## 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

#### 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
Revised: 11 September 2013
First available in Project Euclid: 30 January 2017

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