Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Modular forms and Donaldson invariants for 4-manifolds with $b_+=1$


Author: Lothar Göttsche
Journal: J. Amer. Math. Soc. 9 (1996), 827-843
MSC (1991): Primary 57R55; Secondary 11F11, 14D20
DOI: https://doi.org/10.1090/S0894-0347-96-00212-3
MathSciNet review: 1362873
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257–315. MR 1066174, DOI https://doi.org/10.1016/0040-9383%2890%2990001-Z
  • Ellingsrud, G. and Göttsche, L., Variation of moduli spaces and Donaldson invariants under change of polarization, J. reine angew. Math. 467 (1995), 1–49.
  • Ellingsrud, G. and Göttsche, L., Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces, preprint 1995.
  • Ellingsrud, G., Le Potier, J. and Strømme, S. A., Some Donaldson Invariants of $\C \P ^2$, preprint 1995.
  • Robert Friedman and John W. Morgan, On the diffeomorphism types of certain algebraic surfaces. I, J. Differential Geom. 27 (1988), no. 2, 297–369. MR 925124
  • Friedman, R., Qin, Z., Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces, preprint 1994.
  • Fintushel, R., Stern, R.J., The blowup formula for Donaldson invariants, preprint 1994.
  • Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR 1189136
  • Hyun, S., Park, J.S. Holomorphic Yang-Mills Theory and Variation of the Donaldson Invariants, hep-th/9503036.
  • D. Kotschick, ${\rm SO}(3)$-invariants for $4$-manifolds with $b^+_2=1$, Proc. London Math. Soc. (3) 63 (1991), no. 2, 426–448. MR 1114516, DOI https://doi.org/10.1112/plms/s3-63.2.426
  • Kotschick, D., Lisca, P., Instanton Invariants of $\C \P ^2$ via Topology, Math. Ann. 303 (1995), 345-371.
  • D. Kotschick and J. W. Morgan, ${\rm SO}(3)$-invariants for $4$-manifolds with $b^+_2=1$. II, J. Differential Geom. 39 (1994), no. 2, 433–456. MR 1267898
  • P. B. Kronheimer and T. S. Mrowka, Recurrence relations and asymptotics for four-manifold invariants, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 215–221. MR 1246469, DOI https://doi.org/10.1090/S0273-0979-1994-00492-6
  • Leness, T. Blow-up formulae for $SO(3)$-Donaldson invariants, preprint.
  • Wei-Ping Li and Zhenbo Qin, Lower-degree Donaldson polynomial invariants of rational surfaces, J. Algebraic Geom. 2 (1993), no. 3, 413–442. MR 1211994
  • Jun Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom. 37 (1993), no. 2, 417–466. MR 1205451
  • Morgan, J., Ozsváth, P., private communication.
  • Peter S. Ozsváth, Some blowup formulas for ${\rm SU}(2)$ Donaldson polynomials, J. Differential Geom. 40 (1994), no. 2, 411–447. MR 1293659
  • Pidstrigach, V.Y. Tyurin, A.N., in preparation.
  • Zhenbo Qin, Equivalence classes of polarizations and moduli spaces of sheaves, J. Differential Geom. 37 (1993), no. 2, 397–415. MR 1205450
  • Zhenbo Qin, Moduli of stable sheaves on ruled surfaces and their Picard groups, J. Reine Angew. Math. 433 (1992), 201–219. MR 1191606, DOI https://doi.org/10.1515/crll.1992.433.201
  • Robert A. Rankin, Modular forms and functions, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0498390
  • N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory, Nuclear Phys. B 426 (1994), no. 1, 19–52. MR 1293681, DOI https://doi.org/10.1016/0550-3213%2894%2990124-4
  • Stern, R., Computing Donaldson Invariants, Lectures for the Park City Geometry Institute, 1994.
  • Taubes, C.H., in preparation.
  • Witten, E., Monopoles and Four-Manifolds, Math. Res. Lett. 1 (1994), 769–796.
  • Witten, E., On $S$-Duality in Abelian Gauge Theory, hep-th/9505186.
  • Kaneko, M., Zagier, D., A generalized Jacobi theta function and quasimodular forms, preprint.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 57R55, 11F11, 14D20

Retrieve articles in all journals with MSC (1991): 57R55, 11F11, 14D20


Additional Information

Lothar Göttsche
Affiliation: Max–Planck–Institut für Mathematik Gottfried–Claren–Straße 26, D-53225 Bonn, Germany
Address at time of publication: Dipartimento di Matematica, Via Buonarroti 2, I-56127 Pisa, Italy
MR Author ID: 288886
Email: goettsch@dm.unipi.it

Keywords: Blowup formulas, Donaldson invariants, modular forms
Received by editor(s): August 4, 1995
Article copyright: © Copyright 1996 American Mathematical Society