Plurisubharmonic Functions Research Articles

Plurisubharmonic geodesics and interpolating sets

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of $${\mathbb {C}}^n$$ . Namely, two non-pluripolar, polynomially closed, compact subsets of $${\mathbb {C}}^n$$ are interpolated as level sets $$L_t=\{z: u_t(z)=-1\}$$ for the geodesic $$u_t$$ between their relative extremal functions with respect to any ambient bounded domain. The sets $$L_t$$ are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge–Ampere capacities of $$L_t$$ satisfy a dual Brunn–Minkowski inequality.

On convex hulls and pseudoconvex domains generated by q-plurisubharmonic functions, part III

We characterise in this work the q -plurisubharmonic functions in terms of the theory of viscosity solutions. We show that an upper semi-continuous function is q -plurisubharmonic if and only if its complex Hessian has at most q strictly negative eigenvalues in the viscosity sense. This characterisation is then used to prove that the sup-convolution of a (strictly) q -plurisubharmonic function is again (strictly) q -plurisubharmonic on a maybe different set of definition. Finally, we use the supremum convolution to deduce a new characterisation for the q -pseudoconvex subsets in C n .

Hessian boundary measures

We will study certain boundary measures related to [Formula: see text]-subharmonic functions on [Formula: see text]-hyperconvex domains. These measures generalize the boundary measures studied by Wan and Wang (see [Complex Hessian operator and Lelong number for unbounded [Formula: see text]-subharmonic functions, Potential. Anal. 44(1) (2016) 53–69]). For the case of plurisubharmonic functions ([Formula: see text]) the boundary measure has been studied by Cegrell and Kemppe (see [Monge–Ampère boundary measures, Ann. Polon. Math. 96 (2009) 175–196]).

Analytic adjoint ideal sheaves associated to plurisubharmonic functions II

In this article, we will present that the analytic adjoint ideal sheaves associated to plurisubharmonic functions are not coherent.

Courants résidus et opérateurs de Monge–Ampère

We extend to the case when the support is a possibly singular analytic subvariety the Federer theorem on the structure of the residue current. On another hand, we determine the general law of transformation of the distribution γf associated with a holomorphic map f=(f1,…,fN). In such a way, we arrive at the cohomological interpretation of the fundamental class of an effective analytic cycle, which is not necessarily a local complete intersection. By this same law, we obtain a characterization of the pure dimensional algebraic subsets of Cn, which are complete intersections. We also characterize the complete intersections of codimension q in Pn in terms of the solutions of the singular Monge–Ampère equation in Pq. Lastly, we express the condition on the dimension of the poles of the plurisubharmonic function u, so that the Monge–Ampère operator Q∧dcu has measure coefficients, for all closed positive current Q of bidimension (k,k).

(Pluri)Potential Compactifications

Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V ). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

On Extension Properties of Pluricomplex Green Functions

In 1985, Klimek introduced an extremal plurisubharmonic function on bounded domains in C that generalizes the Green’s function of one variable. This function is called the pluricomplex Green function of Ω with logarithmic pole at a and is denoted by gΩ(·, a). The aim of this thesis was to investigate the extension properties of gΩ(·, a). Let Ω be a bounded domain of C and E be a compact subset of Ω such that Ω := Ω\E is connected. In general, gΩ(·, a) cannot be extended as a pluricomplex Green function to any subdomain of Ω that is strictly larger that Ω. In this thesis it was proved that if Ω is a pseudoconvex, bounded complete Reinhardt domain in C and E 63 0 is a strictly logarithmically convex, Reinhardt compact subset of Ω with E ∩ {z1 · · · zn = 0} = ∅, there exists a subdomain Ω % Ω of Ω such that gΩ(z, 0) = gΩ(z, 0) for any z ∈ Ω. It was also shown that in C, one can omit the condition E ∩ {z1z2 = 0} = ∅. The methods required to prove the results heavily use the relation between the plurisubharmonicity of poyradial functions on Reinhardt domains and convexity of related functions. Special classes of convex functions were introduced and discussed for this purpose. These methods were also used to discuss the extension properties of the pluricomplex Green functions when Ω is equal to unit the bidisk in C and in that case a complete solution of the problem was given. EXTENSION PROPERTIES OF PLURICOMPLEX GREEN FUNCTIONS by Sidika Zeynep Ozal B.S., Bilkent University, 2003 M.S., Sabanci University, 2006 Dissertation Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Syracuse University June 2013 c © 2013 Sidika Zeynep Ozal All Rights Reserved

On a Monge-Ampere operator for plurisubharmonic functions with analytic singularities

We study continuity properties of generalized Monge-Amp`ere 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-Amp`ere measure of such a function on a compact Ka hler manifold.

