scholarly journals Extension of the curvature form of the relative canonical line bundle on families of Calabi–Yau manifolds and applications

2021 ◽  
pp. 1-14
Author(s):  
Young-Jun Choi ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Curvature Form ◽  
Canonical Line Bundle ◽  
Calabi Yau Manifolds

scholarly journals Heterotic line bundle models on generalized complete intersection Calabi Yau manifolds

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Magdalena Larfors ◽  
Davide Passaro ◽  
Robin Schneider
Keyword(s):  
Line Bundle ◽  
Complete Intersection ◽  
Particle Physics ◽  
Model Building ◽  
Wilson Line ◽  
Symmetry Groups ◽  
The Standard Model ◽  
Order Of Magnitude ◽  
Systematic Program ◽  
Calabi Yau Manifolds

Abstract The systematic program of heterotic line bundle model building has resulted in a wealth of standard-like models (SLM) for particle physics. In this paper, we continue this work in the setting of generalised Complete Intersection Calabi Yau (gCICY) manifolds. Using the gCICYs constructed in ref. [1], we identify two geometries that, when combined with line bundle sums, are directly suitable for heterotic GUT models. We then show that these gCICYs admit freely acting ℤ2 symmetry groups, and are thus amenable to Wilson line breaking of the GUT gauge group to that of the standard model. We proceed to a systematic scan over line bundle sums over these geometries, that result in 99 and 33 SLMs, respectively. For the first class of models, our results may be compared to line bundle models on homotopically equivalent Complete Intersection Calabi Yau manifolds. This shows that the number of realistic configurations is of the same order of magnitude.

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.

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.

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].

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.

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