On vector bundles of finite order (Q2466357)

This long paper on complex geometry builds a theory of special complex affine manifolds with finite-order holomorphic vector bundles over them that parallels the work [Invent. Math. 28, 1--106 (1975; Zbl 0293.32026)] of \textit{M. Cornalba} and \textit{Ph. Griffiths} for holomorphic line bundles, and shares with it the spirit of an Oka principle with growth conditions. The main difference between the author's definition of the notion of a holomorphic vector bundle \(E\) of finite order and those of Cornalba's and Griffiths's is that here instead of a direct condition on the vector bundle \(E\) we impose a finite-order condition on the line bundle \({\mathcal O}_{{\mathbb P}(E^*)}(1)\) over the projectivization \({\mathbb P}(E^*)\) of the dual bundle \(E^*\), as it is done in parts of algebraic geometry. This definition opens up the way to exploit suitable modifications of the techniques of Cornalba and Griffiths for line bundles, and those of Stein theory such as Hörmander's \(L^2\)-method and injective immersions into affine or projective space. Here we can but state the two main theorems without recalling the somewhat lengthy definitions. Theorem~1. Let \(X\) be a special complex affine variety, and \(E\to X\) a holomorphic vector bundle equipped with a Finsler metric of finite order. Then the sheaf \({\mathcal O}_{\text{f.o.}}(E^*)\) of germs of finite-order holomorphic sections of the dual bundle \(E^*\) is acyclic. Theorem~2. Let \(X\) be a special affine variety of complex dimension \(n\), \(E\to X\) a holomorphic vector bundle of rank \(r+1\) equipped with a Finsler metric of finite order, and \(N>2(n+r)\) an integer. Then there is a holomorphic immersion \(f:{\mathbb P}(E)\to{\mathbb P}^N\) of finite order from the projectivization of \(E\) into projective space \({\mathbb P}^N\) such that the pullback \(f^*{\mathcal O}_{{\mathbb P}^N}(1)\) of the hyperplane line bundle of \({\mathbb P}^N\) is isomorphic to the hyperplane line bundle \({\mathcal O}_{{\mathbb P}(E)}(1)\). The paper is well written and pleasant to read.