A survey on symplectic singularities and symplectic resolutions. (English) Zbl 1116.14008
This is a survey paper concerning the specialized problem in algebraic geometry called symplectic resolution. If a normal variety \(W\) is a symplectic variety, i.e. its smooth part admits a holomorphic symplectic form \(\omega \) and pull-back of this form to any resolution \(\pi :Z\to W\) extends to a holomorphic 2-form \(\Omega \) on \(Z.\) If additionally the extended 2-form \(\Omega \) is a symplectic form then \(\pi \) is called a symplectic resolution. In the paper under review, the author describes basic properties of symplectic normal varieties and symplectic resolutions. Namikawa’s work, quotient singularities, nilpotent orbit closures, birational geometry, symplectic moduli spaces and the list of some conjectures on symplectic resolutions: on finiteness, on deformations, on cohomology, on derived equivalence and on birational geometry.
Reviewer: Stanislaw Janeczko (Warszawa)
MSC:
| 14E05 | Rational and birational maps |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
References:
| [1] | Batyrev, V., Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 1-32 (1998) |
| [2] | Batyrev, V., Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc., 1, 5-33 (1999) · Zbl 0943.14004 · doi:10.1007/PL00011158 |
| [3] | Beauville, A., Fano contact manifolds and nilpotent orbits, Comment. Math. Helv, 73, 566-583 (1998) · Zbl 0946.53046 · doi:10.1007/s000140050069 |
| [4] | Beauville, A., Symplectic singularities, Invent. Math., 139, 541-549 (2000) · Zbl 0958.14001 · doi:10.1007/s002229900043 |
| [5] | Bezrukavnikov, R.; Kaledin, D., McKay equivalence for symplectic resolutions of singularities, Proc. Steklov Inst. Math., 246, 13-33 (2004) · Zbl 1137.14301 |
| [6] | Bialynicki-Birula, A., Some theorems on actions of algebraic groups, Ann. of Math. (2), 98, 480-497 (1973) · Zbl 0275.14007 · doi:10.2307/1970915 |
| [7] | Bottacin, F., Poisson structures on moduli spaces of sheaves over Poisson surfaces, Invent. Math., 121, 421-436 (1995) · Zbl 0829.14019 · doi:10.1007/BF01884307 |
| [8] | Burns, D.; Hu, Y.; Luo, T., Vector bundles and representation theory (Columbia, MO, 2002), 141-149 (2003) · Zbl 1080.14508 |
| [9] | Cho, Y.; Miyaoka, Y.; Shepherd-Barron, N.; Miyaoka; Mori, Higher dimensional birational geometry (Kyoto, 1997), 1-88 (2002) · Zbl 1063.14065 |
| [10] | Choy, J.; Kiem, Y.-H., Nonexistence of crepant resolution of some moduli spaces of sheaves on a K3 surface (2004) |
| [11] | Choy, J.; Kiem, Y.-H., On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface, Math. Z., 252, 557-575 (2006) · Zbl 1375.14046 · doi:10.1007/s00209-005-0866-x |
| [12] | Cohen, A. M., Finite quaternionic reflection groups, J. Algebra, 64, 293-324 (1980) · Zbl 0433.20035 · doi:10.1016/0021-8693(80)90148-9 |
| [13] | Collingwood, D.; Mc Govern, W., Nilpotent orbits in semi-simple Lie algebras (1993) · Zbl 0972.17008 |
| [14] | Druel, S., Singularités symplectiques, J. Algebraic Geom., 13, 427-439 (2004) · Zbl 1068.32018 |
| [15] | Fu, B., Symplectic resolutions for coverings of nilpotent orbits, C. R. Acad. Sci., 336, 159-162 (2003) · Zbl 1068.14055 |
| [16] | Fu, B., Symplectic resolutions for nilpotent orbits, Invent. Math., 151, 167-186 (2003) · Zbl 1072.14058 · doi:10.1007/s00222-002-0260-9 |
| [17] | Fu, B., Symplectic resolutions for nilpotent orbits (II), C. R. Acad. Sci., 337, 277-281 (2003) · Zbl 1073.14547 |
| [18] | Fu, B., Birational geometry in codimension 2 of symplectic resolutions (2004) |
| [19] | Fu, B., Extremal contractions, stratified Mukai flops and Springer maps (2006) · Zbl 1120.14039 |
| [20] | Fu, B., Mukai flops and deformations of symplectic resolutions, Math. Z., 253, 87-96 (2006) · Zbl 1098.14009 · doi:10.1007/s00209-005-0886-6 |
| [21] | Fu, B.; Namikawa, Y., Uniqueness of crepant resolutions and symplectic singularities, Ann. Inst. Fourier, 54, 1-19 (2004) · Zbl 1063.14018 · doi:10.5802/aif.2008 |
| [22] | Ginzburg, V.; Kaledin, D., Poisson deformations of symplectic quotient singularities, Adv. Math., 186, 1-57 (2004) · Zbl 1062.53074 · doi:10.1016/j.aim.2003.07.006 |