Siu’s lemma, optimal $$L^2$$ extension and applications to twisted pluricanonical sheaves

We prove a general version of Siu’s lemma for plurisubharmonic functions with nontrivial multiplier ideal sheaves and use it to prove an optimal $$L^2$$ extension theorem and an optimal $$L^{\frac{2}{m}}$$ extension theorem for Kahler fibrations. These results are used to prove the positivity of twisted relative pluricanonical bundles and their direct images for Kahler fibrations and to answer a comparison question about singular metrics of twisted pluricanonical bundles.

Extension of S-plurisubharmonic currents across zero sets of plurisubharmonic functions

ABSTRACTIn this paper, we study the extension of S-plurisubharmonic currents of bi-dimension across zero sets of plurisubharmonic functions. More precisely, we show that the extension of the current exists for every non-negative plurisubharmonic function g and negative S-plurisubharmonic current T defined outside as soon as the -Hausdorff measure of A vanishes. By some examples we show the astonishing fact that in some sense this result is sharp.

Quaternionic Monge–Ampère operator for unbounded plurisubharmonic functions

In this paper, we generalize the definition of the quaternionic Monge–Ampere operator to some unbounded plurisubharmonic functions, and we prove that the quaternionic Monge–Ampere operator is continuous on the monotonically decreasing sequences of plurisubharmonic functions. After introducing the generalized Lelong number of a positive current, Demailly’s comparison theorems are showed. Moreover, we prove that the quaternionic Lelong–Jensen-type formula also holds for the unbounded plurisubharmonic function.

Weakly Solutions to the Complex Monge–Ampère Equation on Bounded Plurifinely Hyperconvex Domains

Let $$\mu $$ be a non-negative measure defined on bounded $${\mathcal {F}}$$ -hyperconvex domain $$\Omega $$ . We are interested in giving sufficient conditions on $$\mu $$ such that we can find a plurifinely plurisubharmonic function satisfying NP $$(dd^c u)^n =\mu $$ in QB $$(\Omega )$$ .

Pluripotential Numerics

We study the numerical approximation of the fundamental quantities in pluripotential theory, namely the Siciak Zaharjuta extremal plurisubharmonic function $$V_E^*$$ of a compact $$\mathcal {L}$$ -regular set $$E\subset {\mathbb {C}}^n$$ , its transfinite diameter $$\delta (E),$$ and the pluripotential equilibrium measure $$\mu _E:=\left( {{\mathrm{dd^c}}}V_E^*\right) ^n$$ . The developed methods rely on the computation of a polynomial mesh for E, for which a suitable orthonormal polynomial basis can be defined. We prove the convergence of the approximation of $$\delta (E)$$ and the local uniform convergence of our approximation to $$V_E^*$$ on $${\mathbb {C}}^n$$ . Then the convergence of the proposed approximation of $$\mu _E$$ follows. Our algorithms are based on the properties of polynomial meshes and Bernstein–Markov measures. Numerical tests on some simple cases with $$E\subset \mathbb {R}^2$$ show the performance of the proposed methods.

A note on the weighted log canonical thresholds of plurisubharmonic functions

In this note, we give a characterization for the weighted log canonical thresholds of plurisubharmonic functions. As an application, we prove an inequality for weighted log canonical thresholds and Monge–Ampère masses.

On the approximation of weakly plurifinely plurisubharmonic functions

In this note, we study the approximation of singular plurifine plurisubharmonic function u defined on a plurifine domain Ω. Under some condition we prove that u can be approximated by an increasing sequence of plurisubharmonic functions defined on Euclidean neighborhoods of Ω.

A cup product lemma for continuous plurisubharmonic functions

A version of Gromov’s cup product lemma in which one factor is the (1, 0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kähler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a [Formula: see text]-dimensional complex analytic set at some point has the Bochner–Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.

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–Fornaess exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.

Minimum Principle for Plurisubharmonic Functions and Related Topics

This is a survey about some recent developments of the minimum principle for plurisubharmonic functions and related topics.

Analytic adjoint ideal sheaves associated to plurisubharmonic functions

In this article, we will present that the analytic adjoint ideal sheaves associated to plurisubharmonic functions are not coherent.

Geometry and Topology of the Space of Plurisubharmonic Functions

Let $$\Omega $$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $$\Omega $$ . We study the 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.