The space of essential infinitesimal deformations (Q1387260)

An \(n(q+1)\)-dimensional manifold \(M\) is said to be a manifold over \(R(\varepsilon^q)\) if there exists an affinor field \(J\) such that \(J^{q+1} = 0\), \(\text{rank }J= nq\), and \(J\) is integrable, i.e., the coordinates \(J^i_j\) are constant with respect to an atlas of \(M\). A vector field \(X\) on \(M\) is said to be holomorphic if \(L_X J = 0\), where \(L_X\) is the Lie derivative with respect to \(X\) [\textit{V. V. Vishnevskij, A. P. Shirokov} and \textit{V. V. Shurygin}, Spaces over algebras (1985; Zbl 0592.53001); \textit{V. V. Shurygin}, J. Sov. Math. 44, No. 2, 85-98 (1989); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 19, 3-22 (1987; Zbl 0721.53064)]. The objective of this paper is to characterize the space of essential infinitesimal deformations \(D(J)\) of the structure of a manifold over \(R(\varepsilon^q)\). The author constructs a fine resolution of the sheaf of holomorphic vector fields. Using this resolution, she proves that \(D(J)\) is isomorphic to the first cohomology group of \(M\) with coefficients in the sheaf of holomorphic vector fields, and exemplifies the calculation of such a group.