On neighborhoods of projective space bundles over elliptic curves
Takayuki Koike, Laurent Stolovitch
TL;DR
This paper establishes holomorphic full linearization for neighborhoods of a fiberwise $\mathbb{P}^r$-bundle $Y$ over an elliptic curve $C$ embedded in a complex manifold $X$, under the assumptions that the normal bundle $N_{Y/X}$ is a Diophantine unitary flat bundle and the tangent restriction $T_X|_Y$ splits as $T_Y\oplus N_{Y/X}$. The authors develop a global model $M$ containing $C$ and construct a $\mathbb{P}^r$-fibration over $M$, then apply a Stolo-Gong-Tori-type linearization for a neighborhood of $C$ in $M$ to decouple the fiber coordinates; a key technical step shows fiber transitions can be taken independent of fiber coordinates, enabling a biholomorphism between a neighborhood of $Y$ in $X$ and the zero-section neighborhood in $N_{Y/X}$. The work also proves an extension-triviality theorem for vector bundles over neighborhoods of tori under vertical Diophantine conditions, which underpins the reduction to a linearized model. Together, these results extend Grauert-type linearization to new classes of embeddings, combining Ueda-type analysis, patching techniques for projective bundles, and a Newton-scheme approach.
Abstract
We give conditions ensuring that a neighborhood of an embedded projective space bundle over an elliptic curve is holomorphically equivalent to a neighborhood of the zero section of its normal bundle.