| [23] | Gordon, I., Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc., 35, 321-336 (2003) · Zbl 1042.16017 · doi:10.1112/S0024609303001978 |
| [24] | Guralnick, Robert M.; Saxl, J., Generation of finite almost simple groups by conjugates, J. Algebra, 268, 519-571 (2003) · Zbl 1037.20016 · doi:10.1016/S0021-8693(03)00182-0 |
| [25] | Hartshorne, R., Algebraic Geometry (1977) · Zbl 0367.14001 |
| [26] | Hesselink, W., Polarizations in the classical groups, Math. Z., 160, 217-234 (1978) · Zbl 0364.20048 · doi:10.1007/BF01237035 |
| [27] | Hu, Y., Geometric Invariant Theory and Birational Geometry (2005) |
| [28] | Hu, Y.; Yau, S.-T., HyperKähler manifolds and birational transformations, Adv. Theor. Math. Phys., 6, 557-574 (2002) · Zbl 1044.81105 |
| [29] | Huybrechts, D., Compact hyper-Kähler manifolds: basic results, Invent. Math., 135, 63-113 (1999) · Zbl 0953.53031 · doi:10.1007/s002220050280 |
| [30] | Kaledin, D., Symplectic singularities from the Poisson point of view, J. Reine Angew. Math. · Zbl 1121.53056 |
| [31] | Kaledin, D., Symplectic resolutions: deformations and birational maps (2000) |
| [32] | Kaledin, D., McKay correspondence for symplectic quotient singularities, Invent. math., 148, 150-175 (2002) · Zbl 1060.14020 · doi:10.1007/s002220100192 |
| [33] | Kaledin, D., On crepant resolutions of symplectic quotient singularities, Selecta Math. (N.S.), 9, 529-555 (2003) · Zbl 1066.14003 · doi:10.1007/s00029-003-0308-8 |
| [34] | Kaledin, D., Derived equivalence by quantization (2005) |
| [35] | Kaledin, D.; Lehn, M., Local structure of hyperKaehler singularities in O’Grady’s examples (2004) · Zbl 1160.14006 |
| [36] | Kaledin, D.; Lehn, M.; Sorger, C., Singular symplectic moduli spaces, Invent. Math., 164, 591-614 (2006) · Zbl 1096.14037 · doi:10.1007/s00222-005-0484-6 |
| [37] | Kawamata, Y., D-equivalence and K-equivalence, J. Differential Geom., 61, 147-171 (2002) · Zbl 1056.14021 |
| [38] | Kraft, H.; Procesi, C., Closures of conjugacy classes of matrices are normal, Invent. Math., 53, 227-247 (1979) · Zbl 0434.14026 · doi:10.1007/BF01389764 |
| [39] | Markman, E., Brill-Noether duality for moduli spaces of sheaves of \(K3\) surfaces, J. Algebr. Geom., 10, 623-694 (2001) · Zbl 1074.14525 |
| [40] | Mukai, S., Symplectic structure of the moduli space of sheaves on an abelian or \(K3\) surface, Invent. Math., 77, 101-116 (1984) · Zbl 0565.14002 · doi:10.1007/BF01389137 |
| [41] | Namikawa, Y., Deformation theory of singular symplectic n-folds, Math. Ann., 319, 597-623 (2001) · Zbl 0989.53055 · doi:10.1007/PL00004451 |
| [42] | Namikawa, Y., Extension of 2-forms and symplectic varieties, J. Reine Angew. Math., 539, 123-147 (2001) · Zbl 0996.53050 · doi:10.1515/crll.2001.070 |
| [43] | Namikawa, Y., A note on symplectic singularitie (2001) |
| [44] | Namikawa, Y., Birational geometry of symplectic resolutions of nilpotent orbits (2004) |
| [45] | Namikawa, Y., Birational geometry of symplectic resolutions of nilpotent orbits II (2004) |
| [46] | Namikawa, Y., Flops and Poisson deformations of symplectic varieties (2005) · Zbl 1148.14008 |
| [47] | Namikawa, Y., On deformations of Q-factorial symplectic varieties (2005) |
| [48] | O’Grady, K., Desingularized moduli spaces of sheaves on a \(K3\), J. reine angew. Math., 512, 49-117 (1999) · Zbl 0749.14030 · doi:10.1515/crll.1999.056 |
| [49] | O’Grady, K., A new six-dimensional irreducible symplectic variety, J. Algebraic Geom., 12, 435-505 (2003) · Zbl 1018.32028 · doi:10.1090/S1056-3911-03-00323-0 |
| [50] | Panyushev, D., Rationality of singularities and the Gorenstein property for nilpotent orbits, Funct. Anal. Appl., 25, 225-226 (1991) · Zbl 1094.14010 · doi:10.1007/BF01085494 |
| [51] | Verbitsky, M., Holomorphic symplectic geometry and orbifold singularities, Asian J. Math., 4, 553-563 (2000) · Zbl 1036.14007 |
| [52] | Wierzba, J., Contractions of symplectic varieties, J. Algebraic Geom., 12, 507-534 (2003) · Zbl 1094.14010 · doi:10.1090/S1056-3911-03-00318-7 |
| [53] | Wierzba, J.; Wisniewski, J. A., Small contractions of symplectic 4-folds, Duke Math. J., 120, 65-95 (2003) · Zbl 1036.14007 · doi:10.1215/S0012-7094-03-12013-X |
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