Ohsawa-Takegoshi Extension Theorem Research Articles

On the Ohsawa–Takegoshi Extension Theorem

We establish a new extension result for twisted canonical forms defined on a hypersurface with simple normal crossings of a projective manifold. Some of the examples presented in the appendix show that the bounds we obtain for the extension are sharp.

Gaussian Gabor frames, Seshadri constants and generalized Buser–Sarnak invariants

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander’s {overline{partial }}-L^2 estimate with singular weight, Demailly’s Calabi–Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff–Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson–Lempert method and the Ohsawa–Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings’ theta metric formula for the Arakelov Green functions.

Pseudonorms on direct images of pluricanonical bundles

We study pseudonorms on pluricanonical bundles over Stein manifolds. We prove that the pseudonorms determine holomorphic structures of Stein manifolds under certain assumptions. This theorem is based on and a generalization of the result obtained by Deng, Wang, Zhang and Zhou [8] for bounded domains in Cn. We also investigate Stein morphisms and the pseudonorms on direct images of pluricanonical bundles. Our main goal in this paper is to show that the pseudonorms also determine holomorphic structures of Stein morphisms. One important technique is an L2/m-variant of the Ohsawa-Takegoshi extension theorem.

Applications of the Ohsawa–Takegoshi Extension Theorem to Direct Image Problems

AbstractIn the 1st part of the paper, we study a Fujita-type conjecture by Popa and Schnell and obtain an effective bound on the generic global generation of direct images of twisted pluricanonical bundles. We also point out the relation between the Seshadri constant and the optimal bound. In the 2nd part, we give an affirmative answer to a question by Demailly, Peternell, and Schneider in a more general setting. As byproducts of our proofs, we extend a result by Fujino and Gongyo on images of weak Fano manifolds to the Kawamata log terminal settings and refine a theorem by Broustet and Pacienza on the rational connectedness of the image.

Bergman kernel and oscillation theory of plurisubharmonic functions

Based on Harnack’s inequality and convex analysis we show that each plurisubharmonic function is locally BUO (bounded upper oscillation) with respect to polydiscs of finite type but not for arbitrary polydiscs. We also show that each function in the Lelong class is globally BUO with respect to all polydiscs. A dimension-free BUO estimate is obtained for the logarithm of the modulus of a complex polynomial. As an application we obtain an approximation formula for the Bergman kernel that preserves all directional Lelong numbers. For smooth plurisubharmonic functions we derive a new asymptotic identity for the Bergman kernel from Berndtsson’s complex Brunn–Minkowski theory, which also yields a slightly better version of the sharp Ohsawa–Takegoshi extension theorem in some special cases.

A converse of Hörmander’s L2-estimate and new positivity notions for vector bundles

We study conditions of Hormander’s L2-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of the Hormander-type condition, we show a converse of Hormander L2-estimate under some regularity assumptions on an n-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson (1998). We also de_ne new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.

A new proof of Kiselman's minimum principle for plurisubharmonic functions

We give a new proof of Kiselman's minimum principle for plurisubharmonic functions, inspired by Demailly's regularization of plurisubharmonic functions by using Ohsawa–Takegoshi's extension theorem.

A characterization of regular points by Ohsawa–Takegoshi extension theorem

In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related Ohsawa–Takegoshi extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

A proof of the Ohsawa–Takegoshi theorem with sharp estimates

We show how ideas from \cite{2Blocki} and \cite{3Blocki} to prove the Suita conjecture can be adapted to give a proof of the Ohsawa-Takegoshi extension theorem with sharp estimates.

Cauchy–Riemann meet Monge–Ampère

This is a relatively self-contained introduction to recent developments in the $$\bar{\partial }$$ -equation, Ohsawa–Takegoshi extension theorem and applications of pluripotential theory to the Bergman kernel and metric. The main tools are the Hörmander $$L^2$$ -estimate for $$\bar{\partial }$$ and Bedford–Taylor’s theory of the complex Monge–Ampère operator.

An elementary proof of the Ohsawa-Takegoshi extension theorem

An elementary proof of the Ohsawa-Takegoshi extension theorem

Suita conjecture and the Ohsawa-Takegoshi extension theorem

We prove a conjecture of N. Suita which says that for any bounded domain D in ℂ one has $c_{D}^{2}\leq\pi K_{D}$ , where c D (z) is the logarithmic capacity of ℂ∖D with respect to z∈D and K D the Bergman kernel on the diagonal. We also obtain optimal constant in the Ohsawa-Takegoshi extension theorem.

On the Ohsawa–Takegoshi [formula omitted] extension theorem and the twisted Bochner–Kodaira identity

In this Note, we improve the estimate in Ohsawaʼs generalization of the Ohsawa–Takegoshi L 2 extension theorem for holomorphic functions by finding a smaller constant, and apply the result to the Suita conjecture. We also present a remark allowing to generalize the Ohsawa–Takegoshi extension theorem to the case of ∂ ¯ -closed smooth ( n − 1 , q ) -forms. Finally, we prove that the twist factor in the twisted Bochner–Kodaira identity can be a non-smooth plurisuperharmonic function.

Some applications of the Ohsawa–Takegoshi extension theorem

We sketch a proof of the Ohsawa–Takegoshi extension theorem (due to Berndtsson) and then present some applications of this result: optimal lower bound for the Bergman kernel, relation to the Suita conjecture, and the Demailly approximation.

The Ohsawa-Takegoshi Extension Theorem on Some Unbounded Sets

AbstractWe use a method of Berndtsson to obtain a simplification of Ohsawa’s result concerning extension of L2-holomorphic functions. We also study versions of the Ohsawa-Takegoshi theorem for some unbounded pseudoconvex domains, with an application to the theory of Bergman spaces. Using these methods we improve some constants, that arise in related inequalities.

Integral formulas and the ohsawa-takegoshi extension theorem

We construct a semiexplicit integral representation of the canonical solution to the\(\bar \partial \)-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.