Related papers: Multi-circled singularities, Lelong numbers, and i…
In this paper, we combine tools from pluripotential theory and commutative algebra to study singularity invariants of plurisubharmonic functions. We establish several relationships between the singularity invariants of plurisubharmonic…
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…
We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge--Amp\`ere masses. They are characterized in terms of Green functions…
We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are…
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…
Two properties of plurisubharmonic functions are proven. The first result is a Skoda type integrability theorem with respect to a Monge-Amp\`ere mass with H\"older continuous potential. The second one says that locally, a p.s.h. function is…
We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and…
We show that a Skoda complex with a general plurisubharmonic weight function is exact if its 'degree' is sufficiently large. This answers a question of Lazarsfeld and implies that not every integrally closed ideal is equal to a multiplier…
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…
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…
In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic…
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…
In this paper, we study the coherence of a higher rank analogue of a multiplier ideal sheaf. Key tools of the study are H\"ormander's $L^2$-estimate and a singular version of a Demailly--Skoda type result.
A notion of local indicator for a plurisubharmonic function is introduced. The indicator is a certain plurisubharmonic function in the unit polydisc, which controls the behavior of the considered function near a fixed point of its…
We prove a conjecture proposed by Berman-Boucksom-Eyssidieux-Guedj-Zeriahi, affirming that the Demailly-Lelong number can be determined through a combination of intersection numbers given by the divisorial part of the potential and the SNC…
A positive correlation inequality is established for circular-invariant plurisubharmonic functions, with respect to complex Gaussian measures. The main ingredients of the proofs are the Ornstein-Uhlenbeck semigroup, and another natural…
We describe briefly a new approach to some problems related to Teichm\"uller spaces, invariant metrics, and extremal quasiconformal maps. This approach is based on the properties of plurisubharmonic functions, especially of the…
This paper is devoted to stability analysis of continuous-time delay systems based on a set of Lyapunov-Krasovskii functionals. New multiple integral inequalities are derived that involve the famous Jensen's and Wirtinger's inequalities, as…
We study geodesics for plurisubharmonic functions from the Cegrell class ${\mathcal F}_1$ on a bounded hyperconvex domain of ${\mathbb C}^n$ and show that, as in the case of metrics on K\"{a}hler compact menifolds, they linearize an energy…
We give a proof of the openness conjecture of Demailly and Koll\'ar.