Analyticity of sets associated to Lelong numbers and the extension of closed positive currents

1974 ◽  
Vol 27 (1-2) ◽  
pp. 53-156 ◽  
Author(s):  
Yum-Tong Siu
Keyword(s):  
Lelong Numbers ◽  
Positive Currents

scholarly journals Relative non-pluripolar product of currents

2021 ◽  
Author(s):  
Duc-Viet Vu
Keyword(s):  
Kähler Manifold ◽  
Monotonicity Property ◽  
Necessary Condition ◽  
Positive Current ◽  
Lelong Numbers ◽  
Kahler Manifold ◽  
Closed Positive Current ◽  
Positive Currents ◽  
Compact Kähler Manifold

AbstractLet X be a compact Kähler manifold. Let $$T_1, \ldots , T_m$$ T 1 , … , T m be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of $$T_1, \ldots , T_m$$ T 1 , … , T m . We recover the well-known non-pluripolar product of $$T_1, \ldots , T_m$$ T 1 , … , T m when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.

scholarly journals Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junyan Cao ◽  
Henri Guenancia ◽  
Mihai Păun
Keyword(s):  
General Type ◽  
Einstein Metrics ◽  
Kodaira Dimension ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
Fiber Space ◽  
Generic Fiber ◽  
Lelong Numbers ◽  
Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

scholarly journals Polynomial hulls and positive currents

2003 ◽  
Vol 12 (3) ◽  
pp. 317-334 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Mark G. Lawrence
Keyword(s):  
Positive Currents ◽  
Polynomial Hulls

scholarly journals Dimension of ergodic measures and currents on

2019 ◽  
Vol 40 (8) ◽  
pp. 2131-2155
Author(s):  
CHRISTOPHE DUPONT ◽  
AXEL ROGUE
Keyword(s):  
Upper Bound ◽  
Equilibrium Measure ◽  
Pointwise Dimension ◽  
Ergodic Measures ◽  
Positive Currents ◽  
Green Current

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^{2}$ of degree $d\geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $\unicode[STIX]{x1D708}$. We infer several applications. The first one is an upper bound for the lower pointwise dimension of the equilibrium measure, towards a Binder–DeMarco’s formula for this dimension. The second one shows that every current $S$ containing a measure of entropy $h_{\unicode[STIX]{x1D708}}>\log d$ has a directional dimension ${>}2$, which answers a question of de Thélin–Vigny in a directional way. The last one estimates the dimensions of the Green current of Dujardin’s semi-extremal endomorphisms.

Electrical Stimulation of the Ear: Clinical Applications

1983 ◽  
Vol 92 (6) ◽  
pp. 621-622 ◽  
Author(s):  
M. Portmann ◽  
J.-M. Aran ◽  
M. Nègrevergne ◽  
Y. Cazals
Keyword(s):  
Electrical Stimulation ◽  
Round Window ◽  
Clinical Applications ◽  
Positive Currents ◽  
Stimulation Of

Electrical stimulation of the ear in humans was performed with an extracochlear electrode on the round window. With positive currents, suppression of tinnitus could be induced. With negative currents, auditory sensations were evoked. Since electrical stimulation with DC currents may be hazardous in the long term, it cannot yet be proposed for the suppression of tinnitus. However, electrically evoked hearing sensations with AC currents seem to be of definite interest for some totally deaf patients.

scholarly journals An extremal problem for generalized Lelong numbers

2009 ◽  
Vol 266 (2) ◽  
pp. 345-362
Author(s):  
Alexander Rashkovskii
Keyword(s):  
Extremal Problem ◽  
Lelong Numbers
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