scholarly journals Lelong numbers of bidegree (1, 1) currents on multiprojective spaces

2019 ◽  
Vol 295 (3-4) ◽  
pp. 1569-1582
Author(s):  
Dan Coman ◽  
James Heffers
Keyword(s):  
Lelong Numbers ◽  
Multiprojective Spaces

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 WEAKLY UNIFORM RANK TWO VECTOR BUNDLES ON MULTIPROJECTIVE SPACES

2011 ◽  
Vol 84 (2) ◽  
pp. 255-260
Author(s):  
EDOARDO BALLICO ◽  
FRANCESCO MALASPINA
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Splitting Type ◽  
Multiprojective Spaces

AbstractHere we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.

Special arrangements of lines: Codimension 2 ACM varieties in ℙ1 × ℙ1 × ℙ1

2019 ◽  
Vol 18 (04) ◽  
pp. 1950073 ◽  
Author(s):  
Giuseppe Favacchio ◽  
Elena Guardo ◽  
Beatrice Picone
Keyword(s):  
Point Of View ◽  
Arrangements Of Lines ◽  
Multiprojective Spaces ◽  
Combinatorial Algebra

In this paper, we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimension 2 arithmetically Cohen–Macaulay (ACM) varieties in [Formula: see text], called varieties of lines. We also describe their ACM property from a combinatorial algebra point of view.

scholarly journals Multiprojective spaces and the arithmetically Cohen–Macaulay property

2018 ◽  
Vol 166 (3) ◽  
pp. 583-597 ◽  
Author(s):  
GIUSEPPE FAVACCHIO ◽  
JUAN MIGLIORE
Keyword(s):  
Ambient Space ◽  
Inclusion Property ◽  
New Construction ◽  
Multiprojective Spaces

AbstractIn this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1× ℙ1and, more recently, in (ℙ1)r. In ℙ1× ℙ1the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm× ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1× ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.

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

scholarly journals On the geometry of bifurcation currents for quadratic rational maps

2014 ◽  
Vol 35 (5) ◽  
pp. 1369-1379 ◽  
Author(s):  
FRANÇOIS BERTELOOT ◽  
THOMAS GAUTHIER
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Complex Projective Space ◽  
The Self ◽  
Rational Maps ◽  
Two Dimensional ◽  
Lelong Numbers ◽  
Complex Projective

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive $(1,1)$-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

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