Complex Hessian Operator and Generalized Lelong Numbers Associated to a Closed m-Positive Current

2017 ◽  
Vol 12 (2) ◽  
pp. 475-489
Author(s):  
Dongrui Wan
Keyword(s):  
Positive Current ◽  
Lelong Numbers ◽  
Hessian Operator ◽  
Complex Hessian Operator

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.

The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

2021 ◽  
Vol 127 (2) ◽  
pp. 287-316
Author(s):  
Ayoub El Gasmi
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Borel Measure ◽  
Positive Borel Measure ◽  
Hessian Equation ◽  
Hessian Operator ◽  
Hyperconvex Domain ◽  
Complex Hessian Operator

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

Hessian boundary measures

2019 ◽  
Vol 30 (03) ◽  
pp. 1950016
Author(s):  
Van Thien Nguyen
Keyword(s):  
Subharmonic Functions ◽  
Plurisubharmonic Functions ◽  
Lelong Number ◽  
Boundary Measure ◽  
Hessian Operator ◽  
Complex Hessian Operator ◽  
Monge Ampere

We will study certain boundary measures related to [Formula: see text]-subharmonic functions on [Formula: see text]-hyperconvex domains. These measures generalize the boundary measures studied by Wan and Wang (see [Complex Hessian operator and Lelong number for unbounded [Formula: see text]-subharmonic functions, Potential. Anal. 44(1) (2016) 53–69]). For the case of plurisubharmonic functions ([Formula: see text]) the boundary measure has been studied by Cegrell and Kemppe (see [Monge–Ampère boundary measures, Ann. Polon. Math. 96 (2009) 175–196]).

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).

3. On Diamagnetic Rotation

1878 ◽  
Vol 9 ◽  
pp. 85-92 ◽  
Author(s):  
George Forbes ◽  
J. Clerk Maxwell
Keyword(s):  
Small Angle ◽  
Vertical Direction ◽  
Positive Current ◽  
End To End ◽  
Pass Through ◽  
Lines Of Force

Faraday's discovery of the magnetic rotatory polarisation of light may be expressed in the following manner:—Let two electromagnets, in the form of iron tubes, surrounded by helices of wire, be placed end to end, so that in the space between them the lines of force are very intense. Let a rod of dense glass be placed in this space, so that a ray of light may pass through the two tubes and the rod of glass. Let such a ray on entrance be plane-polarised, so that the direction of vibration is in a vertical direction. If the electro-magnet be now magnetised, the emergent ray will be polarised, so that its vibrations are inclined to the vertical at a small angle. The direction in which the line of vibration has been rotated is the same as the direction of the positive current in the helices.

Past positive current negative crossmatch considerations revisited

1994 ◽  
Vol 40 ◽  
pp. 18
Author(s):  
R.H. Kerman ◽  
C.T. Van Buren ◽  
S. Katz ◽  
R. Lewis ◽  
A. Heydari ◽  
...  
Keyword(s):  
Positive Current
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