Entire pluricomplex Green functions and Lelong numbers of projective currents

Abstract In this study, Levinson-type inequalities are generalized by using new Green functions and Montgomery identity for the class of k-convex functions (k ≥ 3). Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, a new functional is introduced based on {\mathfrak{f}} divergence and then some estimates for new functional are obtained. Some inequalities for Shannon entropies are obtained too.

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Relative non-pluripolar product of currents

Annals of Global Analysis and Geometry ◽

10.1007/s10455-021-09780-7 ◽

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.

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Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0028 ◽

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).

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