Alexander Rashkovskii
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.
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In this note we study the existence of the Lelong-Demailly number of a negative plurisubharmonic current with respect to a positive plurisubharmonic function on an open subset of $\C^n$. Then we establish some estimates of the Lelong-Demailly numbers of positive or negative plurisubharmonic currents.
We study the Lelong classes $\mathcal{L}(V),\mathcal{L}^+(V)$ of psh functions on an affine variety $V$. We compute the Monge-Ampère mass of these functions, which we use to define the degree of a polynomial on $V$ in terms of pluripotential theory (the Lelong degree). We compute the Lelong degree explicitly in a specific example. Finally, we derive an affine version of Bézout's theorem.
Equivalence between VMO functions and plurisubharmonic functions with zero Lelong numbers
本文证明了全纯函数与具有零Lelong数的VMO函数之间的等价性，并探讨了残余Monge-Ampère质量与VMO函数的关系。
We introduce a new type of pluricomplex Green function which has a logarithmic pole along a complex subspace A of a complex manifold X. It is the largest negative plurisubharmonic function on X whose Lelong number is at least the Lelong number of log max{|f_1|,...,|f_m|}, where f_1,...,f_m are local generators for the ideal sheaf of A. The pluricomplex Green function with a single logarithmic pole or a finite number of weighted poles is a very special case of our construction. We give several equivalent definitions of this function and study its properties, including boundary behaviour, contin…
We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. In certain cases, the problem is solved explicitly.
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…
We introduce a quantity which measures the singularity of a plurisubharmonic function f relative to another plurisubharmonic function g, at a point a. This quantity, which we denote by $ν_{a,g}(f)$, can be seen as a generalization of the classical Lelong number, in a natural way. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form $\{z: ν_{z,g}(f) \geq c > 0 \}$, are in fact analytic sets, under certain conditions on the weight g.
On Lelong numbers of plurisubharmonic functions on complex spaces
本文引入强局部不可约复空间概念，证明Lelong数与射影质量及重数的关系，并表明上水平集的闭包为子簇。
The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampère mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge-Ampère mass is always decreasing under the symmetrization
We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set V of valuations on R and--by way of a natural Laplace operator defined in terms of the tree structure on V--a positive measure on V. This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampere mass of two pluri…
We discuss several related problems on residual Monge-Ampère masses of plurisubharmonic functions. The note is based on the author's talk at the 27th Congress of Nordic Mathematicians, March 19, 2016.
In this paper we study the existence of the directional Lelong-Demailly numbers of positive plurisubharmonic or plurisuperharmonic currents. We prove the independence of these numbers to the system of coordinates. Moreover these numbers will be given by locally integrable functions.
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…
In this paper, we first establish a Poincaré-Lelong type formula in the almost complex setting. Then, after introducing the notion of $J$-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold $(M,J)$ on a $J$-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.
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…
An Exact Spherical-Solid Mean Formula and Vanishing Criteria for Lelong Numbers
本文建立了Lelong数的精确球-固体均值公式，并据此推导出消失准则、VMO刻画及非线性容量刻画，简化了Biard-Wu全局结果证明。
Our interest is the behavior of weak geodesics between two plurisubharmonic functions on pseudoconvex domains. We characterize the convergence condition along the geodesic between toric psh functions with a pole at origin on a unit ball in $\mathbb{C}^n$ by means of Lelong numbers.
Dinh--Sibony theory of tangent and density currents is a recent but powerful tool to study positive closed currents. Over twenty years ago, Alessandrini and Bassanelli initiated the theory of the Lelong number of a positive plurisubharmonic current in $\mathbb{C}^k$ along a linear subspace. Although the latter theory is intriguing, it has not yet been explored in-depth since then. Introducing the concept of the generalized Lelong numbers and studying these new numerical values, we extend both theories to a more general class of positive plurisubharmonic currents and in a more general context o…
In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong number. We also present a new proof of the related well known integrability criterion due to Skoda.
Let $ψ$ be a plurisubharmonic function on the closed unit ball and $K_{tψ}(z)$ the Bergman kernel on the unit ball with respect to the weight $tψ$. We show that the boundary behavior of $K_{tψ}(z)$ is determined by certain directional Lelong number of $ψ$ for all $t$ smaller than the John-Nirenberg exponent of $ψ$ associated to certain family of nonisotropic balls, which is always positive.
We study the masses charged by $(dd^cu)^n$ at isolated singularity points of plurisubharmonic functions $u$. It is done by means of the local indicators of plurisubharmonic functions. As a consequence, bounds for the masses are obtained in terms of the directional Lelong numbers of $u$, and the notion of the Newton number for a holomorphic mapping is extended to arbitrary plurisubharmonic functions. We also describe the local indicator of $u$ as the logarithmic tangent to $u$.
On the residual Monge-Ampère mass of plurisubharmonic functions with symmetry in $\mathbb{C}^2$
本文研究$\u2102^2$中带孤立奇点的对称多重次调和函数的剩余蒙日-安培质量，证明$S^1$-不变函数原点勒洛数为零时剩余质量为零，解答了特殊情形下的零质量猜想，并给出更一般的剩余质量估计。
First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball $\B \sub \C^n$ with its relative logarithmic capacity in $\C^n$ with respect to the same ball $\B$. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of $\C^n$ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at in…
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 reversed Alexandrov-Fenchel inequality for mixed Monge-Ampère masses of plurisubharmonic functions, which generalizes a result of Demailly and Pham. As applications to convex geometry, this gives a complex analytic proof of the reversed Alexandrov-Fenchel inequality for mixed covolumes, which generalizes recent results in convex geometry of Kaveh-Khovanskii, Khovanskii-Timorin, Milman-Rotem and R. Schneider on reversed (or complemented) Brunn-Minkowski and Alexandrov-Fenchel inequalities. Also for toric plurisubharmonic functions in the Cegrell class, we confirm Demailly's conject…
On the residual Monge-Ampère mass of plurisubharmonic functions with symmetry, II
本文利用Sasakian几何，对具有圆对称性和孤立奇点的多次调和函数，建立了剩余Monge-Ampère质量与Lelong数及最大方向Lelong数之间的估计，部分解决了零质量猜想。
The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds
本文研究了准亚调和函数在紧致凯勒流形上的迹算子，提出新的L^2扩展定理并应用于大线丛的限制体积。
Local-to-global maximality for plurisubharmonic functions with locally analytic singularities
本文证明了具有局部解析奇点的局部极大多次调和函数在满足解析扩张界时即为极大，并通过Monge--Ampère测度消失条件刻画极大性。
Let T be a positive plurisubharmonic current of bidimension (p,p) and let $δ>0$. Assume that the Lelong number of T satisfies $ν(T,a)\geq δ$ on a dense subset of supp(T) (rectifiable currents satisfy this condition). Then $T=ϕ[X]$, where X is a complex subvariety of pure dimension p and $ϕ$ is a weakly plurisubharmonic function on X. We have an analogous result for plurisuperharmonic currents. We also introduce and study a notion of polynomial p-hull.
$α$-Hölder Integrability Exponent
该研究针对多重次调和函数定义了α-Hölder可积性指数，得到了其以Lelong数和交数表示的改进下界，并给出了一类函数该指数的最优下界，推进了相关可积性问题的研究。
Demailly-Lelong numbers on complex spaces
本文证明了 Berman 等人提出的猜想，表明 Demailly-Lelong 数可由势的除子部分与 SNC 除子的相交数组合确定，并建立了奇异复空间上两种 Lelong 数的逐点比较及奇点估计。
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…
Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity
本文研究了与紧凸集相关的多项式及加权极值函数的正则性问题，通过积分、极小和极大连算子证明了相关函数的下半连续性，并推翻了Hölder连续性的已有结论。
Analyticity theorems for parameter-dependent plurisubharmonic functions
研究依赖参数的（拟）多重次调和函数的水平集解析性，证明 Lelong 数上水平集之并通常多重极，并给出复奇点指数次水平集及相对 Bergman 核对数极点的解析性条件与非解析反例。
We show that on almost complex surfaces plurisubharmonic functions can be locally approximated by smooth plurisubharmonic functions. The main tool is the Poletsky type theorem due to U. Kuzman.
A counterexample to the zero-mass conjecture
该研究在至少2维的复欧几里得空间中构造出满足特定条件的负多重次调和函数，得到Guedj-Rashkovskii零质量猜想的反例，否定了该猜想在所有至少2维复维数下的成立性。
We prove in an elementary way that for a Lipschitz domain $D\subset \cn$, all plurisubharmonic functions on $D$ can be regularized near any boundary point.
On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
本文证明管状域上的 $q$-多次调和函数等价于基域上的实 $q$-凸函数，并由此刻画 $q$-伪凸管状域，应用于仿射线性映射图的补集和 Reinhardt 域。
Using techniques from the analysis of PDEs to study the boundary behaviour of functions on domains with low boundary regularity, we extend results by Fornaæss-Wiegerinck (1989) on plurisubharmonic approximation and by Demailly (1987) on the existence on bounded plurisubharmonic exhaustion functions to domains beyond Lipschitz boundary regularity.
