All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
Authors:
Gwyn Bellamy, Alastair Craw, Steven Rayan, Travis Schedler and Hartmut Weiss
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/827
Published electronically:
April 1, 2024
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We demonstrate that the linear quotient singularity for the exceptional subgroup $G$ in $\mathrm {Sp}(4,\mathbb {C})$ of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of $\mathbb {C}^4/G$ as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the $n$-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension $2n-6$; for example, we show that there are 1684 projective crepant resolutions when $n=6$. We also prove that the resulting affine cones are not quotient singularities for $n \geq 6$.
References
- The on-line encyclopedia of integer sequences, OEIS Foundation Inc., 2023, Published electronically at oeis.org.
- Marco Andreatta and Jarosław A. Wiśniewski, 4-dimensional symplectic contractions, Geom. Dedicata 168 (2014), 311–337. MR 3158045, DOI 10.1007/s10711-013-9832-7
- Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
- Arnaud Beauville, Symplectic singularities, Invent. Math. 139 (2000), no. 3, 541–549. MR 1738060, DOI 10.1007/s002229900043
- Gwyn Bellamy, On singular Calogero-Moser spaces, Bull. Lond. Math. Soc. 41 (2009), no. 2, 315–326. MR 2496507, DOI 10.1112/blms/bdp019
- Gwyn Bellamy, Counting resolutions of symplectic quotient singularities, Compos. Math. 152 (2016), no. 1, 99–114. MR 3453389, DOI 10.1112/S0010437X15007630
- Gwyn Bellamy and Alastair Craw, Birational geometry of symplectic quotient singularities, Invent. Math. 222 (2020), no. 2, 399–468. MR 4160872, DOI 10.1007/s00222-020-00972-9
- G. Bellamy, A. Craw, and T. Schedler, Birational geometry of quiver varieties and other GIT quotients, https://arxiv.org/abs/2212.09623, 2022.
- G. Bellamy, A. Craw, and T. Schedler, The semi-invariant ring as the Cox ring of a GIT quotient, 2023, Work in preparation.
- Gwyn Bellamy and Travis Schedler, A new linear quotient of $\textbf {C}^4$ admitting a symplectic resolution, Math. Z. 273 (2013), no. 3-4, 753–769. MR 3030675, DOI 10.1007/s00209-012-1028-6
- Gwyn Bellamy and Travis Schedler, On the (non)existence of symplectic resolutions of linear quotients, Math. Res. Lett. 23 (2016), no. 6, 1537–1564. MR 3621098, DOI 10.4310/MRL.2016.v23.n6.a1
- Gwyn Bellamy and Travis Schedler, Symplectic resolutions of quiver varieties, Selecta Math. (N.S.) 27 (2021), no. 3, Paper No. 36, 50. MR 4265941, DOI 10.1007/s00029-021-00647-0
- Gwyn Bellamy, Johannes Schmitt, and Ulrich Thiel, Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution, Math. Z. 300 (2022), no. 1, 661–681. MR 4359539, DOI 10.1007/s00209-021-02793-9
- Gwyn Bellamy, Johannes Schmitt, and Ulrich Thiel, On parabolic subgroups of symplectic reflection groups, Glasg. Math. J. 65 (2023), no. 2, 401–413. MR 4625992, DOI 10.1017/S0017089522000416
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
- Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster, Quantizations of conical symplectic resolutions, Astérisque 384 (2016), iii–iv (English, with English and French summaries). MR 3594663
- Michael Bulois, Christian Lehn, Manfred Lehn, and Ronan Terpereau, Towards a symplectic version of the Chevalley restriction theorem, Compos. Math. 153 (2017), no. 3, 647–666. MR 3705238, DOI 10.1112/S0010437X16008277
- Gilles Carron, On the quasi-asymptotically locally Euclidean geometry of Nakajima’s metric, J. Inst. Math. Jussieu 10 (2011), no. 1, 119–147. MR 2749573, DOI 10.1017/S1474748010000162
- Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 72877, DOI 10.2307/2372597
- Arjeh M. Cohen, Finite quaternionic reflection groups, J. Algebra 64 (1980), no. 2, 293–324. MR 579063, DOI 10.1016/0021-8693(80)90148-9
- William Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257–293. MR 1834739, DOI 10.1023/A:1017558904030
