Positivity and $\mathbf{L^2}$ Extension

TL;DR

This paper investigates the link between vector bundle positivity and $L^2$ extension problems for holomorphic forms, showing Griffiths positivity is insufficient and providing new proofs for Nakano positivity's sufficiency.

Contribution

It demonstrates that Griffiths positivity alone does not guarantee $L^2$ extension, and offers a novel proof of Nakano positivity's sufficiency using a vector bundle analogue of Berndtsson's result.

Findings

Griffiths positivity is not enough for $L^2$ extension.

Nakano positivity guarantees $L^2$ extension, with a new proof provided.

Established a vector bundle analogue of Berndtsson's theorem.

Abstract

We examine the relationship between positivity of a vector bundle E and the problem of $L^2$ extension of holomorphic E-valued forms of top degree. In particular, we show by example that Griffiths positivity is not enough. Though the sufficiency of Nakano positivity of E has been known for some time, we provide another proof along the lines of Berndtsson and Lempert. For such a proof we establish a vector bundle analogue of a well-known result of Berndtsson. This result is of independent interest and should have many other useful applications.