A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in $\Rn$. Another class of extreme singularities is presented by means of a notion of relative type.
A variant of Siu's analyticity theorem is proved for relative types of plurisubharmonic functions. Some results on propagation of plurisubharmonic singularities and maximality of pluricomplex Green functions with analytic singularities are derived.
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.
This is a survey of results, both classical and recent, on behaviour of plurisubharmonic functions near their $-\infty$-points, together with the related topics for positive closed currents.
We study continuity properties of generalized Monge-Ampère operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a formula for the total mass of the Monge-Ampère measure of such a function on a compact Kähler manifold.
We construct examples of plurisubharmonic functions with isolated singularities at $0\in \mathbb{C}^n$, whose residual Monge-Ampère masses at the origin can not be approximated by masses of analytic approximations obtained via multiplier ideals. This answers negatively a conjecture of Demailly.
We study the complex Monge-Ampère operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Ampère energy. They generalize the classes introduced by U.Cegrell, and give a stratification of the space of (almost) all unbounded plurisubharmonic functions. We give an interpretation of these classes in terms of the speed of decreasing of the Monge-Ampère capacity of sublevel sets and solve associated complex Monge-Ampère equations.
By comparing Green functions of multi-circled plurisubharmonic singularities in the n-domensional complex space to their indicators, we obtain formulas for the higher Lelong numbers and integrability index for such singularities and extend Howald's result on multiplier ideals for monomial ideals to multi-circled singularities. This also leads to an elementary proof of the relations between the k-th Lelong numbers and the integrability index. For k=1 and arbitrary plurisubharmonic functions the inequality is due to Skoda, and for k=n and any plurisubharmonic function with isolated singularity t…
Local-to-global maximality for plurisubharmonic functions with locally analytic singularities
Institute for Artificial Intelligence, VNU University of Engineering and Technology（越南国立大学工程技术大学人工智能研究所）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
We study properties of relative types of plurisubharmonic functions with respect to maximal plurisubharmonic weights. It is shown that they give a general form for upper semicontinuous, tropically additive functionals on plurisubharmonic singularities. We consider some extremal problems whose solutions are Green-like functions that give best possible bounds on a plurisubharmonic function, given the values of its types relative to some of (or all) the weights. An analyticity theorem is proved for the upperlevel sets for the types with respect to exponentially Hölder continuous weights, which le…
We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge--Ampère masses. They are characterized in terms of Green functions for Demailly's approximations, relative types, and valuations. Furthermore, the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals.
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…
We show by an example that the (equivalence class of) singularity of a plurisubharmonic function cannot be determined by the data of its Lelong numbers, in a nontrivial sense. Such an example is provided by Siu-type singular hermitian metrics associated to an effective line bundle. We also show that a Siu-type metric has analytic singularities if and only if the section ring of the line bundle is finitely generated.
Each extreme edge of the Newton diagram of a plurisubharmonic polynomial on $\mathbb{C}^2$ gives rise to a plurisubharmonic polynomial. It is tempting to believe that the union of the extreme edges or the convex hull of said union will do the same. We construct a plurisubharmonic polynomial $P$ on $\mathbb{C}^2$ with precisely two extreme edges $E_1$ and $E_2$, such that neither $E_1\cup{E_2}$ nor $\text{Conv}({E_1\cup{}E_2})$ yields a plurisubharmonic polynomial.
Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex Monge-Ampère operator is well-defined on $u.$ Suppose also that $u$ has a weak singularity, in the sense that the Lelong number of $u$ at $x$ vanish. Is it true that the residual mass of the measure $(dd^c{u})^n$ vanish at $x$? To our knowledge there is no known example that falsifies the posed question. In this paper some partial results are obtained. We find that for a significant subset of plurisubharmonic functions with well defined Monge-Ampère mass vanishing Lelong number does implies…
We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.
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）
The goal of this work is to prove the regularity of certain quasi-plurisubharmonic upper envelopes. Such envelopes appear in a natural way in the construction of hermitian metrics with minimal singularities on a big line bundle over a compact complex manifold. We prove that the complex Hessian forms of these envelopes are locally bounded outside an analytic set of singularities. It is furthermore shown that a parametrized version of this result yields a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge-Ampere operator; applications to geodesics in the space o…
We observe that a slight adjustment of a method of Caffarelli, Li, and Nirenberg yields that plurisubharmonic functions extend across subharmonic singularities as long as the singularities form a closed set of measure zero. This solves a problem posed by Chirka.
We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the last two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x_1,...,x_n] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C^n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we def…
We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on Kähler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross--Witt-Nyström concerning the Hele-Shaw flow on an arbitrary Reimann surface.
