Localization of Atiyah classes (Q1955675)

Let \(M\) be a complex manifold of dimension \(n,\) let \(E\) be a holomorphic vector bundle over \(M,\) and \(F\) a holomorphic subbundle of rank \(r\leq n\) of the tangent bundle \(T_M.\) Let \({\mathcal E}\) and \({\mathcal F}\) be the sheaves of germs of holomorphic sections of \(E\) and \(F,\) respectively. A holomorphic partial connection for \(E\) is a pair \((F,\delta)\) given by a \(\mathbb C\)-linear homomorphism \(\delta: {\mathcal E} \rightarrow {\mathcal F}^*\otimes {\mathcal E}\) satisfying the Leibnitz rule. The main result of the paper under review is the following assertion. Let \(E\) and \((F,\delta)\) be as above. If \(\nabla\) is a connection on \(E\) extending \((F\oplus\overline{T}_M, \delta \oplus \bar{\partial}),\) then \(a^d(\nabla) \equiv 0\) for all \(d> n-r,\) where \(a^d(\nabla)\) denotes the Atiyah form of bidegree \((d,d).\) A similar statement was proved by cohomological arguments in [\textit{P. F. Baum} and \textit{R. Bott}, Essays Topol. Relat. Top., Mem. ded. a Georges de Rham 29--47 (1970; Zbl 0193.52201); \textit{J. B. Carrell} and \textit{D. I. Lieberman}, Math. Ann. 225, 263--273 (1977; Zbl 0365.32020)]. The authors also explain that their result can be considered as an analog of the Bott vanishing theorem for nonsingular distributions. They also prove that if a nonsingular holomorphic distribution \(F\) on \(M\) leaves a complex submanifold \(V\) of \(M\) invariant, then there exists a holomorphic partial connection \(\delta\) for the normal bundle \(N_V\) along \(F|_V\) (cf. [\textit{C. Camacho} and \textit{P. Sad}, Ann. Math. (2) 115, 579--595 (1982; Zbl 0503.32007)]). In two final sections the residue theory of singular distributions is discussed. In particular, for a singular distribution of corank one on \({\mathbb C}^3_{x,y,z}\) defined by the \(1\)-form \(\omega = zdx+zdy-ydz,\) the corresponding Atiyah residues are computed explicitly.