Description:

Title:  Holomorphic tubular neighborhoods

Abstract:  A classical result in differential topology shows that every submanifold of an ambient manifold admits a tubular neighborhood diffeomorphic to a neighborhood of the zero section of its normal bundle. The analogous result in the category of complex manifolds does not hold in general, already for plane algebraic curves of degree at least 2, and deciding in which cases it holds can be a very difficult question. In 1975, Ogus studied the case of certain elliptic curves embedded in the blowup of the plane at 9 points, which have topologically trivial (but not holomorphically torsion) normal bundle. Arnol'd showed in 1976 that almost all of these (in the measure-theoretic sense) admit a holomorphic tubular neighborhood, but not a single one was known which does not admit such a neighborhood.  I will discuss ongoing work with Simion Filip where we construct many such examples, answering Ogus's question.