scholarly journals Holomorphic Line Bundles Over Domains in Cousin Groups and the Algebraic Dimension of Oeljeklaus-Toma Manifolds

2014 ◽  
Vol 58 (2) ◽  
pp. 273-285 ◽  
Author(s):  
Laurent Battisti ◽  
Karl Oeljeklaus
Keyword(s):  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Line Bundles ◽  
Algebraic Dimension ◽  
Finite Dimensional ◽  
On Line ◽  
Holomorphic Line Bundles

AbstractIn this paper we extend results due to Vogt on line bundles over Cousin groups to the case of domains stable by the maximal compact subgroup. This is used to show that the algebraic dimension of Oeljeklaus—Toma manifolds (OT-manifolds) is 0. In the last part we establish that certain Cousin groups, in particular those arising from the construction of OT-manifolds, have finite-dimensional irregularity.

scholarly journals QUANTUM STATES FROM TANGENT VECTORS

2004 ◽  
Vol 19 (31) ◽  
pp. 2339-2352 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Vector Bundles ◽  
Vacuum State ◽  
Quantum States ◽  
Line Bundles ◽  
Holomorphic Vector Bundles ◽  
Finite Dimensional ◽  
Classical Phase Space ◽  
Complex Projective ◽  
Holomorphic Line Bundles ◽  
Tangent Vectors

We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite-dimensional, compact, classical phase spaces [Formula: see text], by explicitly constructing Hilbert-space vector bundles over [Formula: see text]. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle [Formula: see text], plus a complex line bundle [Formula: see text]. Quantum states (except the vacuum) appear as tangent vectors to [Formula: see text]. The vacuum state appears as the fibrewise generator of [Formula: see text]. Holomorphic line bundles [Formula: see text] are classified by the elements of [Formula: see text], the Picard group of [Formula: see text]. In this way [Formula: see text] appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when [Formula: see text] is complex projective space CPn.

scholarly journals Yang–Mills connections on compact complex tori

2015 ◽  
Vol 07 (02) ◽  
pp. 293-307
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Structure Group ◽  
Kähler Structure ◽  
Yang Mills ◽  
Complex Torus ◽  
Complex Tori

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

Some Results on Line Bundles over Susy-Curves

1990 ◽  
pp. 675-680
Author(s):  
C. Bartocci ◽  
U. Bruzzo ◽  
D. Hernández Ruipérez
Keyword(s):  
Line Bundles ◽  
On Line

scholarly journals Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

scholarly journals Equivariant holomorphic maps into the Siegel disc and the metaplectic representation

1997 ◽  
Vol 62 (2) ◽  
pp. 160-174 ◽  
Author(s):  
Jean-Louis Clerc
Keyword(s):  
Symplectic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Invariant Subspaces ◽  
Holomorphic Maps ◽  
Harmonic Polynomials ◽  
Metaplectic Representation ◽  
Siegel Disc

AbstractWe restrict the metaplectic representation to subgroupsGof the symplectic group associated to equivariant holomorphic maps into the Siegel disc. We describe the invariant subspaces of the decomposition, and reduce the problem to the decomposition of a space of ‘harmonic’ polynomials under the action of the maximal compact subgroup ofG.

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