Infinitesimal neighborhoods of infinite-dimensional complex projective spaces (Q1880812)

Let \(V\) be a complex Banach space and \(\mathbb{P}(V)\) be the projective space of all its 1-dimensional subspaces. Let \(Y\) be a complex Banach manifold that contains \(X\) as a codimension \(r\) closed submanifold, \(X^{(n)}\) the infinitesimal neighborhood of \(X\) in \(Y\) of order \(n\geq 1\) and \(E\to X^{(n)}\) a holomorphic vector bundle of rank \(s\); assume that \(A\) admits smooth partitions of unity and that there exists a holomorphic line bundle \(O_Y(1)\) on \(Y\) such that \(O_Y(1)\mid X\simeq O_X(1)\). Then the following main result is established. Theorem 1: In such a situation, the cohomology group \[ H^1(X^{(n)}, E)= 0 \] and there exist unique integers \(a_1\geq\cdots\geq a_s\) such that \[ E\simeq \bigoplus_{1\leq i\leq s} O_X(n) (a_i). \] Among the ingredients of this proof are \textit{L. Lempert's} vanishing and splitting theorems [J. Am. Math. Soc. 11, No. 3, 485--520 (1998; Zbl 0904.32014)].