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 |
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