- William Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), no. 1, 55–79. MR 1957264, DOI 10.1007/s00208-002-0367-8
- William Crawley-Boevey and Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605–635. MR 1620538, DOI 10.1215/S0012-7094-98-09218-3
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
- Anda Degeratu and Rafe Mazzeo, Fredholm theory for elliptic operators on quasi-asymptotically conical spaces, Proc. Lond. Math. Soc. (3) 116 (2018), no. 5, 1112–1160. MR 3805053, DOI 10.1112/plms.12105
- Maria Donten-Bury and Jarosław A. Wiśniewski, On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32, Kyoto J. Math. 57 (2017), no. 2, 395–434. MR 3648055, DOI 10.1215/21562261-3821846
- Jonathan Fisher and Steven Rayan, Hyperpolygons and Hitchin systems, Int. Math. Res. Not. IMRN 6 (2016), 1839–1870. MR 3509942, DOI 10.1093/imrn/rnv172
- Laura Fredrickson, Perspectives on the asymptotic geometry of the Hitchin moduli space, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 018, 20. MR 3921918, DOI 10.3842/SIGMA.2019.018
- Laura Fredrickson, Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss, Asymptotic geometry of the moduli space of parabolic $\textrm {SL}(2,\Bbb C)$-Higgs bundles, J. Lond. Math. Soc. (2) 106 (2022), no. 2, 590–661. MR 4477200, DOI 10.1112/jlms.12581
- Fabio Gavarini, A Brauer algebra-theoretic proof of Littlewood’s restriction rules, J. Algebra 212 (1999), no. 1, 240–271. MR 1670662, DOI 10.1006/jabr.1998.7536
- Leonor Godinho and Alessia Mandini, Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles, Adv. Math. 244 (2013), 465–532. MR 3077880, DOI 10.1016/j.aim.2013.04.026
- M. Grab, Resolutions of quotient singularities and their Cox rings, Ph.D. Thesis, University of Warsaw, 2019.
- Megumi Harada and Nicholas Proudfoot, Hyperpolygon spaces and their cores, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1445–1467. MR 2115372, DOI 10.1090/S0002-9947-04-03522-6
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157
- Jürgen Hausen and Simon Keicher, A software package for Mori dream spaces, LMS J. Comput. Math. 18 (2015), no. 1, 647–659. MR 3418031, DOI 10.1112/S1461157015000212
- Jean-Claude Hausmann and Allen Knutson, Polygon spaces and Grassmannians, Enseign. Math. (2) 43 (1997), no. 1-2, 173–198. MR 1460127
- N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. MR 877637
- Jun Hu and Zhankui Xiao, On tensor spaces for Birman-Murakami-Wenzl algebras, J. Algebra 324 (2010), no. 10, 2893–2922. MR 2725207, DOI 10.1016/j.jalgebra.2010.08.017
- Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
- Yukari Ito and Miles Reid, The McKay correspondence for finite subgroups of $\textrm {SL}(3,\mathbf C)$, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 221–240. MR 1463181
- Dominic Joyce, Quasi-ALE metrics with holonomy $\textrm {SU}(m)$ and $\textrm {Sp}(m)$, Ann. Global Anal. Geom. 19 (2001), no. 2, 103–132. MR 1826397, DOI 10.1023/A:1010778214851
- D. Kaledin, McKay correspondence for symplectic quotient singularities, Invent. Math. 148 (2002), no. 1, 151–175. MR 1892847, DOI 10.1007/s002220100192
- D. Kaledin, On crepant resolutions of symplectic quotient singularities, Selecta Math. (N.S.) 9 (2003), no. 4, 529–555. MR 2031751, DOI 10.1007/s00029-003-0308-8
- D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. 600 (2006), 135–156. MR 2283801, DOI 10.1515/CRELLE.2006.089
- Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
- Steven L. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235–321. MR 2223410
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- Hiroshi Konno, On the cohomology ring of the hyperKähler analogue of the polygon spaces, Integrable systems, topology, and physics (Tokyo, 2000) Contemp. Math., vol. 309, Amer. Math. Soc., Providence, RI, 2002, pp. 129–149. MR 1953356, DOI 10.1090/conm/309/05345
- H. Kraft and C. Procesi, Classical invariant theory: a primer, 1996, Textbook at https://dmi.unibas.ch/fileadmin/user_upload/dmi/Personen/Kraft_Hanspeter/Classical_Invariant_Theory.pdf.
