Positivity and L² Extension

2025/08/03 by Dror Varolin, Varolin, Dror
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2508.01753

Abstract

We examine the relationship between positivity of a vector bundle E and the problem of L² 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.

Citations

• Direct Images and Hilbert Fields
• A Hilbert bundle approach to the sharp strong openness theorem and the Ohsawa-Takegoshi extension theorem
• Extension of L² holomorphic functions
• A proof of the Ohsawa-Takegoshi theorem with sharp estimates
• A solution of an L² extension problem with optimal estimate and applications
• Curvatures of direct image sheaves of vector bundles and applications I

Related

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• Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness