Analytic approximations of plurisubharmonic singularities

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.     

Bordered complex Hessians

We record some basic facts about bordered complex Hessians and logarithmically plurisubharmonic functions. These enable us to prove that a nonnegative bihomogeneous polynomial is plurisubharmonic if and only if it is log-plurisubharmonic; we give a more general version for twice differentiable bihomogeneous functions. The proof relies on the vanishing of the determinant of the bordered complex Hessian; we go on to find general classes of solutions to the nonlinear PDE given by setting the determinant of a bordered complex Hessian equal to zero.     

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.     

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

On the behavior of strictly plurisubharmonic functions near real hypersurfaces

We describe the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in ℂ ⁿ , n≥3. Given a hypersurface we study continuous plurisubharmonic functions which are zero on the hypersurface and have Monge–Ampère mass greater than one in a one-sided neighborhood of the hypersurface. If we can find complex curves which have sufficiently high contact order with the hypersurface then the plurisubharmonic functions we study cannot be globally Lipschitz in the one-sided neighborhood.     

Plurifinely Plurisubharmonic Functions and the Monge Ampère Operator

We will define the Monge-Ampère operator on finite (weakly) plurifinely plurisubharmonic functions in plurifinely open sets U???? ⁿ and show that it defines a positive measure. Ingredients of the proof include a direct proof for bounded strongly plurifinely plurisubharmonic functions, which is based on the fact that such functions can plurifinely locally be written as difference of ordinary plurisubharmonic functions, and an approximation result stating that in the Dirichlet norm weakly plurifinely plurisubharmonic functions are locally limits of plurisubharmonic functions. As a consequence of the latter, weakly plurifinely plurisubharmonic functions are strongly plurifinely plurisubharmonic outside of a pluripolar set.     

Logarithms of moduli of entire functions are nowhere dense in the space of plurisubharmonic functions

We prove that the set of logarithms of moduli of entire functions of several complex variables is nowhere dense in the space of plurisubharmonic functions equipped with a topology that is a generalization of the topology of uniform convergence on compact sets. This topology is generated by a metric in which plurisubharmonic functions form a complete metric space. Thus, the logarithms of moduli of entire functions form a set of the first Baire category. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1602 – 1609, December, 2008.     

Plurisubharmonic and holomorphic functions relative to the plurifine topology

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 it is locally bounded from above in the plurifine topology and f°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 preserved under composition with weakly plurifinely holomorphic maps.     

When does the subadditivity theorem for multiplier ideals hold?

Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.

     

Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.     

Discs in complex manifolds with no bounded plurisubharmonic functions

Roughly speaking: In a complex manifold on which all bounded plurisubharmonic functions are constant, the center of a holomorphic disc and its boundary can be prescribed somewhat arbitrarily.

     

Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values

The aim of the paper is to investigate the Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values. As an application, we study the approximation of negative plurisubharmonic function with given boundary values by an increasing sequence of plurisubharmonic functions defined in larger domains.     

Constancy regions of mixed multiplier ideals in two‐dimensional local rings with rational singularities

The aim of this paper is to study mixed multiplier ideals associated with a tuple of ideals in a two‐dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range.     

On subextension of pluriharmonic and plurisubharmonic functions

The problem of subextension of plurisubharmonic functions is considered. Recently it was shown by Cegrell–Zeriahi, that subextension is always possible for negative plurisubharmonic functions in the energy class ℱ. In this paper we construct, for every hyperconvex domain Ω, a negative plurisubharmonic function in the class ℰ which cannot be subextended. Given any pseudoconvex domain we construct a pluriharmonic function that cannot be subextended.     

Jensen measures and boundary values of plurisubharmonic functions

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions inB-regular domain. This theorem implies that the two classes of Jensen measures coincide inB-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above. The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.     

Weak convergence of currents

We give certain conditions to guarantee weak convergence u _k T _k → uT, where u _k , u are plurisubharmonic functions and T _k , T are positive closed currents. As applications we obtain that convergence in capacity of plurisubharmonic functions u _k implies weak convergence of the complex Monge-Ampère measures (dd ^c u _k ) ⁿ if all of the plurisubharmonic functions u _k are bounded below by one of some sorts of plurisubharmonic functions.     

Multiplier ideals of hyperplane arrangements

In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustata (2004).

     

Bounded Plurisubharmonic Exhaustion Functions and Levi-flat Hypersurfaces

In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich–Forn?ss exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.     

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J, K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics. We prove a quaternionic analogue of A. D. Aleksandrov and ChernLevine-Nirenberg theorems.