Arkiv för Matematik

  • Ark. Mat.
  • Volume 52, Number 2 (2014), 355-365.

A modification of the Hodge star operator on manifolds with boundary

Ryszard L. Rubinsztein

Full-text: Open access

Abstract

For smooth compact oriented Riemannian manifolds M of dimension 4k+2, k≥0, with or without boundary, and a vector bundle F on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of M twisted by F. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when k=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.

Article information

Source
Ark. Mat., Volume 52, Number 2 (2014), 355-365.

Dates
Received: 6 December 2012
Revised: 11 September 2013
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485802679

Digital Object Identifier
doi:10.1007/s11512-013-0190-3

Mathematical Reviews number (MathSciNet)
MR3255144

Zentralblatt MATH identifier
1317.58004

Rights
2013 © Institut Mittag-Leffler

Citation

Rubinsztein, Ryszard L. A modification of the Hodge star operator on manifolds with boundary. Ark. Mat. 52 (2014), no. 2, 355--365. doi:10.1007/s11512-013-0190-3. https://projecteuclid.org/euclid.afm/1485802679


Export citation

References

  • Cappell, S., DeTurck, D., Gluck, H. and Miller, E. Y., Cohomology of harmonic forms on Riemannian manifolds with boundary, Forum Math. 18 (2006), 923–931.
  • DeTurck, D. and Gluck, H., Poincaré duality angles and Hodge decomposition for Riemannian manifolds, Preprint, 2004.
  • Guruprasad, K., Huebschmann, J., Jeffrey, L. and Weinstein, A., Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89 (1997), 377–412.
  • Morrey, C. B., A variational method in the theory of harmonic integrals, II, Amer. J. Math. 78 (1956), 137–170.
  • Weil, A., Remarks on the cohomology of groups, Ann. of Math. 80 (1964), 149–157.