A modification of the Hodge star operator on manifolds with boundary

Ryszard L. Rubinsztein Department of Mathematics, Uppsala University

Geometric Analysis and Geometric Topology Rings and Algebras mathscidoc:1701.15005

Arkiv for Matematik, 52, (2), 355-365, 2012.12
For smooth compact oriented Riemannian manifolds$M$of dimension 4$k$+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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@inproceedings{ryszard2012a,
  title={A modification of the Hodge star operator on manifolds with boundary},
  author={Ryszard L. Rubinsztein},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203638234704066},
  booktitle={Arkiv for Matematik},
  volume={52},
  number={2},
  pages={355-365},
  year={2012},
}
Ryszard L. Rubinsztein. A modification of the Hodge star operator on manifolds with boundary. 2012. Vol. 52. In Arkiv for Matematik. pp.355-365. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203638234704066.
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