Topological Invariants and Differential Geometry

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Chern Class ◽  
Euler Number ◽  
Line Bundles ◽  
Topological Invariants ◽  
Fundamental Groups ◽  
Hermitian Metrics ◽  
Finite Set ◽  
First Chern Class

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.

PRINCIPAL BUNDLES ON RATIONALLY CONNECTED FIBRATIONS OVER ABELIAN VARIETIES

2009 ◽  
Vol 20 (02) ◽  
pp. 167-188
Author(s):  
INDRANIL BISWAS
Keyword(s):  
Algebraic Group ◽  
Chern Class ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
Line Bundles ◽  
Projective Manifold ◽  
Galois Cover ◽  
Rationally Connected ◽  
First Chern Class ◽  
Complex Projective

Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2( ad (EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover [Formula: see text] and a reduction of structure group [Formula: see text] to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent.

Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems

2000 ◽  
Vol 52 (3) ◽  
pp. 582-612 ◽  
Author(s):  
Lisa C. Jeffrey ◽  
Jonathan Weitsman
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Chern Class ◽  
Boundary Component ◽  
Symplectic Geometry ◽  
Line Bundles ◽  
Vanishing Theorems ◽  
Stable Bundles ◽  
Flat Connections ◽  
First Chern Class

AbstractThis paper treats the moduli space g,1(Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component which send the loop around the boundary to an element conjugate to exp Λ, where Λ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over g,1(Λ) and exhibit natural homology cycles representing the Poincaré dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank k and degree d.

scholarly journals The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

2021 ◽  
Vol 9 ◽  
Author(s):  
Patrick Graf ◽  
Martin Schwald
Keyword(s):  
Chern Class ◽  
Cohomology Group ◽  
Canonical Bundle ◽  
Deformation Space ◽  
Projective Varieties ◽  
Quotient Singularities ◽  
Finite Quotient ◽  
First Chern Class ◽  
Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

scholarly journals Chern Class Inequalities on Polarized Manifolds and Nef Vector Bundles

2020 ◽  
Author(s):  
Ping Li ◽  
Fangyang Zheng
Keyword(s):  
Chern Class ◽  
Vector Bundles ◽  
Euler Number ◽  
Type Inequality ◽  
Chern Number ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bisectional Curvature ◽  
Chern Numbers ◽  
Nef Vector Bundles

Abstract This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our 1st main result is a family of sharp Chern class inequalities. Among them the 1st one is a variant of a classical result and the equality case of the 2nd one is a characterization of hypersurfaces. The 2nd main result is a Chern number inequality on it, which includes a reverse Miyaoka–Yau-type inequality. The 3rd main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bounded below by the Euler number. As an application, we classify compact Kähler manifolds with nonnegative bisectional curvature whose Chern numbers are all positive. A conjecture related to the Euler number of compact Kähler manifolds with nonpositive bisectional curvature is proposed, which can be regarded as a complex analogue to the Hopf conjecture.

TOPOLOGICAL PROPERTIES OF CYCLICALLY PRESENTED GROUPS

2003 ◽  
Vol 12 (02) ◽  
pp. 243-268 ◽  
Author(s):  
ALBERTO CAVICCHIOLI ◽  
DUŠAN REPOVŠ ◽  
FULVIA SPAGGIARI
Keyword(s):  
Finite Volume ◽  
High Dimensional ◽  
Fundamental Groups ◽  
Topological Properties ◽  
3 Dimensional ◽  
Hnn Extensions ◽  
Finite Set ◽  
Split Extensions

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

The First Chern Class of the Verlinde Bundles

2015 ◽  
pp. 87-111 ◽  
Author(s):  
Alina Marian ◽  
Dragos Oprea ◽  
Rahul Pandharipande
Keyword(s):  
Chern Class ◽  
First Chern Class

Topological invariance of the Hall conductance and quantization

2015 ◽  
Vol 29 (24) ◽  
pp. 1550135
Author(s):  
Paul Bracken
Keyword(s):  
Chern Class ◽  
Fiber Bundle ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Hall Conductance ◽  
Integer Multiple ◽  
Principal Fiber Bundle ◽  
Topological Invariance ◽  
First Chern Class

It is shown that the Kubo equation for the Hall conductance can be expressed as an integral which implies quantization of the Hall conductance. The integral can be interpreted as the first Chern class of a [Formula: see text] principal fiber bundle on a two-dimensional torus. This accounts for the conductance given as an integer multiple of [Formula: see text]. The formalism can be extended to deduce the fractional conductivity as well.

ASYMPTOTIC BEHAVIOR OF THE GINZBURG-LANDAU FUNCTIONAL ON COMPLEX LINE BUNDLES OVER COMPACT RIEMANN SURFACES

1996 ◽  
Vol 08 (03) ◽  
pp. 457-486
Author(s):  
GIANDOMENICO ORLANDI
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Chern Class ◽  
Riemann Surfaces ◽  
Line Bundles ◽  
Complex Line ◽  
Ginzburg Landau ◽  
Renormalized Energy ◽  
Compact Riemann Surfaces ◽  
Complex Line Bundles

Motivated by the works of F. Bethuel, H. Brezis, F. Hélein [5] and of F. Bethuel, T. Rivière [6], an asymptotic analysis is carried out for minimizers of the Ginzburg-Landau functional depending on a parameter ε, in the more general case of complex line bundles with prescribed Chern class over compact Riemann surfaces. Such a functional describes a 2-dimensional abelian Higgs model and is also related to phenomena in superconductivity. A suitable renormalized energy is defined which characterizes the singularities (degree one vortices) of the limiting configuration.

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