A modification of the Hodge star operator on manifolds with boundary. (English) Zbl 1317.58004

Let \(M\) be a smooth compact oriented Riemannian manifold of dimension \(n = 4k+2\), \(k \geq 0\), with or without a boundary, and \(F\) be a real finite-dimensional vector bundle over \(M\) with an inner product \(B\) and a flat connection \(A\). The tensor product of the usual Hodge star operator \(\widehat{*} : \Omega ^m (M) \rightarrow \Omega ^{n-m} (M)\) on \(M\) with the identity \(\text{Id} _F\) of \(F\) is a Hodge star operator \(* = \widehat{*} \otimes \text{Id} _F : \Omega ^m (F) \rightarrow \Omega ^{n-m} (F)\) on the smooth \(F\)-valued differential forms \(\Omega ^* (F)\) on \(M\). Thus, the Riemannian metric and the orientation on \(M\), combined with the inner product \(B : F \times F \rightarrow {\mathbb R}\) provide an \(L^2\) inner product \((\alpha, \beta) := \int _M B( \alpha \wedge * \beta)\) on \(\Omega ^m (F) \ni \alpha , \beta\) for \(\forall 0 \leq m \leq n\).
The work under review constructs a modification \(J_{\text{par}}\) of the Hodge star operator on the middle-dimensional parabolic cohomology \(H^p _{\text{par}} (M; F)\), \(p = 2k+1\) of \(M\), twisted by \(F\). It shows that \(J_{\text{par}}\) can be interpreted as a canonical complex structure on \(H^p _{\text{par}} (M; F)\), which is orthogonal with respect to the inner product and its associated symplectic form \(\omega (J \text{ }, \text{ })\) on a \(J\)-invariant subspace \(V \subset \Omega ^p (F)\), containing an isomorphic copy of \(H^p_{\text{par}} (M; F)\). In particular, for \(k=0\) the construction provides a canonical almost complex structure on the smooth locus \(\mathcal{M}^{\text{smooth}}\) of a moduli space \(\mathcal{M}\) of flat connections on a Riemann surface \(S\), which is compatible with the standard symplectic form on \(\mathcal{M}\).
In order to formulate precisely, let us recall that the parabolic cohomology of \(M\) with coefficients in \(F\) are defined as the kernel \(H^* _{\text{par}} (M; F) := \ker [ r^* : H^* (M; F) \rightarrow H^* (\partial M; F)]\) of the restriction homomorphism \(r^*\) of the cohomologies of \(M\) with coefficients in \(F\) to the boundary \(\partial M\) of \(M\). One can identify \(H^* _{\text{par}} (M; F) = \text{im} [ j^* H^* (M, \partial M; F) \rightarrow H^* (M; F)]\) with the image of the relative cohomologies \(H^* (M, \partial M; F)\). The flat connection \(A\) on \(F\) gives rise to a uniquely determined differential \(d_A : \Omega ^m (F) \rightarrow \Omega ^{m+1} (F)\), \(d_A ^2 =0\) on the complex \(\Omega ^* (F)\). By the means of the Hodge star \(* : \Omega ^m (F) \rightarrow \Omega ^{n-m} (F)\), one defines the \(L^2\)-adjoint co-differential \(\delta _A = (-1) ^{n(p+1) +1} * d_A * : \Omega ^p (F) \rightarrow \Omega ^{p-1} (F)\) and considers the subspace \(CcC^p := \{ \omega \in \Omega ^p (F) \, | \, d \omega =0, \, \delta \omega =0 \}\) of the closed and co-closed \(F\)-valued \(p\)-forms on \(M\). An arbitrary \(\omega \in \Omega ^p (F)\) decomposes into a sum \(\omega = \omega _{\text{tan}} + \omega _{\text{norm}}\) of its tangential part \(\omega _{\text{tan}}\) and its normal part \(\omega _{\text{norm}}\) along the boundary \(\partial M\) of \(M\). The complex structure \(J = * | _{\Omega ^p (F)} : \Omega ^p (F) \rightarrow \Omega ^p (F)\) restricts to a complex structure \(J : CcC ^p \rightarrow CcC ^p\), interchanging the subspace \(CcC ^p _N := \{ \omega \in CcC^p \, | \, \omega _{\text{norm}} =0 \}\) of \(CcC^p\) with Neumann boundary condition \(\omega _{\text{norm}} =0\) with the subspace \(CcC _D ^p := \{ \omega \in CcC ^p \, | \, \omega _{\text{tan}} =0 \}\) with Dirichlet boundary condition \(\omega _{\text{tan}} =0\). Let \(P_N : CcC^p \rightarrow CcC ^p _N\) be the orthogonal projection on \(CcC_N^p\) and \(\mathcal{P}_N : CcC _D ^p \rightarrow CcC ^p _N\) be the restriction of \(P_N\) to \(CcC ^p _D\). The identification of \(H^p (M, \partial M; F)\) with \(CcC ^p _D\) and of \(H^p (M; F)\) with \(CcC ^p _N\) allows to identify \(H^p _{\text{par}} (M; F)\) with the image \(U := \mathcal{P}_N (CcC ^p _D) \subseteq CcC ^p _N\) of \(\mathcal{P}_N\). According to \(J (CcC ^p _D) = CcC ^p _N\) and \(J (CcC ^p _N) = CcC ^p _D\), the real subspace \(V\) of \(CcC^p\), generated by \(CcC_D ^p\) and \(CcC_N^p\) is \(J\)-invariant and can be viewed as a complex vector space. The inner product \((\text{ }, \text{ }) : \Omega ^p (F) \times \Omega ^p (F) \rightarrow {\mathbb R}\) restricts to an inner product on \(V \subseteq CcC ^p \subset \Omega ^p (F)\), with respect to which the complex structure \(J : V \rightarrow V\) is an isometry. The composition \(G = \pi _U J : U \rightarrow U\) of \(J : U \rightarrow V\) with the orthogonal projection \(\pi _U : V \rightarrow U\) is shown to be a skew-symmetric real operator with respect to \((\text{ }, \text{ })\). After checking that \(J(U) \cap U^{\perp} =0\), the author establishes that \(G ^2 : U \rightarrow U\) is a negatively definite symmetric linear operator and defines \(J_{\text{par}} := (-G^2) ^{ - \frac{1}{2}} G\) as the composition of \(G\) with the inverse of the positive square root \((-G^2) ^{ \frac{1}{2}}\) of the positive definite linear operator \((-G^2) : U \rightarrow U\). The article proves that \(J_{\text{par}} : H^p _{\text{par}} (M; F) \rightarrow H^p _{\text{par}} (M; F)\) with \(J_{\text{par}} ^2 = - \text{Id} \) is compatible with the restriction of the symplectic form \(\omega (u,v) = (Ju, v)\) to \(U \ni u,v\), i.e., \(\omega (J_{\text{par}} (u), J_{\text{par}} (v)) = \omega (u,v)\) for \(\forall u, v \in H^p _{\text{par}} (M; F)\). Moreover, \(\omega (u, J_{\text{par}} (u)) >0\) for all \(u \in H^p _{\text{par}} (M; F) \setminus \{ 0 \}\). In the case of an empty boundary, the parabolic cohomologies \(H^* _{\text{par}} (M; F) := \ker [ r^* : H^* (M; F) \rightarrow H^* ( \partial M; F)] = H^* (M; F)\) coincide with the ordinary ones and \(H^p (M; F)\) can be identified with the space \(CcC ^p _N = CcC ^p _D = U = V = CcC ^p\) of the harmonic \(F\)-valued \(p\)-forms. The modified Hodge star operator \(J_{\text{par}} = G = J\) reduces to the ordinary Hodge star operator \(J\) on \(CcC ^p \simeq H^p (M; F)\).
Let \(S\) be a smooth compact oriented surface with \(s \geq 0\) irreducible boundary components and \(G\) be a compact Lie group, whose Lie algebra \(\mathfrak{g}\) is endowed with a positive definite inner product \((\text{ }, \text{ }) : \mathfrak{g} \times \mathfrak{g} \rightarrow {\mathbb R}\), invariant under the adjoint action of \(G\) on \(\mathfrak{g}\). Denote by \(\mathcal{M} = \mathcal{M} (S; G, C_1, \ldots , C_s)\) the moduli space of the gauge equivalence classes of the flat connections on \(S \times G \rightarrow S\), whose monodromy around the \(i\)-th irreducible component of \(\partial S\) belongs to the conjugacy class \(C_i\) of \(G\) for all \(1 \leq i \leq s\). The smooth points of \(\mathcal{M}\) are represented by group homomorphisms \(\phi : \pi _1 (S) \rightarrow G\), transforming the monodromy around the \(i\)-th component of \(\partial S\) in \(C_i \subset G\). The trivial bundle \(\widetilde{S} \times \mathfrak{g} \rightarrow \widetilde{S}\) with fibre \(\mathfrak{g}\) over the universal cover \(\widetilde{S}\) of \(S\) is acted by the fundamental group \(\pi _1 (S)\) of \(S\) through the adjoint representation of \(\phi ( \pi _1 (S))\) on \(\mathfrak{g}\). Its \(\pi _1 (S)\)-quotient \(\mathfrak{g}_{\phi}\) is a flat vector bundle over \(S = \widetilde{S} / \pi _1 (S)\) with a real valued positive-definite inner product. The tangent space to \(\mathcal{M}\) at a smooth point \([\phi] \in \mathcal{M}^{\text{smooth}}\) can be identified with the parabolic cohomology group \(H^1 _{\text{par}} (S; \mathfrak{g} _{\phi})\). The inner product on \(\mathfrak{g}_{\phi}\) and the wedge product of forms determine a natural symplectic form \(\omega\) on \(\mathcal{M} ^{\text{smooth}}\). For an arbitrary Riemannian metric on \(S\), the construction of the article provides an almost complex structure \(J_{\text{par}}\) on \(\mathcal{M} ^{\text{smooth}}\), which is compatible with \(\omega\).


58A14 Hodge theory in global analysis
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
Full Text: DOI arXiv


[1] Cappell, S., DeTurck, D., Gluck, H. and Miller, E. Y., Cohomology of harmonic forms on Riemannian manifolds with boundary, Forum Math.18 (2006), 923-931. · Zbl 1114.53031 · doi:10.1515/FORUM.2006.046
[2] DeTurck, D. and Gluck, H., Poincaré duality angles and Hodge decomposition for Riemannian manifolds, Preprint, 2004.
[3] Guruprasad, K., Huebschmann, J., Jeffrey, L. and Weinstein, A., Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J.89 (1997), 377-412. · Zbl 0885.58011 · doi:10.1215/S0012-7094-97-08917-1
[4] Morrey, C. B., A variational method in the theory of harmonic integrals, II, Amer. J. Math.78 (1956), 137-170. · Zbl 0070.31402 · doi:10.2307/2372488
[5] Weil, A., Remarks on the cohomology of groups, Ann. of Math.80 (1964), 149-157. · Zbl 0192.12802 · doi:10.2307/1970495
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.