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Dissertations / Theses on the topic 'Moduli space of vector bundles'
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1
Costa, Farràs Laura. "Moduli spaces of vector bundles on algebraic varieties." Doctoral thesis, Universitat de Barcelona, 1998. http://hdl.handle.net/10803/659.
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This thesis seeks to contribute to a deeper understanding of the moduli spaces M-sub X, H (r; c1,., Cmin{r;n}) of rank r, H-stable vector bundles E on an n-dimensional variety X, with fixed Chern classes c-sub1(E) = csub1 H-super2i ( X , Z) , displaying new and interesting geometric properties of M-sub X, H (r; c1,., Cmin{r;n}) which nicely reflect the general philosophy that moduli spaces inherit a lot of .geometrical properties of the underlying variety X.More precisely, we consider a smooth, irreducible, n-dimensional, projective variety X defined over an algebraically closed field k of characteristic zero, H an ample divisor on X, r >/2 an integer and c-subi H-super2i(X,Z) for i = 1, .,min{r,n}. We denote by M-sub X, H (r; c1,., Cmin{r;n}) the moduli space of rank r, vector bundles E on X, H-stable, in the sense of Mumford-Takemoto, with fixed Chern classes c-subi(E) = c-subi for i = 1, . , min{r, n}.
The contents of this Thesis is the following: Chapter 1 is devoted to provide the reader with the general background that we will need in the sequel. In the first two sections, we have collected the main definitions and results concerning coherent sheaves and moduli spaces, at least, those we will need through this work.
The aim of Chapter 2 is to establish the enterions of rationality for moduli spaces of rank two, it-stable vector bundles on a smooth, irreducible, rational surface X that will be used as one of our tools for answering Question (1), who is that follows: "Let X be a smooth, irreducible, rational surface. Fix C-sub1 Pic(X) and 0 « c2 Z. Is there an ample divisor H on X such that M-sub X,H(2; Ci, c2) is rational?"
In Chapter 3 we prove that the moduli space M-sub X,H(2; Ci, c2) of rank two, H-stable, vector bundles E on a smooth, irreducible, rational surface X, with fixed Chern classes C-sub1(E) = C-sub1 Pic(X) and 0 « C-sub2«(E) Z is a smooth, irreducible, rational, quasi-projective variety (Theorem 3.3.7) which solves Question (1).
In Chapter 4 we study moduli spaces (M-sub X,H(2; Ci, c2)) of rank r, H-stable vector bundles on either minimal rational surfaces or on algebraic K3 surfaces.
In Chapter 5 we deal with moduli spaces M-sub x,l (2;Ci,C2) of rank two, L-stable vector bundles E, on P-bundles of arbitrary dimension, with fixed Chern classes.
2
Moraru, Ruxandra. "Moduli spaces of vector bundles on a Hopf surface, and their stability properties." Thesis, McGill University, 2000. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=37786.
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We study the moduli spaces Mn of rank two stable holomorphic SL(2, C )-bundles E over Hopf surfaces H , with c2(E) = n, and their stabilisation properties. We show that one cannot construct stabilisation maps Mn→Mn+1 that are a natural holomorphic counterpart to Taubes's subtraction procedure that is used to construct such maps in the topological case of moduli spaces of connections. We also study the fiber of a map that associates to any holomorphic bundle a graph, and show that, in certain cases, the fiber is the Jacobian of a Riemann surface. We then show that this map is a Lagrangian fibration, with respect to a Poisson structure that we will define on Mn . Finally, we generalize the notion of graph to connections, and show that the graph map thus obtained is not topologically trivial.
3
Lo, Giudice Alessio. "Some topics on Higgs bundles over projective varieties and their moduli spaces." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4100.
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In this thesis we study vector bundles on projective varieties and their moduli spaces. In Chapters 2, 3 and 4 we recall some basic notions as Higgs bundles, decorated bundles and generalized parabolic sheaves and introduce the problem we want to study. In chapter 5, we study Higgs bundles on nodal curves. After moving the problem on the normalization of the curve, starting from a Higgs bundle we obtain a generalized parabolic Higgs bundle. Using decorated bundles we are able to construct a projective moduli space which parametrizes equivalence classes of Higgs bundles on a nodal curve X. This chapter is an extract of a joint work with Andrea Pustetto
Later on Chapter 6 is devoted to the study of holomorphic pairs (or twisted Higgs bundles) on elliptic curve. Holomorphic pairs were introduced by Nitsure and they are a natural generalization of the concept of Higgs bundles. In this Chapter we extend a result of E. Franco, O. Garc\'ia-Prada And P.E. Newstead valid for Higgs bundles to holomorphic pairs.
