Arkiv för Matematik
- Ark. Mat.
- Volume 52, Number 2 (2014), 355-365.
A modification of the Hodge star operator on manifolds with boundary
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.
Ark. Mat., Volume 52, Number 2 (2014), 355-365.
Received: 6 December 2012
Revised: 11 September 2013
First available in Project Euclid: 30 January 2017
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2013 © Institut Mittag-Leffler
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