- P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683. MR 992334
- P. B. Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), no. 3, 685–697. MR 992335
- Manfred Lehn and Christoph Sorger, A symplectic resolution for the binary tetrahedral group, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 429–435 (English, with English and French summaries). MR 3203037
- Noppadol Mekareeya, The moduli space of instantons on an ALE space from 3d $\mathcal N=4$ field theories, J. High Energy Phys. 12 (2015), 174, front matter+29pp. MR 3464628, DOI 10.1007/jhep12(2015)174
- Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math. 77 (1984), no. 1, 101–116. MR 751133, DOI 10.1007/BF01389137
- Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, DOI 10.1215/S0012-7094-94-07613-8
- Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560. MR 1604167, DOI 10.1215/S0012-7094-98-09120-7
- Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. MR 1808477, DOI 10.1090/S0894-0347-00-00353-2
- Yoshinori Namikawa, Induced nilpotent orbits and birational geometry, Adv. Math. 222 (2009), no. 2, 547–564. MR 2538020, DOI 10.1016/j.aim.2009.05.001
- Yoshinori Namikawa, Poisson deformations of affine symplectic varieties, Duke Math. J. 156 (2011), no. 1, 51–85. MR 2746388, DOI 10.1215/00127094-2010-066
- Yoshinori Namikawa, Poisson deformations and birational geometry, J. Math. Sci. Univ. Tokyo 22 (2015), no. 1, 339–359. MR 3329199
- Rikito Ohta, On the relative version of Mori dream spaces, Eur. J. Math. 8 (2022), no. suppl. 1, S147–S181. MR 4452841, DOI 10.1007/s40879-022-00552-6
- Steven Rayan and Laura P. Schaposnik, Moduli spaces of generalized hyperpolygons, Q. J. Math. 72 (2021), no. 1-2, 137–161. MR 4271383, DOI 10.1093/qmath/haaa036
- Miles Reid, La correspondance de McKay, Astérisque 276 (2002), 53–72. Séminaire Bourbaki, Vol. 1999/2000. MR 1886756
- J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80 (French). MR 631309, DOI 10.1515/crll.1981.327.12
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
- Michael Thaddeus, Variation of moduli of parabolic Higgs bundles, J. Reine Angew. Math. 547 (2002), 1–14. MR 1900134, DOI 10.1515/crll.2002.051
- Misha Verbitsky, Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (2000), no. 3, 553–563. MR 1796694, DOI 10.4310/AJM.2000.v4.n3.a4
- Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, NJ, 1939. MR 255
References
- The on-line encyclopedia of integer sequences, OEIS Foundation Inc., 2023, Published electronically at oeis.org.
