Plurisubharmonic Functions Research Articles

Applications of the Stone–Weierstrass theorem in the Calderón problem

We give examples on the use of the Stone–Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. We also study the holomorphic separability condition in terms of plurisubharmonic functions. The Stone–Weierstrass theorem allows us to generalize and simplify earlier results. It also makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.

Geodesic connectivity and rooftop envelopes in the Cegrell classes

AbstractThis study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $$\mathbb {C}^n$$ C n . We establish that solutions possessing comparable singularities to the complex Monge–Ampère equation are identical, affirmatively addressing a longstanding open question raised by Cegrell. This achievement furnishes the most general form of the Bedford–Taylor comparison principle within the Cegrell classes. Building on this foundational result, we explore plurisubharmonic geodesics, broadening the criteria for geodesic connectivity among plurisubharmonic functions with connectable boundary values. Our investigation also delves into the notion of rooftop envelopes, revealing that the rooftop equality condition and the idempotency conjecture are valid under substantially weaker conditions than previously established, a finding made possible by our proven uniqueness result. The paper concludes by discussing the core open problems within the Cegrell classes related to the complex Monge–Ampère equation.

On the finite energy classes of quaternionic plurisubharmonic functions

We investigate the finite p-energy classes Ep of quaternionic plurisubharmonic functions of Cegrell type. We also construct an example to show that the optimal constant in the energy estimate is strictly bigger than 1 for p>0, p≠1. This leads us to the fact that we can not use the variational method to solve the quaternionic Monge–Ampère equation for the classes Ep when p>0, p≠1.

Mini-Workshop: Positivity and Inequalities in Convex and Complex Geometry

The workshop convened researchers from algebraic geometry, convex geometry, and complex geometry to explore themes arising from the Alexandrov–Fenchel and Brunn–Minkowski inequalities. It featured three introductory talks delving into the basics of Lorentzian polynomials, valuations in convex geometry, and plurisubharmonic functions, that served as a foundation for the subsequent research talks. As anticipated, significant overlap emerged among the varied perspectives within these three areas, evident in the presentations and ensuing discussions.

New characterization of plurisubharmonic functions and positivity of direct image sheaves

abstract: We discover a new characterization of plurisubharmonic functions in terms of $L^p$ extension from one point and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics in terms of $L^p$ extensions. As applications, we give a stronger result or new proof of some well-known theorems on the Griffiths positivity of the holomorphic vector bundles and their direct image sheaves associated to certain holomorphic fibrations.

Analyticity theorems for parameter-dependent plurisubharmonic functions

In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous.

Generalized Levi Currents and Singular Loci for Families of Plurisubharmonic Functions

Generalized Levi Currents and Singular Loci for Families of Plurisubharmonic Functions

On compact pseudoconcave sets

Replying to three questions posed by N. Shcherbina, we show that a compact pseudoconcave set can have the core smaller than itself, that the core of a compact set must be pseudoconcave, and that it can be decomposed into compact pseudoconcave sets on which all smooth plurisubharmonic functions are constant.

Lebesgue points of functions in the complex Sobolev space

Let [Formula: see text] be a function in the complex Sobolev space [Formula: see text], where [Formula: see text] is an open subset in [Formula: see text]. We show that the complement of the set of Lebesgue points of [Formula: see text] is pluripolar. The key ingredient in our approach is to show that [Formula: see text] for [Formula: see text] is locally bounded from above by a plurisubharmonic function.

The second variation of the Hodge norm and higher Prym representations

AbstractLet denote a rational cohomology class, and let denote its Hodge norm. We recover the result that is a plurisubharmonic function on the Teichmüller space , and characterize complex directions along which the complex Hessian of vanishes. Moreover, we find examples of such that is not strictly plurisubharmonic. As part of this construction, we find an unbranched covering such that the subgroup of generated by lifts of simple curves from is strictly contained in . Finally, combining the characterization theorem with the Riemann–Roch, and the Li–Yau [Invent. Math. 69 (1982), no. 2, 269–291] gonality estimate, we show that geometrically uniform covers of satisfy the Putman–Wieland Conjecture about the induced Higher Prym representations.

On the residual Monge–Ampère mass of plurisubharmonic functions with symmetry in $${{\mathbb {C}}}^2$$

On the residual Monge–Ampère mass of plurisubharmonic functions with symmetry in $${{\mathbb {C}}}^2$$

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Quasi-monotone convergence of plurisubharmonic functions

The complex Monge-Ampère operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded quasi-plurisubharmonic functions by various authors. As this operator is not continuous for the L1-topology, several stronger topologies have been introduced over the last decades to remedy this, while maintaining efficient compactness criteria. The purpose of this note is to show that these stronger topologies are essentially equivalent to the natural quasi-monotone topology that we introduce and study here.

On an injectivity theorem for log-canonical pairs with analytic adjoint ideal sheaves

As an application of the residue functions corresponding to the lc-measures developed by the authors, the proof of the injectivity theorem on compact Kähler manifolds for plt pairs by Matsumura is improved in this article to allow multiplier ideal sheaves of plurisubharmonic functions with neat analytic singularities in the coefficients of the relevant cohomology groups. With the use of a refined version of analytic adjoint ideal sheaves, a plan towards a solution to the generalised version of Fujino’s conjecture (i.e. an injectivity theorem on compact Kähler manifolds for lc pairs with multiplier ideal sheaves) is laid down and, in addition to the result for plt pairs, a proof for lc pairs in dimension 2 2 , which is also an improvement of Matsumura’s result, is given.

Cone of Maximal Subextensions of the Plurisubharmonic Functions

Cone of Maximal Subextensions of the Plurisubharmonic Functions

Some results on the comparison principle for the weighted log canonical thresholds

The purpose of this paper is to establish some results on the comparison principle for the weighted log canonical thresholds of plurisubharmonic functions in case the weight is a measure of the form [Formula: see text] and [Formula: see text] with [Formula: see text], where [Formula: see text] is the Lebesgue measure in [Formula: see text].

Plurisubharmonic Interpolation and Plurisubharmonic Geodesics

We give a short survey on plurisubharmonic interpolation, with a focus on the possibility of connecting two given plurisubharmonic functions by plurisubharmonic geodesics.

A Note on the Weighted Log Canonical Threshold of Toric Plurisubharmonic Functions

A Note on the Weighted Log Canonical Threshold of Toric Plurisubharmonic Functions

Quasi-Projective Manifolds Uniformized by Carathéodory Hyperbolic Manifolds and Hyperbolicity of Their Subvarieties

Abstract Let $M$ be a Carathéodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete Kähler, we show that $M$ admits the Bergman metric. When $M$ is strongly Carathéodory hyperbolic and is the universal covering of a quasi-projective manifold $X$, the Bergman metric can be estimated in terms of a Poincaré-type metric on $X$. It is also proved that any quasi-projective (resp. projective) subvariety of $X$ is of log-general type (resp. general type), a result consistent with a conjecture of Lang.

Continuity of the Perron–Bremermann envelope of plurisubharmonic functions

Continuity of the Perron–Bremermann envelope of plurisubharmonic functions