Finally the last Chapter describes a joint work with Professor Ugo Bruzzo. We study Higgs bundles over varieties with nef tangent bundle. In particular generalizing a result of Nitsure we prove that if a Higgs bundle $(E,\phi)$ over the variety X with nef tangent remains semisatble when pulled-back to any smooth curve then it discrimiant vanishes.
4
Gronow, Michael Justin. "Extension maps and the moduli spaces of rank 2 vector bundles over an algebraic curve." Thesis, Durham University, 1997. http://etheses.dur.ac.uk/5081/.
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Let SUc(2,Ʌ) be the moduli space of rank 2 vector bundles with determinant Ʌ on an algebraic curve C. This thesis investigates the properties of a rational map PU(_d,A) →(^c,d) SUc(2, A) where PU(_d,A) is a projective bundle of extensions over the Jacobian J(^d)(C). In doing so the degree of the moduli space SUc(2, Oc) is calculated for non- hyperelliptic curves of genus four (3.4.2). Information about trisecants to the Kummer variety K C SUc(2,Oc) is obtained in sections 4.3 and 4.4. These sections describe the varieties swept out by these trisecants in the fibres of PU1,o(_c) → J(^1)(C) for curves of genus 3, 4 and 5. The fibres of over ϵ(_d) over E ϵ SUc{2,A) are then studied. For certain values of d these correspond to the family of maximal line subbundles of E. These are either zero or one dimensional and a complete description of when these families are smooth is given (5.4.9), (5.4.10). In the one dimensional case its genus is also calculated (if connected) (5.5.5). Finally a correspondence on the curve fibres is shown to exist (5.6.2) and its degree is calculated (5.6.5). This in turn gives some information about multisecants to projective curves (5.7.4), (5.7.7).
5
Dyer, Ben. "NC-algebroid thickenings of moduli spaces and bimodule extensions of vector bundles over NC-smooth schemes." Thesis, University of Oregon, 2018. http://hdl.handle.net/1794/23168.
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We begin by reviewing the theory of NC-schemes and NC-smoothness, as introduced by Kapranov in \cite{Kapranov} and developed further by Polishchuk and Tu in \cite{PT}.
For a smooth algebraic variety $X$ with a torsion-free connection $\nabla$, we study modules over the NC-smooth thickening $\tw \O_X$ of $X$ constructed in \cite{PT} via NC-connections. In particular we show that the NC-vector bundle $\tw E_{\bar\nabla}$ constructed via mNC-connections in \cite{PT} from a vector bundle $(E,\bar\nabla)$ with connection additionally admits a bimodule extension at least to nilpotency degree 3.
Next, in joint work with A. Polishchuk \cite{DP}, we show that the gap, as first noticed in \cite{PT}, in the proof from \cite{Kapranov} that certain functors are representable by NC-smooth thickenings of moduli spaces of vector bundles is unfixable. Although the functors do not represent NC-smooth thickenings, they lead to a weaker structure of \textit{NC-algebroid thickening}, which we define. We also consider a similar construction for families of quiver representations, in particular upgrading some of the quasi-NC-structures of \cite{Toda1} to NC-smooth algebroid thickenings.
This thesis includes unpublished co-authored material.
6
Kaur, Inder [Verfasser]. "The C₁ conjecture for the moduli space of stable vector bundles with fixed determinant on a smooth projective curve / Inder Kaur." Berlin : Freie Universität Berlin, 2017. http://d-nb.info/1131629337/34.
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Sanna, Giangiacomo. "Rational curves and instantons on the Fano threefold Y_5." Doctoral thesis, SISSA, 2014. http://hdl.handle.net/20.500.11767/3867.
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This thesis is an investigation of the moduli spaces of instanton bundles on the Fano threefold Y_5 (a linear section of Gr(2,5)). It contains new proofs of classical facts about lines, conics and cubics on Y_5, and about linear sections of Y_5.
The main original results are a Grauert-Mülich theorem for the splitting type of instantons on conics, a bound to the splitting type of instantons on lines and an SL_2-equivariant description of the moduli space in charge 2 and 3.
Using these results we prove the existence of a unique SL_2-equivariant instanton of minimal charge and we show that for all instantons of charge 2 the divisor of jumping lines is smooth. In charge 3, we provide examples of instantons with reducible divisor of jumping lines. Finally, we construct a natural compactification for the moduli space of instantons of charge 3, together with a small resolution of singularities for it.
