scholarly journals All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions

1998 ◽  
Vol 139 (2) ◽  
pp. 194-239 ◽  
Author(s):  
Dimitrios I. Dais ◽  
Martin Henk ◽  
Günter M. Ziegler
Keyword(s):  
Crepant Resolutions

scholarly journals Folding of Hitchin Systems and Crepant Resolutions

2021 ◽  
Author(s):  
Florian Beck ◽  
Ron Donagi ◽  
Katrin Wendland
Keyword(s):  
Fixed Point ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Trace Back ◽  
Dynkin Diagrams ◽  
Or Groups ◽  
Hitchin Systems ◽  
Singular Fibers ◽  
Crepant Resolutions ◽  
Intermediate Jacobian

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.

scholarly journals Non-commutative crepant resolutions for some toric singularities. II

2020 ◽  
Vol 14 (1) ◽  
pp. 73-103
Author(s):  
Špela Špenko ◽  
Michel Van den Bergh
Keyword(s):  
Crepant Resolutions

scholarly journals Crepant resolutions of ℂ3∕Z4 and the generalized Kronheimer construction (in view of the gauge/gravity correspondence)

2019 ◽  
Vol 145 ◽  
pp. 103467
Author(s):  
Ugo Bruzzo ◽  
Anna Fino ◽  
Pietro Fré ◽  
Pietro Antonio Grassi ◽  
Dimitri Markushevich
Keyword(s):  
Crepant Resolutions ◽  
Gauge Gravity

scholarly journals Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

2014 ◽  
Vol 328 (1) ◽  
pp. 83-130 ◽  
Author(s):  
Kwokwai Chan ◽  
Cheol-Hyun Cho ◽  
Siu-Cheong Lau ◽  
Hsian-Hua Tseng
Keyword(s):  
Crepant Resolutions

scholarly journals On hypersurface quotient singularities of dimension 4

2004 ◽  
Vol 2004 (48) ◽  
pp. 2547-2581
Author(s):  
Li Chiang ◽  
Shi-Shyr Roan
Keyword(s):  
Abelian Group ◽  
Hilbert Scheme ◽  
Alternating Group ◽  
Detailed Structure ◽  
Special Group ◽  
Standard Representation ◽  
Quotient Singularities ◽  
Crepant Resolutions

We consider geometrical problems on Gorenstein hypersurface orbifolds of dimensionn≥4through the theory of Hilbert scheme of group orbits. For a linear special groupGacting onℂn, we study theG-Hilbert schemeHilbG(ℂn)and crepant resolutions ofℂn/GforGtheA-type abelian groupAr(n). Forn=4, we obtain the explicit structure ofHilbAr(4)(ℂ4). The crepant resolutions ofℂ4/Ar(4)are constructed through their relation withHilbAr(4)(ℂ4), and the connections between these crepant resolutions are found by the “flop” procedure of 4-folds. We also make some primitive discussion onHilbG(ℂn)forGthe alternating group𝔄n+1of degreen+1with the standard representation onℂn; the detailed structure ofHilb𝔄4(ℂ3)is explicitly constructed.

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 Genus zero Gopakumar-Vafa invariants from open strings

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Andrés Collinucci ◽  
Andrea Sangiovanni ◽  
Roberto Valandro
Keyword(s):  
String Theory ◽  
Open String ◽  
Genus Zero ◽  
Zero Modes ◽  
Open Strings ◽  
Crepant Resolutions ◽  
M Theory

Abstract We propose a new way to compute the genus zero Gopakumar-Vafa invariants for two families of non-toric non-compact Calabi-Yau threefolds that admit simple flops: Reid’s Pagodas, and Laufer’s examples. We exploit the duality between M-theory on these threefolds, and IIA string theory with D6-branes and O6-planes. From this perspective, the GV invariants are detected as five-dimensional open string zero modes. We propose a definition for genus zero GV invariants for threefolds that do not admit small crepant resolutions. We find that in most cases, non-geometric T-brane data is required in order to fully specify the invariants.

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