DIMENSIONAL REDUCTION, ${\rm SL} (2, {\mathbb C})$-EQUIVARIANT BUNDLES AND STABLE HOLOMORPHIC CHAINS

2001 ◽  
Vol 12 (02) ◽  
pp. 159-201 ◽  
Author(s):  
LUIS ÁLVAREZ-CÓNSUL ◽  
OSCAR GARCÍA-PRADA
Keyword(s):  
Dimensional Reduction ◽  
Existence Of Solutions ◽  
Projective Line ◽  
Reduction Procedure ◽  
Complex Projective Line ◽  
Holomorphic Chains ◽  
Holomorphic Bundles ◽  
Compact Kähler Manifold ◽  
Complex Projective ◽  
Central Tool

In this paper we study gauge theory on [Formula: see text]-equivariant bundles over X × ℙ1, where X is a compact Kähler manifold, ℙ1 is the complex projective line, and the action of [Formula: see text] is trivial on X and standard on ℙ1. We first classify these bundles, showing that they are in correspondence with objects on X — that we call holomorphic chains — consisting of a finite number of holomorphic bundles ℰi and morphisms ℰi → ℰi-1. We then prove a Hitchin–Kobayashi correspondence relating the existence of solutions to certain natural gauge-theoretic equations and an appropriate notion of stability for an equivariant bundle and the corresponding chain. A central tool in this paper is a dimensional reduction procedure which allow us to go from X × ℙ1 to X.

scholarly journals Principal co-Higgs bundles on ℙ1

2020 ◽  
Vol 63 (2) ◽  
pp. 512-530 ◽  
Author(s):  
Indranil Biswas ◽  
Oscar García-Prada ◽  
Jacques Hurtubise ◽  
Steven Rayan
Keyword(s):  
Moduli Space ◽  
Borel Subgroup ◽  
Algebraic Groups ◽  
Projective Line ◽  
Higgs Bundles ◽  
Complex Projective Line ◽  
Theoretic Characterization ◽  
Complex Projective

AbstractFor complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

scholarly journals Unobstructedness of hyperkähler twistor spaces

2021 ◽  
Author(s):  
Ana-Maria Brecan ◽  
Tim Kirschner ◽  
Martin Schwald
Keyword(s):  
Open Neighborhood ◽  
Projective Line ◽  
Twistor Spaces ◽  
Period Map ◽  
Universal Family ◽  
Complex Projective Line ◽  
Complex Projective

AbstractA family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

QUANTUM COHOMOLOGIES OF SYMMETRIC PRODUCTS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250005 ◽  
Author(s):  
YONG SEUNG CHO
Keyword(s):  
Moduli Space ◽  
Product Space ◽  
Complex Projective Space ◽  
Projective Line ◽  
Symmetric Product ◽  
Witten Invariant ◽  
Symmetric Products ◽  
Complex Projective Line ◽  
Complex Projective ◽  
Invariant Properties

In this paper we investigate the quantum cohomologies of symmetric products of Kähler manifolds. To do this we study the moduli space of product space and symmetric group action on it, Gromov–Witten invariant and relative Gromov–Witten invariant. Also we investigate the relations between symmetric invariant properties on the products space and the corresponding ones on the symmetric product. As an example we examine the symmetric product of k copies complex projective line ℙ1, which is the k-dimensional complex projective space ℙk.

scholarly journals D-modules of pure Gaussian type and enhanced ind-sheaves

2021 ◽  
Author(s):  
Andreas Hohl
Keyword(s):  
Laplace Transform ◽  
Projective Line ◽  
Irregular Singularity ◽  
Stokes Phenomenon ◽  
Differential Systems ◽  
Gaussian Type ◽  
D Modules ◽  
Complex Projective Line ◽  
Complex Projective

AbstractDifferential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier–Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of Sabbah.

scholarly journals Algebraic families of groups and commuting involutions

2018 ◽  
Vol 29 (04) ◽  
pp. 1850030 ◽  
Author(s):  
Dan Barbasch ◽  
Nigel Higson ◽  
Eyal Subag
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Projective Line ◽  
Real Structure ◽  
The Real ◽  
The Family ◽  
Complex Projective Line ◽  
Complex Projective ◽  
Real Forms

Let [Formula: see text] be a complex affine algebraic group, and let [Formula: see text] and [Formula: see text] be commuting anti-holomorphic involutions of [Formula: see text]. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms [Formula: see text] and [Formula: see text].

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