Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that $φ$ is a plurisubharmonic (psh) function on $X$ which is dominated by a continuous psh exhaustion function $u$ of $M$. Given any number $c>1$, we show that $φ$ admits a psh extension to $M$ which is dominated by $cu$ on $M$.
We use this result to prove that any $ω$-psh function on a subvariety of the complex projective space is the restriction of a global $ω$-psh function, where $ω$ is the Fubini-Study Kähler form.
A notion of indicator for a plurisubharmonic function u of logarithmic growth in C^n is introduced and studied. It is applied to evaluation of the total Monge-Ampère measure (dd^cu)^n({C}^n). Upper bounds for the measure are obtained in terms of growth characteristics of u. When u=log|f| for a polynomial mapping f with isolated zeros, the indicator generates the Newton polyhedron of f whose volume bounds the number of the zeros.
We prove a Thullen type extension theorem of plurisubharmonic functions across a closed complete pluripolar set, which generalizes a theorem of Siu. Our approach depends on an Ohsawa-Takegoshi type extension theorem for a single point in a bounded complete Kähler domain, which is of independent interest.
We prove several results showing that plurisubharmonic functions with various bounds on their Monge-Ampere masses on a bounded hyperconvex domain always admit global plurisubharmonic subextension with logarithmic growth at infinity.
We show subellipticity of the d-bar Neumann problem on domains with Lipschitz boundary in the presence of plurisubharmonic functions with Hessians of algebraic growth. In particular, a subelliptic estimate holds near a point where the boundary is piecewise smooth of finite type.
Monge-Ampere currents generated by plurisubharmonic functions of logarithmic growth are studied. Upper bounds for their total masses are obtained in terms of growth characteristics of the functions. In particular, this gives a plurisubharmonic version of D.Bernstein's theorem on the number of zeros of polynomial mappings in terms of the Newton polyhedra, and a representation for Demailly's generalized degrees of plurisubharmonic functions.
Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the directional Lelong numbers of the functions and the Newton diagrams of the weights. An extension of Demailly's approximation theorem is proved as well.
Let $X$ be an algebraic subvariety of $\mathbb C^n$ and $\bar X$ be its closure in $\mathbb P^n.$ In their paper \cite{CGZ} Coman-Guedj-Zeriahi proved that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on $\mathbb C^n$ when the germs $(\bar X,a)$ in $\mathbb P^n$ are irreducible for all $a\in \bar X\setminus X.$ In this paper we consider $X$ for which the germ $(\bar X,a)$ is reducible for some $a\in \bar X\setminus X$ and we give a necessary and sufficient condition for $X$ so that any plurisubharmonic function with…
We give characterizations of (quasi-)plurisubharmonic functions in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic 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 id…
On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions
CAS Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院、中国科学院吴文俊数学重点实验室）
In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are th…
The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler mani…
Let $(V,ω)$ be a compact Kähler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $ω$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an analytic subvariety, we prove that any $ω|_X$-plurisubharmonic function on $X$ extends to a $ω$-plurisubharmonic function on $V$. This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental Kähler class in the real Neron-Severi space $NS_{\mathbb R}(V)$ has this extension pr…
In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmonic functions related to the jumping numbers, which were posed by Demailly-Kollár and Jonsson-Mustată respectively. We give a new proof of a lower semicontinuity conjecture posed by Demailly-Kollár without using the ACC conjecture. Other applications by combining with well-known r…
The concept of proximate order is widely used in the theories of entire, meromorphic, subharmonic and plurisubharmonic functions. We give a general interpretation of this concept as a proximate growth function relative to a model growth function. If a function is the proximate growth function with respect to the identity function on the positive semi-axis, then the logarithm of this function is the classical proximate order. Our definition uses only one condition. This form of definition is also new for the classical proximate order.
