Abstract:
In present article are construction general theory of single-valued functions and $q$-differentials on variable finite Riemann surfaces begun. We constructed all form elementary $q$-differentials. We found dimensions and constructed explicit basis in important two factor spaces. In particular, we found dimension and constructed basis in first holomorphic cohomology group de Rham for 1-differentials.
Keywords:Teichmueller space of finite Riemann surfaces, $q$-differentials, vector bundles, Jacobi manifolds.
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