Open Access Published by De Gruyter Open Access

Volume 5 Issue 1

January 2018

Issue of Complex Manifolds

Contents
Journal Overview

Open Access February 2, 2018

A generalized Schwarz lemma for two domains related to μ-synthesis

Sourav Pal, Samriddho Roy

Page range: 1-8

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Abstract

We present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock E. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc G 2 . In either case, our results generalize all previous results in this direction for E and G 2 .

Open Access February 2, 2018

Group actions, non-Kähler complex manifolds and SKT structures

Mainak Poddar, Ajay Singh Thakur

Page range: 9-25

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Abstract

We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the principal bundle admits SKT metrics. This generalizes recent results of Grantcharov et al. We study the Picard group and the algebraic dimension of the total space in some cases. We also use a slightly generalized version of the construction to obtain (non-Kähler) complex structures on tangential frame bundles of complex orbifolds.

Open Access February 2, 2018

Contact manifolds, Lagrangian Grassmannians and PDEs

Olimjon Eshkobilov, Gianni Manno, Giovanni Moreno, Katja Sagerschnig

Page range: 26-88

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Abstract

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

Open Access February 16, 2018

A Family of Complex Nilmanifolds with in finitely Many Real Homotopy Types

Adela Latorre, Luis Ugarte, Raquel Villacampa

Page range: 89-102

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Abstract

We find a one-parameter family of non-isomorphic nilpotent Lie algebras ga, with a > [0,∞), of real dimension eight with (strongly non-nilpotent) complex structures. By restricting a to take rational values, we arrive at the existence of infinitely many real homotopy types of 8-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.

Open Access March 24, 2018

Examples of solvmanifolds without LCK structures

Hiroshi Sawai

Page range: 103-110

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Abstract

The purpose in this paper is to construct solvmanifolds without LCK structures such that the complex structure is left-invariant

Open Access May 18, 2018

Benenti Tensors: A useful tool in Projective Differential Geometry

Gianni Manno, Andreas Vollmer

Page range: 111-121

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Abstract

Two metrics are said to be projectively equivalent if they share the same geodesics (viewed as unparametrized curves). The degree of mobility of a metric g is the dimension of the space of the metrics projectively equivalent to g. For any pair of metrics (g, ḡ) on the same manifold one can construct a (1, 1)- tensor L(g, ḡ) called the Benenti tensor. In this paper we discuss some geometrical properties of Benenti tensors when (g, ḡ) are projectively equivalent, particularly in the case of degree of mobility equal to 2.

Open Access June 21, 2018

On the stability of harmonic maps under the homogeneous Ricci flow

Rafaela F. do Prado, Lino Grama

Page range: 122-132

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Abstract

In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow.We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.

Open Access July 5, 2018

Convexity theorems for the gradient map on probability measures

Leonardo Biliotti, Alberto Raffero

Page range: 133-145

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Abstract

Given a Kähler manifold (Z, J, ω) and a compact real submanifold M ⊂ Z, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group G on the space of probability measures on M. In particular, we prove convexity results for such map when G is Abelian and we investigate how to extend them to the non-Abelian case.

Open Access July 31, 2018

Picard Group and Fundamental Group of the Moduli of Higgs Bundles on Curves

Sujoy Chakraborty, Arjun Paul

Page range: 146-149

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Abstract

Let X be an irreducible smooth projective curve of genus g ≥ 2 over ℂ. Let M G, Higgs δ be a connected reductive affine algebraic group over ℂ. Let Higgs be the moduli space of semistable principal G-Higgs bundles on X of topological type δ∈π1(G). In this article,we compute the fundamental group and Picard group of

Open Access August 1, 2018

Complex structures on the complexification of a real Lie algebra

Takumi Yamada

Page range: 150-157

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Abstract

Let g = a+b be a Lie algebra with a direct sum decomposition such that a and b are Lie subalgebras. Then, we can construct an integrable complex structure J̃ on h = ℝ (g ℂ ) from the decomposition, where ℝ (g ℂ ) is a real Lie algebra obtained from g ℂ by the scalar restriction. Conversely, let J̃ be an integrable complex structure on h = ℝ (g ℂ ). Then, we have a direct sum decomposition g = a + b such that a and b are Lie subalgebras. We also investigate relations between the decomposition g = a + b and dim H s.t ∂̄ J̃ (h ℂ ).

Open Access October 5, 2018

A Dual of the Chow Transformation

Michel Méo

Page range: 158-194

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Abstract

We define a dual of the Chow transformation of currents on the complex projective space. This transformation factorizes a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear diferential operator. In such a way we complete the general scheme of integral geometry for the Chow transformation. On another hand we prove the existence of a well defined closed positive conormal current associated to every closed positive current on the projective space. This is a consequence of the existence of a dual current, defined on the dual projective space. This allows us to extend to the case of a closed positive current the known inversion formula for the conormal of the Chow divisor of an effective algebraic cycle.

Open Access November 10, 2018

Stable Higgs bundles over positive principal elliptic fibrations

Indranil Biswas, Mahan Mj, Misha Verbitsky

Page range: 195-201

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Abstract

Let M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B 0 , where B 0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B 0 , Π ⃰ ɸ X ), where (B0, ɸ X ) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

About this journal

Special Issues:

Generalized Geometry

Complex Manifolds is a fully peer-reviewed open access electronic journal that publishes cutting-edge research on complex manifolds and related results from differential geometry, algebraic geometry and complex analysis. We strive to present a forum where all aspects of these problems can be discussed.

Complex Manifolds is devoted to the publication of results on these and related topics:

Hermitian geometry, Kähler and hyperkähler geometry
Calabi-Yau metrics, PDE's on complex manifolds
Generalized complex geometry
Deformations of complex structures
Twistor theory
Geometric flows on complex manifolds
Almost complex geometry
Quaternionic geometry
Geometric theory of analytic functions
Holomorphic dynamics
Several complex variables
Dolbeault cohomology
CR geometry

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