Vanishing Theorems on Complex Manifolds

AbstractIn this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

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Regular Holonomic $D$-modules and Distributions on Complex Manifolds

Complex Analytic Singularities ◽

10.2969/aspm/00810199 ◽

2018 ◽

Cited By ~ 2

Author(s):

Masaki Kashiwara

Keyword(s):

Complex Manifolds ◽

D Modules

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A characterization of complex manifolds biholomorphic to a circular domain

Mathematische Zeitschrift ◽

10.1007/bf01164158 ◽

1985 ◽

Vol 189 (3) ◽

pp. 343-363 ◽

Cited By ~ 12

Author(s):

Giorgio Patrizio

Keyword(s):

Circular Domain ◽

Complex Manifolds

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Generalized vanishing theorems for Yukawa couplings in heterotic compactifications

Journal of High Energy Physics ◽

10.1007/jhep05(2021)085 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Lara B. Anderson ◽

James Gray ◽

Magdalena Larfors ◽

Matthew Magill ◽

Robin Schneider

Keyword(s):

Vector Bundles ◽

Geometric Approach ◽

Low Energy ◽

Geometric Features ◽

Vanishing Theorems ◽

Vanishing Theorem ◽

Alternative Derivation ◽

Yukawa Couplings ◽

Geometric Methods ◽

Higher Dimensional

Abstract Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results used differential geometric methods to explain the origin of some of this structure [1, 2]. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved descended from vector bundles on those larger manifolds. In this paper, we utilize an algebro-geometric approach to provide an alternative derivation of some of these results, and are thus able to generalize them to a much wider arena than has been considered before. For example, we consider cases where the vector bundles of interest do not descend from bundles on the ambient space. In such a manner we are able to highlight the ubiquity with which textures of vanishing Yukawa couplings can be expected to arise in heterotic compactifications, with multiple different constraints arising from a plethora of different geometric features associated to the gauge bundle.

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On contact structures of real and complex manifolds

Tohoku Mathematical Journal ◽

10.2748/tmj/1178243807 ◽

1963 ◽

Vol 15 (3) ◽

pp. 227-252 ◽

Cited By ~ 9

Author(s):

Seizi Takizawa

Keyword(s):

Contact Structures ◽

Complex Manifolds

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Schottky groups acting on homogeneous rational manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0065 ◽

2019 ◽

Vol 2019 (753) ◽

pp. 23-56 ◽

Cited By ~ 1

Author(s):

Christian Miebach ◽

Karl Oeljeklaus

Keyword(s):

Deformation Theory ◽

Group Actions ◽

Picard Group ◽

Schottky Group ◽

Fundamental Groups ◽

Complex Manifolds ◽

Schottky Groups ◽

Algebraic Dimension ◽

Geometric Invariants ◽

Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

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Higher-page Bott–Chern and Aeppli cohomologies and applications

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0014 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Dan Popovici ◽

Jonas Stelzig ◽

Luis Ugarte

Keyword(s):

Positive Integer ◽

Spectral Sequence ◽

Complex Manifolds ◽

Serre Duality ◽

Frölicher Spectral Sequence ◽

Compact Complex Manifolds

Abstract For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

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