- Marco Andreatta and Jarosław A. Wiśniewski, 4-dimensional symplectic contractions, Geom. Dedicata 168 (2014), 311–337. MR 3158045, DOI 10.1007/s10711-013-9832-7
- Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR 3307753
- Arnaud Beauville, Symplectic singularities, Invent. Math. 139 (2000), no. 3, 541–549. MR 1738060, DOI 10.1007/s002229900043
- Gwyn Bellamy, On singular Calogero-Moser spaces, Bull. Lond. Math. Soc. 41 (2009), no. 2, 315–326. MR 2496507, DOI 10.1112/blms/bdp019
- Gwyn Bellamy, Counting resolutions of symplectic quotient singularities, Compos. Math. 152 (2016), no. 1, 99–114. MR 3453389, DOI 10.1112/S0010437X15007630
- Gwyn Bellamy and Alastair Craw, Birational geometry of symplectic quotient singularities, Invent. Math. 222 (2020), no. 2, 399–468. MR 4160872, DOI 10.1007/s00222-020-00972-9
- G. Bellamy, A. Craw, and T. Schedler, Birational geometry of quiver varieties and other GIT quotients, https://arxiv.org/abs/2212.09623, 2022.
- G. Bellamy, A. Craw, and T. Schedler, The semi-invariant ring as the Cox ring of a GIT quotient, 2023, Work in preparation.
- Gwyn Bellamy and Travis Schedler, A new linear quotient of $\mathbf {C}^4$ admitting a symplectic resolution, Math. Z. 273 (2013), no. 3-4, 753–769. MR 3030675, DOI 10.1007/s00209-012-1028-6
- Gwyn Bellamy and Travis Schedler, On the (non)existence of symplectic resolutions of linear quotients, Math. Res. Lett. 23 (2016), no. 6, 1537–1564. MR 3621098, DOI 10.4310/MRL.2016.v23.n6.a1
- Gwyn Bellamy and Travis Schedler, Symplectic resolutions of quiver varieties, Selecta Math. (N.S.) 27 (2021), no. 3, Paper No. 36, 50. MR 4265941, DOI 10.1007/s00029-021-00647-0
- Gwyn Bellamy, Johannes Schmitt, and Ulrich Thiel, Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution, Math. Z. 300 (2022), no. 1, 661–681. MR 4359539, DOI 10.1007/s00209-021-02793-9
- Gwyn Bellamy, Johannes Schmitt, and Ulrich Thiel, On parabolic subgroups of symplectic reflection groups, Glasg. Math. J. 65 (2023), no. 2, 401–413. MR 4625992, DOI 10.1017/S0017089522000416
- Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468. MR 2601039, DOI 10.1090/S0894-0347-09-00649-3
- Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster, Quantizations of conical symplectic resolutions, Astérisque 384 (2016), iii–iv (English, with English and French summaries). MR 3594663
- Michael Bulois, Christian Lehn, Manfred Lehn, and Ronan Terpereau, Towards a symplectic version of the Chevalley restriction theorem, Compos. Math. 153 (2017), no. 3, 647–666. MR 3705238, DOI 10.1112/S0010437X16008277
- Gilles Carron, On the quasi-asymptotically locally Euclidean geometry of Nakajima’s metric, J. Inst. Math. Jussieu 10 (2011), no. 1, 119–147. MR 2749573, DOI 10.1017/S1474748010000162
- Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 72877, DOI 10.2307/2372597
- Arjeh M. Cohen, Finite quaternionic reflection groups, J. Algebra 64 (1980), no. 2, 293–324. MR 579063, DOI 10.1016/0021-8693(80)90148-9
- William Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257–293. MR 1834739, DOI 10.1023/A:1017558904030
- William Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), no. 1, 55–79. MR 1957264, DOI 10.1007/s00208-002-0367-8
- William Crawley-Boevey and Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605–635. MR 1620538, DOI 10.1215/S0012-7094-98-09218-3
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
- Anda Degeratu and Rafe Mazzeo, Fredholm theory for elliptic operators on quasi-asymptotically conical spaces, Proc. Lond. Math. Soc. (3) 116 (2018), no. 5, 1112–1160. MR 3805053, DOI 10.1112/plms.12105
- Maria Donten-Bury and Jarosław A. Wiśniewski, On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32, Kyoto J. Math. 57 (2017), no. 2, 395–434. MR 3648055, DOI 10.1215/21562261-3821846
- Jonathan Fisher and Steven Rayan, Hyperpolygons and Hitchin systems, Int. Math. Res. Not. IMRN 6 (2016), 1839–1870. MR 3509942, DOI 10.1093/imrn/rnv172
- Laura Fredrickson, Perspectives on the asymptotic geometry of the Hitchin moduli space, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 018, 20. MR 3921918, DOI 10.3842/SIGMA.2019.018
- Laura Fredrickson, Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss, Asymptotic geometry of the moduli space of parabolic $\mathrm {SL}(2,\mathbb {C})$-Higgs bundles, J. Lond. Math. Soc. (2) 106 (2022), no. 2, 590–661. MR 4477200, DOI 10.1112/jlms.12581
- Fabio Gavarini, A Brauer algebra-theoretic proof of Littlewood’s restriction rules, J. Algebra 212 (1999), no. 1, 240–271. MR 1670662, DOI 10.1006/jabr.1998.7536
- Leonor Godinho and Alessia Mandini, Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles, Adv. Math. 244 (2013), 465–532. MR 3077880, DOI 10.1016/j.aim.2013.04.026
- M. Grab, Resolutions of quotient singularities and their Cox rings, Ph.D. Thesis, University of Warsaw, 2019.
