Integral formulas and the ohsawa-takegoshi extension theorem

Abstract

ABSTRACT. We construct a semiexplicit integral representation of the canonical solution to the∂-equation with respect to a plurisubharmonic weight function in a pseudoconvex domain. The construction is based on a construction related to the Ohsawa-Takegoshi extension theorem combined with a method to construct weighted integral representations due to M Andersson.

Quantitative extensions of pluricanonical forms and closed positive currents

“…Under these circumstances, the extension theorem established in [32] (and subsequently developed in [1], [2], [13], [27], [33], [34], [41], [46]) states as follows: let u be a holomorphic section of the bundle K X + L |X 0 which is L 2 with respect to h L , i.e.…”

Ohsawa-Takegoshi Extension Theorem for Compact Kähler Manifolds and Applications

“…The L 2 extension theorem by Ohsawa-Takegoshi is a tool of fundamental importance in algebraic and analytic geometry. After the crucial contribution of [OT87,Ohs88], this result has been generalized by many authors in various contexts, including [Man93], [Dem00], [Siu04], [Ber05], [Che11], [Yi12], [ZGZ12], [Blo13], [GZ15a], [Dem15], [BL16].…”

The Ohsawa-Takegoshi Extension Theorem on Some Unbounded Sets

“…It is worth mentioning, that if we apply Berndtsson's argument (see [Be1], [Be2]) directly, by passing to the infimum, we get the constant to be 4π (c β (Ω,0)) 2 , where c β (Ω, 0) is the analytic capacity.…”

“…Most of the research is carried in the direction of changing D ∩ H to varieties, or stating the theorem in the setting of bundles. It seems that the case, when one keeps H to be a hyperplane, but relaxes the condition of boundedness of D, is far less studied, although in many papers Ohsawa-Takegoshi type theorems are proved for unbounded D's, with various restrictions on the domain (see e.g., [Be2], [D-H]). Since it is clear that one cannot expect an Ohsawa-Takegoshi type estimate for any unbounded pseudoconvex D, it seems interesting to characterize these, on which some version of the theorem holds.…”