×

Moduli of vector bundles on projective surfaces: Some basic results. (English) Zbl 0869.14005

The author studies two important questions concerning the moduli space of vector bundles on projective surfaces. Let \(S\) be a polarized projective surface and let \(M(r, c_1, c_2)\) be the moduli space of rank \(r\) stable vector bundles of first Chern class \(c_1\in \text{Pic} (S)\) and second Chern class \(c_2 \in H^4 (S,\mathbb{Z})\). The first question is whether \(M(r, c_1, c_2)\) has the expected dimension and is smooth at its general point. The second question is whether it is irreducible. The answer to both questions are as follows: There is a constant \(C\) depending on \(S\), on the polarization of \(S\) and on \(r\) and \(c_1\) such that whenever \(c_2\geq C\), then \(M(r,c_1, c_2)\) is smooth at its general point, it has the expected dimension and is irreducible. The case \(r=2\) was known for quite a while and the general case was established recently. The draw back of the previous works are that they only established the existence of such constant but gave no effective bound of it. In this paper, the author gives an effective bound of such constant. The estimate is close to the optimal bound in some special cases.
Reviewer: Jun Li (Stanford)

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J10 Families, moduli, classification: algebraic theory
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] [AB] M.F. Atiyah, R. Bott: The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London, Series A.308, 523-615 (1983) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017
[2] [BPV] W. Barth, C. Peters, A. Van de Ven: Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete3. Folge-Band4, (1984)
[3] [Bo] E. Bombieri: Canonical models of surfaces of general type. Publ. Math. Inst. Hautes Etud. Sci.42, 171-219 (1973) · Zbl 0259.14005 · doi:10.1007/BF02685880
[4] [D] S.K. Donaldson: Polynomial invariants for smooth four-manifolds. Topology29, 257-315 (1990) · Zbl 0715.57007 · doi:10.1016/0040-9383(90)90001-Z
[5] [DL] J. M. Drezet, J. Le Potier: Fibrés stables et fibrés exceptionnels sur le plan projectif. Ann. Sci. Ec. Norm. Sup. 4 e série,18, 193-244 (1985)
[6] [DN] J.M. Drezet, M.S. Narasimhan: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math.97, 53-94 (1989) · Zbl 0689.14012 · doi:10.1007/BF01850655
[7] [F1] H. Flenner: Restrictions of semistable bundles on projective varieties. Comment. Math. Helv.59, 635-650 (1984) · Zbl 0599.14015 · doi:10.1007/BF02566370
[8] [Fr] R. Friedman: Vector bundles on surfaces (to be published)
[9] [G1] D. Gieseker: On the moduli of vector bundles on an algebraic surface. Ann. Math.106, 45-60 (1977) · Zbl 0381.14003 · doi:10.2307/1971157
[10] [G2] D. Gieseker: A degeneration of the moduli space of stable bundles. J. Differ. Geom.19, 173-206 (1984) · Zbl 0557.14008
[11] [GL1] D. Gieseker, J. Li: Irreducibility of moduli of rank two vector bundles. (to appear in J. Diff. Geom.)
[12] [GL2] D. Gieseker, J. Li: Moduli of vector bundles over surfaces I (Preprint)
[13] [G] A. Grothendieck: Techniques de construction et théorèmes d’existence en géometrie algebrique IV: les schémas de Hilbert. Sém. Bourbaki221, (1960)
[14] [HL] A. Hirschowitz, Y. Laszlo: A propos de l’existence de fibrés stables sur les surfaces (Preprint)
[15] [I] A. Iarrobino: Punctual Hilbert schemes. Bull. Am. Math. Soc.78, 819-823 (1972) · Zbl 0268.14002 · doi:10.1090/S0002-9904-1972-13049-0
[16] [LP] J. Le Potier: Espaces de modules de faisceaux semi-stables sur le plan projectif (Preprint) School ?Vector bundles on surfaces?-CIMI and Europroj, Nice-Sophia-Antipolis June 1993
[17] [Li] J. Li: Algebraic geometric interpretation of Donaldson’s polynomial invariants. J. Differ. Geom.37, 417-466 (1993) · Zbl 0809.14006
[18] [LQ] W.P. Li, Z. Qin: Stable vector bundles on algebraic surfaces (Preprint)
[19] [Lu] D. Luna: Slices Étales. Bull. Soc. Math. France, Mémoire33, (1973)
[20] [Ma] M. Maruyama: Moduli of stable sheaves II. J. Math. Kyoto Univ.18-3, 557-614 (1978) · Zbl 0395.14006
[21] [MO] J. Morgan, K.G. O’Grady: Differential topology of complex surfaces Elliptic surfaces withp g =1: smooth classification. Lecture Notes in Mathematics 1545, Springer-Verlag
[22] [Mu] S. Mukai: On the moduli space of bundles onK3 surfaces I, Vector bundles on algebraic varieties. Tata Institute of fundamental research studies in mathematics, Oxford University Press, 1987
[23] [O] K.G. O’Grady: The irreducible components of moduli spaces of vector bundles on surfaces. Invent. Math.112, 585-613 (1993) · Zbl 0816.14018 · doi:10.1007/BF01232448
[24] [S] C. Simpson: Moduli of representations of the fundamental group of a smooth projective variety I. (Preprint) · Zbl 0891.14005
[25] [Zuo] K. Zuo: Generic smoothness of the moduli of rank two stable bundles over an algebraic surface. (preprint MPI/90-7)
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.