On the existence of crepant resolutions of Gorenstein abelian quotient singularities in dimensions ≥4

2006 ◽  
pp. 125-193 ◽  
Author(s):  
Dimitrios I. Dais ◽  
Martin Henk ◽  
Günter M. Ziegler
Keyword(s):  
Quotient Singularities ◽  
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 crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities

1998 ◽  
Vol 33 (3-4) ◽  
pp. 208-265 ◽  
Author(s):  
Dimitrios I. Dais ◽  
Utz-Uwe Haus ◽  
Martin Henk
Keyword(s):  
Quotient Singularities ◽  
Crepant Resolutions ◽  
Cyclic Quotient Singularities

scholarly journals The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

2021 ◽  
Vol 9 ◽  
Author(s):  
Patrick Graf ◽  
Martin Schwald
Keyword(s):  
Chern Class ◽  
Cohomology Group ◽  
Canonical Bundle ◽  
Deformation Space ◽  
Projective Varieties ◽  
Quotient Singularities ◽  
Finite Quotient ◽  
First Chern Class ◽  
Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

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.

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