Geometry and Topology Seminar

Roberto Albesiano, University of Waterloo

A degeneration approach to Skoda’s division theorem

Let h1, …, hr be fixed holomorphic sections of E* ⊗ G → X, where E,G are holomorphic line bundles over a Stein manifold X. Is it always possible to write a holomorphic section g of G as a linear combination g = h1 ⊗ f1 + … + hr ⊗ fr , with f1, …, fr holomorphic sections of E? In 1972, H. Skoda proved a theorem addressing this question and giving L2 bounds on the minimal-L2-norm solution. I will sketch a new proof of a Skoda-type theorem inspired by a degeneration argument of B. Berndtsson and L. Lempert. In particular, we will see how to obtain L2 bounds on the solution (f1, …, fr) with minimal L2 norm by deforming a weight on the space of all linear combinations v1 ⊗ f1 + … + vr ⊗ fr to single out the linear combination h1 ⊗ f1 + … + hr ⊗ fr we are interested in.

MC 5417