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.

Hermitian–Einstein metrics and jumping lines forSp(n+1)-homogeneous bundles over ℙ2n+1

1993 ◽  
Vol 114 (3) ◽  
pp. 443-451
Author(s):  
Al Vitter
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Vector Bundles ◽  
Einstein Metrics ◽  
Einstein Metric ◽  
Holomorphic Vector Bundles ◽  
Kähler Form ◽  
Points Of View ◽  
The Stability ◽  
Complex Projective

Stable holomorphic vector bundles over complex projective space ℙnhave been studied from both the differential-geometric and the algebraic-geometric points of view.On the differential-geometric side, the stability ofE-→ ℙncan be characterized by the existence of a unique hermitian–Einstein metric onE, i.e. a metric whose curvature matrix has trace-free part orthogonal to the Fubini–Study Kähler form of ℙn(see [6], [7], and [13]). Very little is known about this metric in general and the only explicit examples are the metrics on the tangent bundle of ℙnand the nullcorrelation bundle (see [9] and [10]).

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 Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor

2013 ◽  
Vol 112 (1) ◽  
pp. 61
Author(s):  
George H. Hitching
Keyword(s):  
Moduli Space ◽  
Tangent Cone ◽  
Vector Bundles ◽  
Smooth Curve ◽  
Line Bundles ◽  
Singularity Theorem ◽  
Theta Divisor ◽  
Stable Point ◽  
Roch Theorem ◽  
Complex Projective

Let $E$ and $F$ be vector bundles over a complex projective smooth curve $X$, and suppose that $0 \to E \to W \to F \to 0$ is a nontrivial extension. Let $G \subseteq F$ be a subbundle and $D$ an effective divisor on $X$. We give a criterion for the subsheaf $G(-D) \subset F$ to lift to $W$, in terms of the geometry of a scroll in the extension space ${\mathbf{P}} H^{1}(X, \mathrm{Hom}(F, E))$. We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank $r$ and slope $g-1$ over $X$, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over $X$. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope $g-1$ and arbitrary rank.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

Vector bundles as direct images of line bundles

1994 ◽  
Vol 104 (1) ◽  
pp. 191-200 ◽  
Author(s):  
A. Hirschowitz ◽  
M. S. Narasimhan
Keyword(s):  
Vector Bundles ◽  
Line Bundles
Sign in / Sign up

Export Citation Format

Share Document