In the first part of this paper, we study the properties of some particular plurisubharmonic functions, namely the toric ones. The main result of this part is a precise description of their multiplier ideal sheaves, which generalizes the algebraic case studied by Howald. In the second part, almost entirely independent of the first one, we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting. This enables us to give an analogue of Howald's theorem for adjoint ideals attached to monomial ideals. Finally, using the local Ohsawa-Takegoshi-Manivel theo…
Let $n \geq 4$ and let $Ω$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $φ$ be a negative exhaustive smooth plurisubharmonic function on $Ω$. We show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}φ)^{n-3}$ can be extended to the holomorphic function on $Ω$.
Let $n \geq 3$ and $Ω$ be a bounded domain in $\mathbb{C}^n$ with a smooth negative plurisubharmonic exhaustion function $φ$. As a generalization of Y. Tiba's result, we prove that any holomorphic function on a connected open neighborhood of the support of $(i\partial \bar \partial φ)^{n-2}$ in $Ω$ can be extended to the whole domain $Ω$. To prove it, we combine an $L^2$ version of Serre duality and Donnelly-Fefferman type estimates on $(n,n-1)$- and $(n,n)$- forms.
We utilize the Legendre-Fenchel transform and weak geodesics for plurisubharmonic functions to construct a weight function that can be used in the Berndtsson-Lempert method, to give an Ohsawa-Takegoshi extension type of result. Theorem 4.1 and Theorem 0.1 in \cite{OT2017} (Theorem \ref{Theorem A} and \ref{Theorem B} below) follow as two special cases of this result, thus answering affirmatively a question posed by Ohsawa in remark 4.1 in \cite{OT2017}, on the Berndtsson-Lempert method.
Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f: W:=(A\times Y) \cup (X\times B)\longrightarrow Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W}:=\left\lbrace(z,w)\in X\times Y:\ \widetildeω(z,A,X)+\widetildeω(w,B,Y)<1 \right\rbrace$ such that $\hat{f}=f$ on $W\cap \hat{W},$ where $\widetildeω(\cdot,A,…
Let D be a strictly convex domain and X be a singular analytic subset of C^2 such that the intersection of X and D is non empty. We give conditions under which a function holomophic on the intersection of X and D can be extended holomorphically to D with growth estimates of BMO or L^q type. The extension is given by mean of integral representation formulas and new residual currents.
This paper generalises the result of Jean-Pierre Demailly on his Ohsawa--Takegoshi-type $L^2$ extension theorem, which guarantees holomorphic extensions for some sections $f$ on analytic subspaces $Y$ defined by multiplier ideal sheaves of plurisubharmonic functions. The result in this paper provides $L^2$ estimates for the extended sections even if those $f$ do not vanish on the singular sets of the reduced loci of the subspaces $Y$, at least for the case when the ambient manifold $X$ is compact and the relevant metrics have only neat analytic singularities. This is achieved by replacing the …
Let $D$ be a strictly pseudoconvex domain and $X$ be a singular analytic set of pure dimension $n-1$ in $C^n$ such that $X\cap D\neq \emptyset$ and $X\cap bD$ is transverse. We give sufficient conditions for a function holomorphic on $D\cap X$ to admit a holomorphic extension which belongs to $L^q(D),$ $q\in [1,+\infty[$, or to $BMO(D)$. The extension is given by mean of integral representation formulas and residue currents.
The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc $\mathbb D\subset \mathbb C$ in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the…
This note addresses the quantitative aspect of the bilipschitz extension problem. The main result states that any bilipschitz embedding of $\mathbb R$ into $\mathbb R^2$ can be extended to a bilipschitz self-map of $\mathbb R^2$ with a linear bound on the distortion.
While the asymptotic Borel mapping, sending a function into its series of asymptotic expansion in a sector, is known to be surjective for arbitrary openings in the framework of ultraholomorphic classes associated with sequences of rapid growth, there is no general procedure to construct extension operators in this case. We do provide such operators in complex sectors for some particular classes considered by S.~Pilipovi{ć}, N.~Teofanov and F.~Tomi{ć} in the ultradifferentiable setting. Although these classes are, in their words, "beyond Gevrey regularity", in some cases they keep the property …
