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A new cohomology on special kinds of complex manifolds. (English) Zbl 1054.57031
Let \(\mathcal V\) and \(\mathcal V'\) be vector spaces. For an arbitrary mapping \(f:\mathcal V^{k-1}\to\mathcal V'\) (\(k>1\)), I. B. Risteski, K. G. Trenčevski and V. C. Covachev defined a mapping \(\Psi f:\mathcal V^k\to\mathcal V'\) in [New York J. Math. 5, 139–142 (1999; Zbl 0933.39047)] by \((\Psi f)((X_1,X_2,\dots,X_{k-1},X_k) =(-1)^{k-1}f(X_1,X_2,\dots,X_{k-1})-f(X_2,X_3,\dots,X_k) +\sum_{i=1}^{k-1}(-1)^{i+1}f(X_1,X_2,\dots,X_i+X_{i+1},\dots,X_{k-1},X_k)\). If \(k=1\), then \(\Psi f=0\). They proved for an arbitrary mapping \(f:\mathcal V^{k-1}\to\mathcal V'\) that \((\Psi\circ\Psi)f(X_1,\dots,X_k,X_{k+1})=0\). By using these previously mentioned results a new cohomology on complex manifolds admitting flat linear connection is constructed. Especially, this cohomology is determined on the \(s\)-dimensional complex torus.
57R19 Algebraic topology on manifolds and differential topology
53C05 Connections (general theory)