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

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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.

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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.

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