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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 $$\mathbb T^m$$.

##### MSC:
 58J10 Differential complexes 35F35 Systems of linear first-order PDEs
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