A modification of the Hodge star operator on manifolds with boundary

TL;DR

This paper introduces a modified Hodge star operator on manifolds with boundary, providing a canonical complex structure on certain cohomology groups and moduli spaces, compatible with symplectic forms.

Contribution

It constructs a new operator on parabolic cohomology that induces a natural complex structure compatible with existing symplectic forms.

Findings

Defines a modified Hodge star operator for manifolds with boundary.

Establishes a canonical complex structure on parabolic cohomology groups.

Provides a compatible almost complex structure on moduli spaces of flat connections.

Abstract

If M is a smooth compact oriented Riemannian manifold of dimension n=4k+2, with or without boundary, and F is a vector bundle on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the parabolic cohomology H^{2k+1}_{par}(M;F). The operator gives a canonical complex structure on H^{2k+1}_{par}(M;F) compatible with the symplectic form \omega given by the wedge product of forms in the middle dimension. In case when k=0 that gives a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface with or without boundary and monodromies along boundary components belonging to fixed conjugacy classes. The almost complex structure is compatible with the standard symplectic form \omega on the moduli space.