Fattening complex manifolds: Curvature and Kodaira-Spencer maps (Q1184376)

The fattening of a complex manifold \((X,{\mathcal O})\) of order \(m\) and codimension \(k\) is a ringed space \(X^{(m)}=(X,{\mathcal O}_{(m)})\), where \({\mathcal O}_{(m)}\) is locally isomorphic to \[ {\mathcal O}_{m,k}\equiv{\mathcal O}[\zeta^ 1,\ldots,\zeta^ k]/(\zeta^ 1,\ldots,\zeta^ k)^{m+1}. \] Such a situation arises for example when \(X\subset Y\) is a closed submanifold of codimension \(k\) in complex manifold \(Y\). Then \[ {\mathcal O}_{(m)}=({\mathcal O}_ y/I^{m+1})|_ X \] where \(I\) is an ideal sheaf of \(X\). In this case \(X^{(m)}=(X,{\mathcal O}_{(m)})\) is called \(m\)-th infinitesimal neighborhood of \(X\). The paper gives an obstruction-theoretic classification of all fattenings of a given complex manifold.