scholarly journals Homogeneous variational problems and Lagrangian sections

2016 ◽  
Vol 24 (2) ◽  
pp. 115-123
Author(s):  
D. J. Saunders
Keyword(s):  
Line Bundle ◽  
Tangent Bundle ◽  
Regularity Condition ◽  
Variational Problems ◽  
Canonical Line Bundle ◽  
Homogeneous Section ◽  
Parametrized Family

Abstract We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

Equivariant Forms: Structure and Geometry

2013 ◽  
Vol 56 (3) ◽  
pp. 520-533 ◽  
Author(s):  
Abdelkrim Elbasraoui ◽  
Abdellah Sebbar
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Modular Forms ◽  
Differential Forms ◽  
Cross Ratio ◽  
Canonical Line Bundle ◽  
The Cross

Abstract.In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of SL2(ℤ) by means of the cross-ratio, weight 2 modular forms, quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

Double Covers and Metastable Immersions of Spheres

1974 ◽  
Vol 26 (1) ◽  
pp. 145-176 ◽  
Author(s):  
Robert Wells
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Projective Space ◽  
Unit Sphere ◽  
Real Line ◽  
Double Cover ◽  
Sphere Bundle ◽  
Canonical Line Bundle ◽  
Projective Space Bundle ◽  
Double Covers

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

scholarly journals Factorization of Generalized Theta Functions Revisited

2017 ◽  
Vol 24 (01) ◽  
pp. 1-52
Author(s):  
Xiaotao Sun
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Theta Functions ◽  
Vanishing Theorems ◽  
Smooth Curves ◽  
Canonical Line Bundle

This survey is based on my lectures given in the last few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorems, we show, without using vanishing theorems, a new observation that [Formula: see text] is independent of all of the choices for any smooth curves. The estimate of various codimensions and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

scholarly journals AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS

2017 ◽  
Vol 18 (06) ◽  
pp. 1331-1340 ◽  
Author(s):  
Dawei Chen
Keyword(s):  
Line Bundle ◽  
Algebraic Curve ◽  
Invariant Manifolds ◽  
Affine Variety ◽  
Affine Invariant ◽  
Algebraic Varieties ◽  
Hurwitz Spaces ◽  
Smooth Curves ◽  
Canonical Line Bundle ◽  
Teichmuller Dynamics

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper, we study affine-related properties of strata of $k$ -differentials on smooth curves which parameterize sections of the $k$ th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$ , then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

EXAMPLES OF A NON-VANISHING CONJECTURE OF KAWAMATA

2007 ◽  
Vol 18 (05) ◽  
pp. 527-533
Author(s):  
YU-LIN CHANG
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Symmetric Spaces ◽  
Sufficient Conditions ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Compact Type ◽  
Hermitian Symmetric Spaces ◽  
Canonical Line Bundle ◽  
Positive Holomorphic Line Bundle

Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H0(M, K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow–Siu [10] surfaces, Kähler threefolds given by Deraux [3] and examples of Zheng [17].

Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients

2018 ◽  
Vol 30 (2) ◽  
pp. 419-432
Author(s):  
Sai-Kee Yeung
Keyword(s):  
Line Bundle ◽  
Projective Planes ◽  
Canonical Line Bundle ◽  
Ball Quotients ◽  
Very Ampleness ◽  
Minimal Type

AbstractThe purpose of this note is to show that{2K}of any smooth compact complex 2-ball quotient is very ample, except possibly for four pairs of fake projective planes of minimal type, whereKis the canonical line bundle. For the four pairs of fake projective planes, the sections of{2K_{M}}give an embedding ofMexcept possibly for at most two points onM.

scholarly journals On the Negativity of Moduli Spaces for Polarized Manifolds

2021 ◽  
Author(s):  
Kang Zuo
Keyword(s):  
Line Bundle ◽  
Moduli Spaces ◽  
Complex Analysis ◽  
Projective Varieties ◽  
Canonical Line Bundle ◽  
Characteristic Varieties ◽  
Almost All ◽  
Complex Projective ◽  
Made In

AbstractGiven a log base space (Y, S), parameterizing a smooth family of complex projective varieties with semi-ample canonical line bundle, we briefly recall the construction of the deformation Higgs sheaf and the comparison map on (Y, S) made in the work by Viehweg–Zuo. While almost all hyperbolicities in the sense of complex analysis such as Brody, Kobayashi, big Picard and Viehweg hyperbolicities of the base U = Y ∖ S (under some technical assumptions) follow from the negativity of the kernel of the deformation Higgs bundle we pose a conjecture on the topological hyperbolicity on U. In order to study the rigidity problem we then introduce the notions of the length and characteristic varieties of a family f : X → Y, which provide an infinitesimal characterization of products of sub log pairs in (Y, S) and an upper bound for the number of subvarieties appearing as factors in such a product. We formulate a conjecture on a characterization of non-rigid families of canonically polarized varieties.

Sign in / Sign up

Export Citation Format

Share Document