arXiv subjects

Zbigniew Błocki

Publications and source records attributed to Zbigniew Błocki.

5 recordsLinked to original sources

arXiv2026

Sharp $C^{1,\bar1}$ estimates in Kähler quantization and non-pluripolar Radon measures

Let $K_φ$ denote the weighted Bergman kernel associated to a plurisubharmonic function $φ$. We obtain upper bounds and positive lower bounds for the Bergman metric $i\partial \bar{\partial} \log K_φ$, expressed solely in terms of upper bounds and positive lower bounds of $i\partial \bar{\partial}φ$. Our approach applies in both local and compact Kähler settings. As an immediate application we obtain the optimal $C^{1,α}$-convergence for the quantization of Kähler currents with bounded coefficients. We also show that any non-pluripolar Radon measure on a compact Kähler manifold admits a quantization.

math.DG↗

arXiv2017

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

We study continuity properties of generalized Monge-Ampère 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-Ampère measure of such a function on a compact Kähler manifold.

math.CV↗

arXiv2017

One dimensional estimates for the Bergman kernel and logarithmic capacity

Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non simply connected domains, generalizing a recent example of Fornæss.

math.CV↗

arXiv2014

On the Suita conjecture for some convex ellipsoids in $\mathbb C^2$

It has been recently shown that for a convex domain $Ω$ in $\mathbb C^n$ and $w\inΩ$ the function $F_Ω(w):=\big(K_Ω(w)λ(I_Ω(w))\big)^{1/n}$, where $K_Ω$ is the Bergman kernel on the diagonal and $I_Ω(w)$ the Kobayashi indicatrix, satisfies $1\leq F_Ω\leq 4$. While the lower bound is optimal, not much more is known about the upper bound. In general it is quite difficult to compute $F_Ω$ even numerically and the highest value of it obtained so far is $1.010182\dots$ In this paper we present precise, although rather complicated formulas for the ellipsoids $Ω=\{|z_1|^{2m}+|z_2|^2<1\}$ (with $m\geq 1/2$) and all $w$, as well as for $Ω=\{|z_1|+|z_2|<1\}$ and $w$ on the diagonal. The Bergman kernel for those ellipsoids had been known, the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case the function $λ(I_Ω(w))$ is not $C^{3,1}$.

math.CV↗

arXiv2014

Estimates for the Bergman Kernel and the Multidimensional Suita Conjecture

We study the lower bound for the Bergman kernel in terms of volume of sublevel sets of the pluricomplex Green function. We show that it implies a bound in terms of volume of the Azukawa indicatrix which can be treated as a multidimensional version of the Suita conjecture. We also prove that the corresponding upper bound holds for convex domains and discuss it in bigger detail on some convex complex ellipsoids.

math.CV↗