8
Fernández, Vargas Néstor. "Fibres vectoriels sur des courbes hyperelliptiques." Thesis, Rennes 1, 2018. http://www.theses.fr/2018REN1S051/document.
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Cette thèse est dédiée à l'étude des espaces de modules de fibrés sur une courbe algébrique et lisse sur le corps des nombres complexes. Le texte est composé de deux parties : Dans la première partie, je m'intéresse à la géométrie liée aux classifications de fibrés quasi-paraboliques de rang 2 sur une courbe elliptique 2-pointée, à isomorphisme près. Les notions d'indécomposabilité, simplicité et stabilité de fibrés donnent lieu à des espaces de modules qui classifient ces objets. La structure projective de ces espaces est décrite explicitement, et on prouve un théorème de type Torelli qui permet de retrouver la courbe elliptique 2-pointée. Cet espace de modules est aussi mis en relation avec l'espace de modules de fibrés quasi-paraboliques sur une courbe rationnelle 5-pointée, qui apparaît naturellement comme revêtement double de l'espace de modules de fibrés quasi-paraboliques sur la courbe elliptique 2-pointée. Finalement, on démontre explicitement la modularité des automorphismes de cet espace de modules. Dans la deuxième partie, j'étudie l'espace de modules de fibrés semistables de rang 2 et déterminant trivial sur une courbe hyperelliptique. Plus précisément, je m'intéresse à l'application naturelle donnée par le fibré déterminant, générateur du groupe de Picard de cet espace de modules. Cette application s'identifie à l'application theta, qui est de degré 2 dans notre cas. On définit une fibration de cet espace de modules vers un espace projective dont la fibre générique est birationnelle à l'espace de modules de courbes rationnelles 2g-épointées, et on décrit la restriction de theta aux fibres de cette fibration. On montre que cette restriction est, à une transformation birationnelle près, une projection osculatoire centrée en un point. En utilisant une description due à Kumar, on démontre que la restriction de l'application theta à cette fibration ramifie sur la variété de Kummer d'une certaine courbe hyperelliptique de genre g – 1This thesis is devoted to the study of moduli spaces of vector bundles over a smooth algebraic curve over field of complex numbers. The text consist of two main parts : In the first part, I investigate the geometry related to the classifications of rank 2 quasi-parabolic vector bundles over a 2-pointed elliptic curves, modulo isomorphism. The notions of indecomposability, simplicity and stability give rise to the corresponding moduli spaces classifying these objects. The projective structure of these spaces is explicitely described, and we prove a Torelli theorem that allow us to recover the 2-pointed elliptic curve. I also explore the relation with the moduli space of quasi-parabolic vector bundles over a 5-pointed rational curve, appearing naturally as a double cover of the moduli space of quasi-parabolic vector bundles over the 2-pointed elliptic curve. Finally, we show explicitely the modularity of the automorphisms of this moduli space. In the second part, I study the moduli space of semistable rank 2 vector bundles with trivial determinant over a hyperelliptic curve C. More precisely, I am interested in the natural map induced by the determinant line bundle, generator of the Picard group of this moduli space. This map is identified with the theta map, which is of degree 2 in our case. We define a fibration from this moduli space to a projective space whose generic fiber is birational to the moduli space of 2g-pointed rational curves, and we describe the restriction of the map theta to the fibers of this fibration. We show that this restriction is, up to a birational map, an osculating projection centered on a point. By using a description due to Kumar, we show that the restriction of the map theta to this fibration ramifies over the Kummer variety of a certain hyperelliptic curve of genus g - 1
9
Zelaci, Hacen. "Espaces de modules de fibrés vectoriels anti-invariants sur les courbes et blocs conformes." Thesis, Université Côte d'Azur (ComUE), 2017. http://www.theses.fr/2017AZUR4063/document.
