Densité des fonctions plurisousharmoniques

Regularization of closed positive currents of type (1,1) by the flow of a Chern connection

“…-Following an idea of Kiselman [Ki79], we let ψ c,ε be the Legendre transform ψ c,ε (x) = inf |w|<1 Ψ(x, εw) + ε 1 − |w| 2 − c log |w| , where Ψ(x, w) = Ψ(x, w) + |w| is the function defined in Remark 4.7. It is clear that ψ c,ε is increasing in ε and that lim ε→0 ψ c,ε (x) = Ψ(x, 0 + ) = Ψ(x, 0 + ) = ψ(x).…”

“…What is perhaps most remarkable is that we have been ultimately able to find the complete Taylor expansion of the exponential convolution kernel (Proposition 3.8); this is indeed possible because we use a modified exponential map which is made fiberwise quasi-holomorphic (see Section 2). Finally, we apply Kiselman's singularity attenuation technique [Ki78], [Ki79] in combination with our main estimates to define a partial regularization process for closed (1, 1)-currents: in that way, the Lelong numbers can be killed up to any given level (Theorem 6.1).…”

Lelong numbers with respect to regular plurisubharmonic weights

“…When u = log |f |, f being a holomorphic function with f (x) = 0, ν(u, x) is just the multiplicity of the zero of f at the point x. The Lelong number can also be calculated as ν(u, x) = lim r→−∞ r −1 sup{u(z) : |z − x| ≤ e r } = lim r→−∞ r −1 M(u, x, r), (1.1) where M(u, x, r) is the mean value of u over the sphere |z − x| = e r , see [6]. Various results on Lelong numbers and their applications to complex analysis can be found in [9], [11], [5], [10].…”

Cauchy–Riemann meet Monge–Ampère

“…It was later simplified and generalized by Kiselman [75,77] (see also [66]) and Demailly [44]. It was Demailly [45] who found a surprisingly simple proof of the Siu theorem using the Ohsawa-Takegoshi theorem.…”