New symplectic \(V\)-manifolds of dimension four via the relative compactified Prymian. (English) Zbl 1138.14032
Three new examples of \(4\)-dimensional irreducible symplectic \(V\)-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-\(3\) curves with involution, and the third one is obtained from a Prymian by Mukai’s flop. They have the same singularities as two of Fujiki’s examples, namely, \(28\) isolated singular points analytically equivalent to the Veronese cone of degree \(8\), but a different Euler number. The family of curves used in this construction forms a linear system on a \(K3\) surface with involution. The structure morphism of both Prymians to the base of the family is a Lagrangian fibration in abelian surfaces with polarization of type \((1,2)\). No example of such fibration is known on nonsingular irreducible symplectic varieties.
Reviewer: Atanas Iliev (Sofia)
MSC:
| 14K30 | Picard schemes, higher Jacobians |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 53D30 | Symplectic structures of moduli spaces |
References:
| [1] | DOI: 10.1016/0001-8708(80)90043-2 · Zbl 0427.14015 · doi:10.1016/0001-8708(80)90043-2 |
| [2] | DOI: 10.1090/S0894-0347-02-00396-X · Zbl 1032.14003 · doi:10.1090/S0894-0347-02-00396-X |
| [3] | M. Artin, Global Analysis (Papers in Honor of K. Kodaira) (University of Tokyo Press, Tokyo, 1969) pp. 21–71. |
| [4] | W. Barth, Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Mathematics 10 (North-Holland, Amsterdam, 1987) pp. 41–84. |
| [5] | A. Beauville, Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progress in Mathematics 39 (Birkhauser, Boston, 1983) pp. 1–26. |
| [6] | Beauville A., J. Differential Geom. 18 pp 755– |
| [7] | A. Beauville, Problems in the Theory of Surfaces and Their Classification (Cortona, 1988) (Academic Press, London, 1991) pp. 25–31. |
| [8] | Brun J., Compositio Math. 53 pp 325– |
| [9] | Bloch S., Compositio Math. 39 pp 47– |
| [10] | DOI: 10.4134/JKMS.2007.44.1.035 · Zbl 1126.14051 · doi:10.4134/JKMS.2007.44.1.035 |
| [11] | DOI: 10.1007/s00209-005-0866-x · Zbl 1375.14046 · doi:10.1007/s00209-005-0866-x |
| [12] | P. R. Cook, Algebraic Geometry (Catania, 1993/Barcelona, 1994), Lecture Notes in Pure and Applied Mathematics 200 (Dekker, New York, 1998) pp. 37–47. |
| [13] | DOI: 10.1353/ajm.1999.0018 · Zbl 0956.14014 · doi:10.1353/ajm.1999.0018 |
| [14] | J.M. Drézet, Algebraic Group Actions and Quotients (Hindawi Publ. Corp., Cairo, 2004) pp. 39–89. |
| [15] | A. Fujiki, Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progress in Mathematics 39 (Birkhäuser Boston, Boston, MA, 1983) pp. 71–250. |
| [16] | DOI: 10.1007/BFb0080482 · Zbl 0212.26101 · doi:10.1007/BFb0080482 |
| [17] | Hassett B., Int. J. Math. 11 pp 1163– |
| [18] | Hassett B., Geom. Funct. Anal. 11 pp 1201– |
| [19] | DOI: 10.1007/978-3-663-11624-0 · doi:10.1007/978-3-663-11624-0 |
| [20] | Huybrechts D., J. Differential Geom. 45 pp 488– · Zbl 0917.53010 · doi:10.4310/jdg/1214459840 |
| [21] | Huybrechts D., Math. Ann. 326 pp 499– · Zbl 1023.14015 · doi:10.1007/s00208-003-0433-x |
| [22] | DOI: 10.1002/cpa.3160420403 · Zbl 0689.58020 · doi:10.1002/cpa.3160420403 |
| [23] | DOI: 10.1007/s00222-005-0484-6 · Zbl 1096.14037 · doi:10.1007/s00222-005-0484-6 |
| [24] | DOI: 10.2307/1971369 · Zbl 0592.14011 · doi:10.2307/1971369 |
| [25] | DOI: 10.1007/BFb0075462 · doi:10.1007/BFb0075462 |
| [26] | DOI: 10.1007/BF01265678 · Zbl 0812.57004 · doi:10.1007/BF01265678 |
| [27] | DOI: 10.1017/CBO9780511662652.016 · doi:10.1017/CBO9780511662652.016 |
| [28] | DOI: 10.1090/S1056-3911-06-00437-1 · Zbl 1156.14030 · doi:10.1090/S1056-3911-06-00437-1 |
| [29] | D. Luna, Sur les Groupes Algébriques (Soc. Math. France, Paris, 1973) pp. 81–105. |
| [30] | DOI: 10.1007/s00229-006-0631-4 · Zbl 1102.14031 · doi:10.1007/s00229-006-0631-4 |
| [31] | DOI: 10.1070/IM2003v067n01ABEH000421 · Zbl 1075.14040 · doi:10.1070/IM2003v067n01ABEH000421 |
| [32] | M. Maruyama, Vector Bundles on Algebraic Varieties, Pap. Colloq., Bombay 1984, Tata Institute of Fundamental Research Studies in Mathematics 11 (1987) pp. 275–339. |
| [33] | Moĭšezon B. G., Izv. Akad. Nauk. SSSR Ser. Mat. 30 pp 133– |
| [34] | Morrison D., Lectures delivered at the Scuola Matematica Interuniversitaria, in: The Geometry of K3 Surfaces (1988) |
| [35] | DOI: 10.1007/BF01389137 · Zbl 0565.14002 · doi:10.1007/BF01389137 |
| [36] | S. Mukai, Vector bundles on algebraic varieties (Bombay, 1984), Tata Institute of Fundamental Research Studies in Mathematics 11 (1987) pp. 341–413. |
| [37] | O’Grady K. G., J. Algebr. Geom. 6 pp 599– |
| [38] | O’Grady K. G., J. Reine Angew. Math. 512 pp 49– |
| [39] | DOI: 10.1090/S1056-3911-03-00323-0 · Zbl 1068.53058 · doi:10.1090/S1056-3911-03-00323-0 |
| [40] | DOI: 10.1215/S0012-7094-06-13413-0 · Zbl 1105.14051 · doi:10.1215/S0012-7094-06-13413-0 |
| [41] | DOI: 10.1007/BF01451409 · Zbl 0566.58028 · doi:10.1007/BF01451409 |
| [42] | DOI: 10.1007/BFb0068691 · doi:10.1007/BFb0068691 |
| [43] | Sawon J., Turk. J. Math. 27 pp 197– |
| [44] | Sawon J., J. Algebr. Geom. 16 pp 447– |
| [45] | DOI: 10.2307/2373709 · Zbl 0301.14011 · doi:10.2307/2373709 |
| [46] | DOI: 10.1007/BF02698887 · Zbl 0891.14005 · doi:10.1007/BF02698887 |
| [47] | DOI: 10.1007/s002080100255 · Zbl 1066.14013 · doi:10.1007/s002080100255 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
