Open Access
October 2014 A modification of the Hodge star operator on manifolds with boundary
Ryszard L. Rubinsztein
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Ark. Mat. 52(2): 355-365 (October 2014). DOI: 10.1007/s11512-013-0190-3


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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Ryszard L. Rubinsztein. "A modification of the Hodge star operator on manifolds with boundary." Ark. Mat. 52 (2) 355 - 365, October 2014.


Received: 6 December 2012; Revised: 11 September 2013; Published: October 2014
First available in Project Euclid: 30 January 2017

zbMATH: 1317.58004
MathSciNet: MR3255144
Digital Object Identifier: 10.1007/s11512-013-0190-3

Rights: 2013 © Institut Mittag-Leffler

Vol.52 • No. 2 • October 2014
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