##
**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 \(\omega\).

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 \(\omega\).

Reviewer: Azniv Kasparian (Sofia)

### MSC:

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) |

### References:

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[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 |

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