scholarly journals A Takayama-type extension theorem

2008 ◽  
Vol 144 (2) ◽  
pp. 522-540 ◽  
Author(s):  
Dror Varolin
Keyword(s):  
Line Bundle ◽  
Normal Bundle ◽  
Extension Theorem ◽  
Canonical Bundle ◽  
Holomorphic Sections

AbstractWe prove a theorem on the extension of holomorphic sections of powers of adjoint bundles from submanifolds of complex codimension 1 having non-trivial normal bundle. The first such result, due to Takayama, considers the case where the canonical bundle is twisted by a line bundle that is a sum of a big and nef line bundle and a $\mathbb {Q}$-divisor that has Kawamata log terminal singularities on the submanifold from which extension occurs. In this paper we weaken the positivity assumptions on the twisting line bundle to what we believe to be the minimal positivity hypotheses. The main new idea is an L2 extension theorem of Ohsawa–Takegoshi type, in which twisted canonical sections are extended from submanifolds with non-trivial normal bundle.

scholarly journals ADJUNCTION FOR VARIETIES WITH A ℂ* ACTION

2020 ◽  
Author(s):  
ELEONORA A. ROMANO ◽  
JAROSŁAW A. WIŚNIEWSKI
Keyword(s):  
Fixed Point ◽  
Line Bundle ◽  
Normal Bundle ◽  
Ample Line Bundle ◽  
Projective Manifold ◽  
Fano Manifolds ◽  
Adjunction Theory ◽  
Rank 2 ◽  
General Orbit ◽  
Complex Projective

Abstract Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

scholarly journals A note on Berezin-Toeplitz quantization of the Laplace operator

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Alberto Della Vedova
Keyword(s):  
Line Bundle ◽  
Laplace Operator ◽  
Adjoint Operator ◽  
Holomorphic Sections ◽  
The Laplace Operator ◽  
Hodge Manifold

AbstractGiven a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.

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 A geometric approach to Orlov’s theorem

2012 ◽  
Vol 148 (5) ◽  
pp. 1365-1389 ◽  
Author(s):  
Ian Shipman
Keyword(s):  
Normal Bundle ◽  
Geometric Approach ◽  
Singular Locus ◽  
Matrix Factorizations ◽  
Canonical Bundle ◽  
Algebraic Construction ◽  
Famous Theorem ◽  
Projective Hypersurface ◽  
Coherent Sheaves ◽  
Formal Neighborhood

AbstractA famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov’s theorem in the Calabi–Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X⊂ℙ be a projective hypersurface. Segal has already established an equivalence between Orlov’s category of graded matrix factorizations and the category of graded D-branes on the canonical bundle Kℙ to ℙ. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on Kℙ and Dbcoh(X). This can be achieved directly, as well as by deforming Kℙ to the normal bundle of X⊂Kℙ and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasiprojective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.

scholarly journals CONVERGENCE OF FUBINI–STUDY CURRENTS FOR ORBIFOLD LINE BUNDLES

2013 ◽  
Vol 24 (07) ◽  
pp. 1350051 ◽  
Author(s):  
DAN COMAN ◽  
GEORGE MARINESCU
Keyword(s):  
Line Bundle ◽  
Asymptotic Distribution ◽  
Line Bundles ◽  
Singular Line ◽  
Distribution Of Zeros ◽  
Bergman Kernels ◽  
Holomorphic Sections ◽  
Asymptotic Distribution Of Zeros

We discuss positive closed currents and Fubini–Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini–Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.

scholarly journals PROJECTIVE EMBEDDINGS AND LAGRANGIAN FIBRATIONS OF KUMMER VARIETIES

2009 ◽  
Vol 20 (05) ◽  
pp. 557-572
Author(s):  
YUICHI NOHARA
Keyword(s):  
Line Bundle ◽  
Ample Line Bundle ◽  
Theta Functions ◽  
Torus Action ◽  
Natural Basis ◽  
Tensor Power ◽  
Hausdorff Topology ◽  
Lagrangian Fibration ◽  
Holomorphic Sections ◽  
Gromov Hausdorff Topology

It is known that holomorphic sections of an ample line bundle L (and its tensor power Lk) over an Abelian variety A are given by theta functions. Moreover, a natural basis of the space of holomorphic sections of L or Lk is related to a certain Lagrangian fibration of A. In our previous paper, we studied projective embeddings of A defined by these basis for Lk. For a natural torus action on the ambient projective space, it is proved that its moment map, restricted to A, approximates the Lagrangian fibration of A for large k, with respect to the "Gromov–Hausdorff topology". In this paper, we prove that the same is true for the Kummer variety associated to A.

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 Convexité holomorphe intermédiaire des revetements d'un domaine pseudoconvexe

1997 ◽  
Vol 56 (2) ◽  
pp. 285-290
Author(s):  
S. Asserda
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Pseudoconvex Domain ◽  
Holomorphic Line Bundle ◽  
Hermitian Metric ◽  
Holomorphic Sections ◽  
Positive Holomorphic Line Bundle ◽  
Holomorphic Covering

Let M be a complex manifold and L → M be a positive holomorphic line bundle over M equipped with a Hermitian metric h of class C2. If D ⊂⊂ M is a pseudoconvex domain which is relatively compact in M then there exists an integer r0 such that for every r ≥ r0 and for every connected holomorphic covering π: the covering is holomorphically convex with respect to holomorphic sections of .

scholarly journals HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

2018 ◽  
Vol 238 ◽  
pp. 104-136
Author(s):  
TAKAYUKI KOIKE
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Normal Bundle ◽  
Holomorphic Foliation ◽  
Compact Complex Manifold ◽  
Sufficient Condition ◽  
Existence Problem ◽  
Hermitian Metric ◽  
Flat Normal Bundle ◽  
Compact Submanifold

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

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