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 $$\omega$$.

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)
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References:

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