri ◽  
Dustin Ross
Keyword(s):  
Vector Space ◽  
Crepant Resolution ◽  
Complete Proof ◽  
Open Strings ◽  
Witten Theory ◽  
Crepant Resolutions ◽  
Symplectic Vector Space ◽  
Crepant Resolution Conjecture ◽  
Minimal Resolutions ◽  
Type Statement

AbstractIn the present paper, we formulate a Crepant Resolution Correspondence for open Gromov–Witten invariants (OCRC) of toric Lagrangian branes inside Calabi–Yau 3-orbifolds by encoding the open theories into sections of Givental’s symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan–Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates–Corti–Iritani–Tseng and Ruan, we furthermore propose (1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and (2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold {A_{n}}-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov–Witten theory of this family of targets.

scholarly journals CONSISTENCY CONDITIONS FOR DIMER MODELS

2012 ◽  
Vol 54 (2) ◽  
pp. 429-447 ◽  
Author(s):  
RAF BOCKLANDT
Keyword(s):  
Crepant Resolution ◽  
Dimer Model ◽  
Consistency Conditions ◽  
Crepant Resolutions ◽  
Dimer Models

AbstractDimer models are a combinatorial tool to describe certain algebras that appear as noncommutative crepant resolutions of toric Gorenstein singularities. Unfortunately, not every dimer model gives rise to a noncommutative crepant resolution. Several notions of consistency have been introduced to deal with this problem. In this paper, we study the major different notions in detail and show that for dimer models on a torus, they are all equivalent.

scholarly journals Variation of geometric invariant theory quotients and derived categories

2019 ◽  
Vol 2019 (746) ◽  
pp. 235-303 ◽  
Author(s):  
Matthew Ballard ◽  
David Favero ◽  
Ludmil Katzarkov
Keyword(s):  
Moduli Spaces ◽  
Invariant Theory ◽  
General Framework ◽  
Derived Categories ◽  
Rational Curves ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Coherent Sheaves ◽  
Orthogonal Decompositions ◽  
The Relationship

Abstract We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed to give applications of this complete description. In this setting, we verify a question posed by Kawamata: we show that D-equivalence and K-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne–Mumford stacks. This recovers Kawamata’s theorem that all projective toric Deligne–Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover and extend Orlov’s σ-model/Landau–Ginzburg model correspondence.

scholarly journals Crepant resolutions and open strings II

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Andrea Brini ◽  
Renzo Cavalieri
Keyword(s):  
Three Dimensional ◽  
Type A ◽  
Projective Planes ◽  
Closed String ◽  
The Body ◽  
The Other ◽  
Crepant Resolution ◽  
Open Strings ◽  
The Family ◽  
Crepant Resolutions

We recently formulated a number of Crepant Resolution Conjectures (CRC) for open Gromov-Witten invariants of Aganagic-Vafa Lagrangian branes and verified them for the family of threefold type A-singularities. In this paper we enlarge the body of evidence in favor of our open CRCs, along two different strands. In one direction, we consider non-hard Lefschetz targets and verify the disk CRC for local weighted projective planes. In the other, we complete the proof of the quantized (all-genus) open CRC for hard Lefschetz toric Calabi-Yau three dimensional representations by a detailed study of the G-Hilb resolution of $[C^3/G]$ for $G=\mathbb{Z}_2 \times \mathbb{Z}_2$. Our results have implications for closed-string CRCs of Coates-Iritani-Tseng, Iritani, and Ruan for this class of examples. Comment: v2: typos fixed, minor changes. v3: some minor points have been clarified, further typos fixed. v4: version accepted for publication on EPIGA

scholarly journals Calabi–Yau Quotients of Hyperkähler Four-folds

2019 ◽  
Vol 71 (1) ◽  
pp. 45-92 ◽  
Author(s):  
Chiara Camere ◽  
Alice Garbagnati ◽  
Giovanni Mongardi
Keyword(s):  
Linear Systems ◽  
Hilbert Scheme ◽  
General Setting ◽  
K3 Surface ◽  
Crepant Resolution ◽  
Hodge Numbers ◽  
Crepant Resolutions

AbstractThe aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non-symplectic involution $\unicode[STIX]{x1D6FC}$. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$, and the involution $\unicode[STIX]{x1D6FC}$ is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold $Y_{S}$, which is the crepant resolution of $X/\unicode[STIX]{x1D6FC}$, with the Calabi–Yau 4-fold $Z_{S}$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_{S}$ to $Y_{S}$.

scholarly journals Counting the Number of Elements in the Mutation Classes of $\tilde A_n-$Quivers

10.37236/585 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Janine Bastian ◽  
Thomas Prellberg ◽  
Martin Rubey ◽  
Christian Stump
Keyword(s):  
Equivalence Classes ◽  
Alternative Proof ◽  
Derived Equivalence ◽  
Tilted Algebras ◽  
Dynkin Type ◽  
Explicit Formulae

In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type $\tilde A_n$ in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type $\tilde A_n$. As a by-product, this provides an alternative proof for the number of quivers mutation equivalent to a quiver of Dynkin type $D_n$ which was first determined by Buan and Torkildsen.

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