scholarly journals The Determinant Line Bundle for Fredholm Operators: Construction, Properties, and Classification

2016 ◽  
Vol 118 (2) ◽  
pp. 203 ◽  
Author(s):  
Aleksey Zinger
Keyword(s):  
Line Bundle ◽  
Line Bundles ◽  
Fredholm Operators ◽  
Explicit Formulas ◽  
Determinant Line Bundles ◽  
Determinant Line Bundle

We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possible systems of compatible determinant line bundles and compare the conventions and approaches used elsewhere in the literature.

scholarly journals Remarks on Determinant Line Bundles, Chern-Simons Forms and Invariants

2002 ◽  
Vol 91 (1) ◽  
pp. 5 ◽  
Author(s):  
Johan L. Dupont ◽  
Flemming Lindblad Johansen
Keyword(s):  
Line Bundle ◽  
Line Bundles ◽  
Manifolds With Boundary ◽  
Principal Bundles ◽  
Determinant Line Bundles ◽  
Chern Simons ◽  
Determinant Line Bundle

We study generalized determinant line bundles for families of principal bundles and connections. We explore the connections of this line bundle and give conditions for the uniqueness of such. Furthermore we construct for families of bundles and connections over manifolds with boundary, a generalized Chern-Simons invariant as a section of a determinant line bundle.

scholarly journals Determinants, Pfaffians and Quasi-Free Representations of the CAR Algebra

1998 ◽  
Vol 10 (05) ◽  
pp. 705-721 ◽  
Author(s):  
Mauro Spera ◽  
Tilmann Wurzbacher
Keyword(s):  
Hilbert Space ◽  
Line Bundle ◽  
Line Bundles ◽  
Complex Structures ◽  
Square Root ◽  
Purification Procedure ◽  
Algebraic Construction ◽  
Infinite Dimensional ◽  
Weil Type ◽  
Determinant Line Bundle

In this paper we apply the theory of quasi-free states of CAR algebras and Bogolubov automorphisms to give an alternative C*-algebraic construction of the Determinant and Pfaffian line bundles discussed by Pressley and Segal and by Borthwick. The basic property of the Pfaffian of being the holomorphic square root of the Determinant line bundle (after restriction from the Hilbert space Grassmannian to the Siegel manifold, or isotropic Grassmannian, consisting of all complex structures on an associated Hilbert space) is derived from a Fock–anti-Fock correspondence and an application of the Powers–Størmer purification procedure. A Borel–Weil type description of the infinite dimensional Spin c- representation is obtained, via a Shale–Stinespring implementation of Bogolubov transformations.

scholarly journals Deligne pairing and Quillen metric

2014 ◽  
Vol 25 (14) ◽  
pp. 1450122 ◽  
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Positive Integer ◽  
Line Bundle ◽  
Ample Line Bundle ◽  
Hermitian Structure ◽  
Line Bundles ◽  
Canonical Isomorphism ◽  
Surjective Morphism ◽  
Kähler Form ◽  
Hermitian Structures ◽  
Determinant Line Bundle

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,…,L〉 → S with the determinant line bundle [Formula: see text] (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen–Mumford expansion, J. Differential Geom. 78 (2008) 475–496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles [Formula: see text] and 〈L,…,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,…,L〉 → S and [Formula: see text] is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91–96] between a Deligne paring and a certain determinant line bundle.

ON THE FUNCTORIALITY OF THE CHERN–SIMONS LINE BUNDLE AND THE DETERMINANT LINE BUNDLE

2006 ◽  
Vol 08 (06) ◽  
pp. 715-735 ◽  
Author(s):  
HAJIME FUJITA
Keyword(s):  
Line Bundle ◽  
Hilbert Spaces ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Mapping Class ◽  
Line Bundles ◽  
Cyclic Subgroups ◽  
Chern Simons ◽  
Determinant Line Bundle ◽  
Closed Oriented Surface

We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.

scholarly journals Canonical holomorphic sections of determinant line bundles

2019 ◽  
Vol 2019 (746) ◽  
pp. 67-116 ◽  
Author(s):  
Jens Kaad ◽  
Ryszard Nest
Keyword(s):  
Finite Rank ◽  
Main Tool ◽  
Line Bundles ◽  
Fredholm Operators ◽  
Homology Groups ◽  
Invariance Properties ◽  
Holomorphic Sections ◽  
Finite Rank Perturbation ◽  
Determinant Line Bundles ◽  
Fredholm Complexes

Abstract We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

The strength for line bundles

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Edoardo Ballico ◽  
Emanuele Ventura
Keyword(s):  
Line Bundle ◽  
Algebraic Variety ◽  
Commutative Algebra ◽  
Upper Bounds ◽  
Line Bundles ◽  
Homogeneous Polynomials

We introduce the strength for sections of a line bundle on an algebraic variety. This generalizes the strength of homogeneous polynomials that has been recently introduced to resolve Stillman's conjecture, an important problem in commutative algebra. We establish the first properties of this notion and give some tool to obtain upper bounds on the strength in this framework. Moreover, we show some results on the usual strength such as the reducibility of the set of strength two homogeneous polynomials.

scholarly journals Projective surfaces with K-very ample line bundles of degree ≤ 4K + 4

1994 ◽  
Vol 136 ◽  
pp. 57-79 ◽  
Author(s):  
Edoardo Ballico ◽  
Andrew J. Sommese
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Smooth Surface ◽  
Negative Integer ◽  
Line Bundles ◽  
Restriction Map ◽  
Projective Surface ◽  
Projective Surfaces

A line bundle, L, on a smooth, connected projective surface, S, is defined [7] to be k-very ample for a non-negative integer, k, if given any 0-dimensional sub-scheme with length , it follows that the restriction map is onto. L is 1-very ample (respectively 0-very ample) if and only if L is very ample (respectively spanned at all points by global sections). For a smooth surface, S, embedded in projective space by | L | where L is very ample, L being k-very ample is equivalent to there being no k-secant Pk−1 to S containing ≥ k + 1 points of S.

scholarly journals Numerical dimension and a Kawamata–Viehweg–Nadel-type vanishing theorem on compact Kähler manifolds

2014 ◽  
Vol 150 (11) ◽  
pp. 1869-1902 ◽  
Author(s):  
Junyan Cao
Keyword(s):  
Line Bundle ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Line Bundles ◽  
Kahler Manifolds ◽  
Vanishing Theorem ◽  
Kahler Manifold ◽  
Singular Metrics ◽  
Compact Kähler Manifold

AbstractLet $X$ be a compact Kähler manifold and let $(L,{\it\varphi})$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension for pseudo-effective line bundles with singular metrics, and then discuss the properties of this numerical dimension. Finally, we prove a very general Kawamata–Viehweg–Nadel-type vanishing theorem on an arbitrary compact Kähler manifold.

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