Stable vector bundles on algebraic surfaces

Li, Wei-Ping; Qin, Zhenbo

doi:10.1090/S0002-9947-1994-1260204-7

Stable vector bundles on algebraic surfaces
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by Wei-Ping Li and Zhenbo Qin PDF

Trans. Amer. Math. Soc. 345 (1994), 833-852 Request permission

Abstract:

We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of a certain moduli space of stable bundles on a rational ruled surface.

References

I. V. Artamkin, Deformation of torsion-free sheaves on an algebraic surface, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 435–468 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 449–485. MR 1072690, DOI 10.1070/IM1991v036n03ABEH002030
I. V. Artamkin, Stable bundles with $c_1=0$ on rational surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 227–241 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 2, 231–246. MR 1062511, DOI 10.1070/IM1991v036n02ABEH002019
W. Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63–91. MR 460330, DOI 10.1007/BF01389784
J. Eric Brosius, Rank-$2$ vector bundles on a ruled surface. I, Math. Ann. 265 (1983), no. 2, 155–168. MR 719134, DOI 10.1007/BF01460796
J. Eric Brosius, Rank-$2$ vector bundles on a ruled surface. II, Math. Ann. 266 (1983), no. 2, 199–214. MR 724737, DOI 10.1007/BF01458442
S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257–315. MR 1066174, DOI 10.1016/0040-9383(90)90001-Z
J.-M. Drezet and J. Le Potier, Fibrés stables et fibrés exceptionnels sur $\textbf {P}_2$, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 193–243 (French, with English summary). MR 816365
David Gieseker, A construction of stable bundles on an algebraic surface, J. Differential Geom. 27 (1988), no. 1, 137–154. MR 918460

Irreducibility of moduli of rank two vector bundles on algebraic surfaces

Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Hans Jürgen Hoppe and Heinz Spindler, Modulräume stabiler $2$-Bündel auf Regelflächen, Math. Ann. 249 (1980), no. 2, 127–140 (German). MR 578720, DOI 10.1007/BF01351410
Klaus Hulek, Stable rank-$2$ vector bundles on $\textbf {P}_{2}$ with $c_{1}$ odd, Math. Ann. 242 (1979), no. 3, 241–266. MR 545217, DOI 10.1007/BF01420729

On the classification of stable rank

vector bundles over the projective plane

Moduli spaces of stable rank two vector bundles and computations of Donaldson polynomials over rational ruled surfaces

Masaki Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ. 18 (1978), no. 3, 557–614. MR 509499, DOI 10.1215/kjm/1250522511

Vector bundles on

and torsion sheaves on the dual plane

The smooth classification of fake K

’s and similar surfaces

Ch. Okonek and A. Van de Ven, Stable bundles, instantons and $C^\infty$-structures on algebraic surfaces, Several complex variables, VI, Encyclopaedia Math. Sci., vol. 69, Springer, Berlin, 1990, pp. 197–249. MR 1095092
Zhenbo Qin, Moduli spaces of stable rank-$2$ bundles on ruled surfaces, Invent. Math. 110 (1992), no. 3, 615–626. MR 1189492, DOI 10.1007/BF01231346
Clifford Henry Taubes, Self-dual connections on $4$-manifolds with indefinite intersection matrix, J. Differential Geom. 19 (1984), no. 2, 517–560. MR 755237
Kang Zuo, Generic smoothness of the moduli spaces of rank two stable vector bundles over algebraic surfaces, Math. Z. 207 (1991), no. 4, 629–643. MR 1119961, DOI 10.1007/BF02571412