Dirichlet problem for plurisubharmonic functions on bounded quasi B-regular domains in $\mathbb{C}^n$
本文研究了有界域上具有非连续非有界边界数据的极大多重次调和函数的存在性与唯一性，通过探讨 Perron-Bremermann 包络的连续性，实现了对可能上方无界的多重次调和函数的连续逼近。
We show that in $\mathbb{C}^2$ if the set of strongly regular points are closed in the boundary of a smooth bounded pseudoconvex domain, then the domain is c-regular, that is, the plurisubharmonic upper envelopes of functions continuous up to the boundary are continuous on the closure of the domain. Using this result we prove that smooth bounded pseudoconvex Reinhardt domains in $\mathbb{C}^{2}$ are $c$-regular.
Log truncated threshold and zero mass conjecture
本文引入对数截断阈值概念，推导了多重次调和函数剩余Monge-Ampère质量的最优估计，为Guedj和Rashkovskii的零质量猜想提供了新方法，并统一加强了相关已知结果。
Given a positive closed (1,1)-current $T$ defined on the regular locus of a projective variety $X$ with bounded mass near the singular part of $X$ and $Y$ an irreducible algebraic subset of $X$, we present uniform estimates for the locus inside $Y$ where the Lelong numbers of $T$ are larger than the generic Lelong number of $T$ along $Y$.
In this paper, we extend some recent results of Guedj-Lu-Zeriahi about psh envelopes of bounded functions on bounded domains in $\mathbb{C}^n$. We also present a result on the regularity of psh envelopes.
We prove that the maximum of two smooth strictly plurisubharmonic functions on an almost complex manifold can be uniformly approximated by smooth strictly plurisubharmonic functions.
A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if f o h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preser…
In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact Kähler manifold. We prove that a precise bound on the complex Monge-Ampère mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Ampère measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Ampè…
In this paper, we prove the $C^{1, 1}$-regularity of the plurisubharmonic envelope of a $C^{1,1}$ function on a compact Hermitian manifold. We also present examples to show this regularity is sharp.
Regularity of geodesics in the spaces of convex and plurisubharmonic functions II
本研究延续凸与多重次调和函数空间测地线的探究，证明光滑严格凸函数间测地线的最优正则性，拓展$C^{1,α}$范畴正则性理论，并探讨环面不变严格多重次调和函数间测地线的正则性。
We study the plurisubharmonic envelopes of functions in the setting of domains in $\mathbb C^n$. In particular we prove a complex analogue of a result of De Philippis and Figalli concerning the optimal regularity of such envelopes in smooth strictly pseudoconvex domains.
In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1} regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.
On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $ϕ$ such that $ϕ^*=ϕ_*$ on the closure of the domain. For $ϕ$ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope $P ϕ$ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge--Ampère equation \[ (dd^cu)^n = μ\] being uniquely solvable for continuous boundary data implies that …
The generalized Lelong numbers and intersection theory
本文推广了正多重次调和电流沿Kähler子流形的广义Lelong数理论，引入水平维数并建立Siu型上半连续性，同时推广Dinh-Sibony类并给出正闭电流相交及其连续性的有效充分条件。
Characterization of Sobolev regularity of plurisubharmonic functions
本文刻画了由全纯芽构造的多次调和函数的局部Sobolev正则性，给出尖锐判据，并应用于Calderón-Zygmund理论反例与Monge-Ampère定义域刻画。
Plurisubharmonic functions with discontinuous boundary behavior
本文研究带不连续边界数据的复 Monge–Ampère Dirichlet 问题，在 b-多重极不连续集与 B-正则域条件下证明解的存在唯一性及正则性，并说明高维中这些条件并非必要。
Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous opera…
Let $X$ be a compact Kähler manifold. We prove that the Kähler-Ricci flow starting from arbitrary closed positive $(1,1)$-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has positive Lelong numbers for short time if the initial current does. We also prove that the flow is unique when starting from currents with zero Lelong numbers.
Let $Ω$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $Ω$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local Kähler-Einstein metrics.
Let $\mathcal J$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log|f|+\mathcal O(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal J$. We give a meaning to the Monge-Ampère products $(dd^c G)^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{\mathcal J}:=\mathbf 1_Z(dd^c G)^k$ at $x$ coincide with the so-called Segre numbers of $\mathcal J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we s…
We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw sever…
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