KASS seminar

Abstract: The importance of the L2 Extension Theorem for sections of holomorphic line bundles is by now widely recognized. The theorem was first proved by Ohsawa and Takegoshi, and since then reproved, modified and extended by many mathematicians, including some of the locals in Chalmers. However, probably for good reasons at the time, the case of holomorphic vector bundles received much less attention. Things began to change with the work of Bo Berndtsson on the positivity of the L2 metric for families of Hilbert spaces, and the question of positivity for holomorphic vector bundles has attracted some new interest.

While it is relatively easy to see that the L2 Extension Theorem holds in the higher rank case with essentially the same proof(s) if the positivity is taken in the sense of Nakano, there have been some relatively recent links between L2 extension and Griffiths positivity.

After briefly reviewing positivity of vector bundles, I will present two results.

First, I will show by example that Griffiths positivity is not enough to obtain the L2 Extension Theorem for vector bundles.

Then I will present a result on L2 Extension with a Griffiths positivity condition. This result will neither imply nor be implied by the standard L2 Extension Theorem (with Nakano positivity hypotheses.)