UC Irvine

In a C∞ geometry situation, given a submanifold X embedded in another manifold Y , X ⊆ Y , we could ask a question about the existence of the an infinitesimal neighborhood of X, and the Tubular Neighborhood Theorem guarantees the existence of a tubular neighborhood U ⊆ Y of X which admits a projection to X. Moreover, given a vector bundle E over X, we could easily extend the bundle E even to the neighborhood U found, by taking the pullback with respect to the projection.

In the holomorphic or algebraic setting, we consider a smooth subvariety X in a smooth variety Y , X ↪ Y , over a field k of characteristic zero. Here we have a formal neighborhood O(∞) of X instead, taking a form of a structure called an L∞- algebroid, which can be described in three different ways: ˇCech , Dolbeault, and formal geometry. For a vector bundle E over X, we assume that E extends to the l-th formal neighborhood of X in Y for k > l. We study cohomological obstruction theory and find out necessary conditions to extending E further to the k-th neighborhood in three different approaches introduced.