- Megumi Harada and Nicholas Proudfoot, Hyperpolygon spaces and their cores, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1445–1467. MR 2115372, DOI 10.1090/S0002-9947-04-03522-6
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 463157
- Jürgen Hausen and Simon Keicher, A software package for Mori dream spaces, LMS J. Comput. Math. 18 (2015), no. 1, 647–659. MR 3418031, DOI 10.1112/S1461157015000212
- Jean-Claude Hausmann and Allen Knutson, Polygon spaces and Grassmannians, Enseign. Math. (2) 43 (1997), no. 1-2, 173–198. MR 1460127
- N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. MR 877637
- Jun Hu and Zhankui Xiao, On tensor spaces for Birman-Murakami-Wenzl algebras, J. Algebra 324 (2010), no. 10, 2893–2922. MR 2725207, DOI 10.1016/j.jalgebra.2010.08.017
- Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786494, DOI 10.1307/mmj/1030132722
- Yukari Ito and Miles Reid, The McKay correspondence for finite subgroups of $\mathrm {SL}(3,\mathbf C)$, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 221–240. MR 1463181
- Dominic Joyce, Quasi-ALE metrics with holonomy $\mathrm {SU}(m)$ and $\mathrm {Sp}(m)$, Ann. Global Anal. Geom. 19 (2001), no. 2, 103–132. MR 1826397, DOI 10.1023/A:1010778214851
- D. Kaledin, McKay correspondence for symplectic quotient singularities, Invent. Math. 148 (2002), no. 1, 151–175. MR 1892847, DOI 10.1007/s002220100192
- D. Kaledin, On crepant resolutions of symplectic quotient singularities, Selecta Math. (N.S.) 9 (2003), no. 4, 529–555. MR 2031751, DOI 10.1007/s00029-003-0308-8
- D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. 600 (2006), 135–156. MR 2283801, DOI 10.1515/CRELLE.2006.089
- Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
- Steven L. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235–321. MR 2223410
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- Hiroshi Konno, On the cohomology ring of the hyperKähler analogue of the polygon spaces, Integrable systems, topology, and physics (Tokyo, 2000) Contemp. Math., vol. 309, Amer. Math. Soc., Providence, RI, 2002, pp. 129–149. MR 1953356, DOI 10.1090/conm/309/05345
- H. Kraft and C. Procesi, Classical invariant theory: a primer, 1996, Textbook at https://dmi.unibas.ch/fileadmin/user_upload/dmi/Personen/Kraft_Hanspeter/Classical_Invariant_Theory.pdf.
