On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature

2010 ◽  
Vol 62 (1) ◽  
pp. 3-18
Author(s):  
Boudjemâa Anchouche
Keyword(s):  
Asymptotic Behavior ◽  
Line Bundle ◽  
Ricci Curvature ◽  
Projective Variety ◽  
Volume Form ◽  
Positive Ricci Curvature ◽  
Standard Type ◽  
Number Of Components ◽  
Holomorphic Sections ◽  
Volume Forms

AbstractLet (X, g) be a complete noncompact Kähler manifold, of dimension n ≥ 2, with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that X can be compactified, i.e., X is biholomorphic to a quasi-projective variety. The aim of this paper is to prove that the L2 holomorphic sections of the line bundle K−qXand the volume form of the metric g have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric g and of the Fubini-Study metric induced on X. In the case of dimC X = 2, we establish a relation between the number of components of the divisor D and the dimension of the.

scholarly journals Growth of balls of holomorphic sections on projective toric varieties

2016 ◽  
Vol 27 (04) ◽  
pp. 1650028
Author(s):  
Mounir Hajli
Keyword(s):  
Asymptotic Behavior ◽  
Line Bundle ◽  
Unit Ball ◽  
Toric Variety ◽  
Toric Varieties ◽  
Holomorphic Sections ◽  
Complex Projective ◽  
Toric Divisor

Let [Formula: see text] be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety [Formula: see text]. Given a continuous toric metric [Formula: see text] on [Formula: see text], we define the energy at equilibrium of [Formula: see text] where [Formula: see text] is the weight of the metrized toric divisor [Formula: see text]. We show that this energy describes the asymptotic behavior as [Formula: see text] of the volume of the [Formula: see text]-norm unit ball induced by [Formula: see text] on the space of global holomorphic sections [Formula: see text].

scholarly journals Skoda division of line bundle sections and pseudo-division

2016 ◽  
Vol 27 (05) ◽  
pp. 1650042 ◽  
Author(s):  
Dano Kim
Keyword(s):  
Line Bundle ◽  
Projective Variety ◽  
Line Bundles ◽  
Finite Generation ◽  
Division Theorem ◽  
Holomorphic Sections ◽  
Effective Nullstellensatz

We first present a Skoda-type division theorem for holomorphic sections of line bundles on a projective variety which is essentially the most general, compared to previous ones. Then we revisit Geometric Effective Nullstellensatz and observe that even this general Skoda division is far from sufficient to yield stronger GEN such as ‘vanishing order [Formula: see text] division’, which could be used for finite generation of section rings by the basic finite generation lemma. To resolve this problem, we develop a notion of pseudo-division and show that it can replace the usual division in the finite generation lemma. We also give a vanishing order 1 pseudo-division result when the line bundle is ample.

scholarly journals Asymptotics of random Betti tables

2015 ◽  
Vol 2015 (702) ◽  
Author(s):  
Lawrence Ein ◽  
Daniel Erman ◽  
Robert Lazarsfeld
Keyword(s):  
Asymptotic Behavior ◽  
Line Bundle ◽  
Normal Distribution ◽  
Gaussian Distribution ◽  
Projective Variety ◽  
Betti Numbers ◽  
Fixed Number ◽  
The Ideal

AbstractThis paper is motivated by the question of understanding the asymptotic behavior of the Betti numbers of the resolution of the ideal of a projective variety as the positivity of the embedding line bundle grows. We present a conjecture asserting that these invariants approach a Gaussian distribution, and we verify this in the case of curves. Then we work out the asymptotics of “random” Betti tables with a fixed number of rows, sampled according to a uniform choice of Boij–Söderberg coefficients. This analysis suggests that the normal distribution of Betti numbers is in any event the typical behavior from a probabilistic viewpoint.

scholarly journals LINEAR SYSTEMS ON IRREGULAR VARIETIES

2019 ◽  
Vol 19 (6) ◽  
pp. 2087-2125 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Continuous Function ◽  
Line Bundle ◽  
Linear Systems ◽  
Projective Variety ◽  
Geometrical Properties ◽  
Complex Projective Variety ◽  
Wide Range ◽  
Image Position ◽  
Irregular Varieties ◽  
Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

scholarly journals Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

2018 ◽  
Vol 2019 (19) ◽  
pp. 6089-6112
Author(s):  
Shu Kawaguchi ◽  
Kazuhiko Yamaki
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Smooth Projective Variety ◽  
Discrete Valuation Ring ◽  
Complete Discrete Valuation Ring ◽  
System L ◽  
Semistable Model ◽  
Adjoint Bundle

Abstract Let R be a complete discrete valuation ring of equi-characteristic zero with fraction field K. Let X be a connected smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model ${\mathscr {X}}$ of X over R and a relatively ample line bundle ${\mathscr {L}}$ over ${\mathscr {X}}$ with $\left .{{\mathscr {L}}}\right \vert_{{X}} \cong L$. Let $S({\mathscr {X}})$ be the skeleton associated to ${\mathscr {X}}$ in the Berkovich analytification Xan of X. In this article, we study when $S({\mathscr {X}})$ is faithfully tropicalized into tropical projective space by the adjoint linear system |L⊗m ⊗ ωX|. Roughly speaking, our results show that if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S({\mathscr {X}})$.

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