KASS seminar

Abstract: Let h₁, …, h_r 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 + … + h_r ⊗ f_r , with f₁, …, f_r 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 minimal solution f = (f₁, …, f_r) by deforming a weight on the space of all linear combinations v₁ ⊗ f₁ + … + v_r ⊗ f_r to single out the linear combination h₁ ⊗ f₁ + … + h_r ⊗ f_r we are interested in.