- P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683. MR 992334
- P. B. Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), no. 3, 685–697. MR 992335
- Manfred Lehn and Christoph Sorger, A symplectic resolution for the binary tetrahedral group, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 429–435 (English, with English and French summaries). MR 3203037
- Noppadol Mekareeya, The moduli space of instantons on an ALE space from 3d $\mathcal N=4$ field theories, J. High Energy Phys. 12 (2015), 174, front matter+29pp. MR 3464628, DOI 10.1007/jhep12(2015)174
- Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math. 77 (1984), no. 1, 101–116. MR 751133, DOI 10.1007/BF01389137
- Hiraku Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. MR 1302318, DOI 10.1215/S0012-7094-94-07613-8
- Hiraku Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515–560. MR 1604167, DOI 10.1215/S0012-7094-98-09120-7
- Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. MR 1808477, DOI 10.1090/S0894-0347-00-00353-2
- Yoshinori Namikawa, Induced nilpotent orbits and birational geometry, Adv. Math. 222 (2009), no. 2, 547–564. MR 2538020, DOI 10.1016/j.aim.2009.05.001
- Yoshinori Namikawa, Poisson deformations of affine symplectic varieties, Duke Math. J. 156 (2011), no. 1, 51–85. MR 2746388, DOI 10.1215/00127094-2010-066
- Yoshinori Namikawa, Poisson deformations and birational geometry, J. Math. Sci. Univ. Tokyo 22 (2015), no. 1, 339–359. MR 3329199
- Rikito Ohta, On the relative version of Mori dream spaces, Eur. J. Math. 8 (2022), no. suppl. 1, S147–S181. MR 4452841, DOI 10.1007/s40879-022-00552-6
- Steven Rayan and Laura P. Schaposnik, Moduli spaces of generalized hyperpolygons, Q. J. Math. 72 (2021), no. 1-2, 137–161. MR 4271383, DOI 10.1093/qmath/haaa036
- Miles Reid, La correspondance de McKay, Astérisque 276 (2002), 53–72. Séminaire Bourbaki, Vol. 1999/2000. MR 1886756
- J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80 (French). MR 631309, DOI 10.1515/crll.1981.327.12
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
- Michael Thaddeus, Variation of moduli of parabolic Higgs bundles, J. Reine Angew. Math. 547 (2002), 1–14. MR 1900134, DOI 10.1515/crll.2002.051
- Misha Verbitsky, Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (2000), no. 3, 553–563. MR 1796694, DOI 10.4310/AJM.2000.v4.n3.a4
- Hermann Weyl, The classical groups. Their invariants and representations, Princeton University Press, Princeton, NJ, 1939. MR 255
Additional Information
Gwyn Bellamy
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom
MR Author ID:
857289
ORCID:
0000-0002-7045-4177
Email:
gwyn.bellamy@glasgow.ac.uk
Alastair Craw
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
MR Author ID:
683635
Email:
a.craw@bath.ac.uk
Steven Rayan
Affiliation:
Department of Mathematics and Statistics and Centre for Quantum Topology and Its Applications (quanTA), University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon SK, S7N 5E6, Canada
MR Author ID:
940734
ORCID:
0000-0003-0273-1598
Email:
rayan@math.usask.ca
Travis Schedler
Affiliation:
Imperial College London, Huxley Building, South Kensington Campus, London SW7 2AZ, United Kingdom
MR Author ID:
655427
ORCID:
0000-0001-7301-2932
Email:
t.schedler@imperial.ac.uk
Hartmut Weiss
Affiliation:
Mathematisches Seminar, Christian-Albrechts-Universität Kiel, Heinrich-Hecht-Platz 6, Kiel D-24118, Germany
MR Author ID:
777371
ORCID:
0000-0003-1928-2725
Email:
weiss@math.uni-kiel.de
Received by editor(s):
June 9, 2022
Received by editor(s) in revised form:
September 5, 2023
Published electronically:
April 1, 2024
Additional Notes:
The first and second authors were partially supported by Research Project Grant RPG-2021-149 from the Leverhulme Trust. The third author was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant. The fifth author was supported by the Deutsche Forschungsgemeinschaft (DFG) within SPP 2026 “Geometry at infinity”.
Article copyright:
© Copyright 2024
University Press, Inc.
