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

Dror Varolin

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arXiv · 2508.01753

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

Dror Varolin

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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.

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Dror Varolin. 2025-08-03. Positivity and $\mathbf{L^2}$ Extension. https://arxiv.org/abs/2508.01753

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