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Cohomology relative to a system of closed forms on the torus. (English) Zbl 1368.58011
Consider on the torus \(\mathbb T^{m+n}\doteq \mathbb T^m_t\times \mathbb T^n_x\), \(t=(t_1,\dots,t_m)\), \(x=(x_1,\dots, x_n)\), a differential complex induced by the operator \[ \mathbb L_\Omega=d_t+\sum_{j=1}^n \omega^j\wedge \frac{\partial}{\partial x_j}, \] where \(\omega^1,\dots, \omega^n\) are real-valued, closed, one-forms on \(\mathbb T^m_t,\) and \(d_t\) is the exterior derivative.
Let \(A=(a_{kj})\) denote the \(m\times n\) matrix given by the periods of the one-forms above, that is, \(a_{kj}=\int_{\gamma_k}\omega^j\), where \(\gamma_1, \dots , \gamma_m\) are loops in \(\mathbb T^m\) such that their homotopy classes form a basis of the first homology group \(H_1(\mathbb T^m).\)
It is assumed that \(A\) satisfies the following Diophantine condition: there exist \(C>0\) and \(\rho>0\) such \[ \max\{|k_\ell+a_\ell\cdot J|, 1\leq \ell\leq m\}\geq C(|K|+|J|)^{-\rho}, \] for all tuples of integers \(K\in\mathbb T^m\) and \(J\in\mathbb T^n\) such that \(K+AJ\not=0,\) where \(a_\ell \) is the \(\ell\)-th row of \(A\) and \(k_\ell\) is the \(\ell\)-th coordinate of \(K\).
Let \(N=N(A)\) be the subgroup of \(\mathbb Z^m\) given by \(A(\mathbb Z^n)\cap \mathbb Z^m\). Let \(\{h_1,\dots, h_r\}\subset \mathbb Z^n\) be a set of generators of \(N\), where \(r\) is the rank of \(N\).
Let \(\Psi:\mathbb T^m\rightarrow \mathbb T^r\) be the linear map given by the matriz \((AH)^T\), where \(H\) is the \(n\times r\) matrix having \(h_1,\dots, h_r\) as its columns.
The main result of this nice paper is that the \(p\)-th cohomology group relative to \(\mathbb L_\Omega\), denoted by \(H^p_\Omega(\mathbb T^{m+n})\), is isomorphic to \(H^p(\mathbb T^m)\otimes_\Psi C^\infty(\mathbb T^r) \), the space generated by the forms \(\alpha\otimes \eta\) with \(\alpha \in \Psi^*(C^\infty(\mathbb T^r))\) and \(\eta\in H^p(\mathbb T^m)\), the \(p\)-th De-Rham cohomology group of \(\mathbb T^m\).

58J10 Differential complexes
35F35 Systems of linear first-order PDEs
Full Text: DOI
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