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Soit X une courbe projective lisse et irréductible munie d'une involution σ. Dans cette thèse, nous étudions les fibrés vectoriels invariants and anti-invariants sur X sous l'action induite par σ. On introduit la notion de modules σ-quadratiques et on l'utilise, avec GIT, pour construire ces espaces de modules, puis on en étudie certaines propriétés. Ces espaces de modules correspondent aux espaces de modules de G-torseurs parahoriques sur la courbe X/σ , pour certains schémas en groupes parahoriques G de type Bruhat-Tits, qui sont twistés dans le cas des anti-invariants. Nous développons les systèmes de Hitchin sur ces espaces de modules et on les utilise pour dériver une classification de leurs composantes connexes en les dominant par des variétés de Prym. On étudie aussi le fibré déterminant sur les espaces de modules des fibrés vectoriels anti-invariants. Dans certains cas, ce fibré en droites admet certaines racines carrées appelées fibrés Pfaffiens. On montre que les espaces des sections globales des puissances de ces fibrés en droites (les espaces des fonctions thêta généralisées) peuvent être canoniquement identifier avec les blocs conformes associés aux algèbres de Kac-Moody affines twistées de type A(2)Let X be a smooth irreducible projective curve with an involution σ. In this dissertation, we studythe moduli spaces of invariant and anti-invariant vector bundles over X under the induced action of σ. We introduce the notion of σ-quadratic modules and use it, with GIT, to construct these moduli spaces, and than we study some of their main properties. It turn out that these moduli spaces correspond to moduli spaces of parahoric G-torsors on the quotient curve X/σ, for some parahoric Bruhat-Tits group schemes G, which are twisted in the anti-invariant case.We study the Hitchin system over these moduli spaces and use it to derive a classification of theirconnected components using dominant maps from Prym varieties. We also study the determinant of cohomology line bundle on the moduli spaces of anti-invariant vector bundles. In some cases this line bundle admits some square roots called Pfaffian of cohomology line bundles. We prove that the spaces of global sections of the powers of these line bundles (spaces of generalized theta functions) can be canonically identified with the conformal blocks for some twisted affine Kac-Moody Lie algebras of type A(2)
10
Koeppe, Thomas. "Moduli of bundles on local surfaces and threefolds." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/33315.
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In this thesis we study the moduli of holomorphic vector bundles over a non-compact complex space X, which will mainly be of dimension 2 or 3 and which contains a distinguished rational curve ℓ ⊂ X. We will consider the situation in which X is the total space of a holomorphic vector bundle on CP1 and ℓ is the zero section. While the treatment of the problem in this full generality requires the study of complex analytic spaces, it soon turns out that a large part of it reduces to algebraic geometry. In particular, we prove that in certain cases holomorphic vector bundles on X are algebraic. A key ingredient in the description of themoduli are numerical invariants that we associate to each holomorphic vector bundle. Moreover, these invariants provide a local version of the second Chern class. We obtain sharp bounds and existence results for these numbers. Furthermore, we find a new stability condition which is expressed in terms of these numbers and show that the space of stable bundles forms a smooth, quasi-projective variety.
11
Hausel, TamaÌs. "Geometry of the moduli space of Higgs bundles." Thesis, University of Cambridge, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397444.
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Fan, Chun-Lin. "Extensions of stable rank-3 vector bundles on ruled surface /." View abstract or full-text, 2004. http://library.ust.hk/cgi/db/thesis.pl?MATH%202004%20FAN.
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Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2004.Includes bibliographical references (leaves 20-21). Also available in electronic version. Access restricted to campus users.
13
Green, Michael Douglas 1965. "Manifolds, Vector Bundles, and Stiefel-Whitney Classes." Thesis, University of North Texas, 1990. https://digital.library.unt.edu/ark:/67531/metadc504181/.
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The problem of embedding a manifold in Euclidean space is considered. Manifolds are introduced in Chapter I along with other basic definitions and examples. Chapter II contains a proof of the Regular Value Theorem along with the "Easy" Whitney Embedding Theorem. In Chapter III, vector bundles are introduced and some of their properties are discussed. Chapter IV introduces the Stiefel-Whitney classes and the four properties that characterize them. Finally, in Chapter V, the Stiefel-Whitney classes are used to produce a lower bound on the dimension of Euclidean space that is needed to embed real projective space.
14
Pustetto, Andrea. "Semistability and Decorated Bundles." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4093.
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This thesis is devoted to the study of semistability condition of type t=(a,b,c,N) decorated bundles and sheaves in order to better understand and simplify it. We approach the problem in two different ways: on one side we “enclose” the above semistability condition between a stronger semistability condition (\epsilon-semistability) and a weaker one (k-semistability), on the other side we try, and succeed for the case of a = 2, to bound the length of weighted filtrations on which one checks the semistability condition.
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Wegner, Dimitri [Verfasser], and Christopher [Akademischer Betreuer] Deninger. "On moduli of vector bundles on p-adic curves and attached representations / Dimitri Wegner ; Betreuer: Christopher Deninger." Münster : Universitäts- und Landesbibliothek Münster, 2014. http://d-nb.info/1138282774/34.
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MOSSA, ROBERTO. "Balanced metrics on complex vector bundles and the diastatic exponential of a symmetric space." Doctoral thesis, Università degli Studi di Cagliari, 2011. http://hdl.handle.net/11584/266274.
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This thesis deals with two different subjects: balanced metrics on complex vector bundles and the diastatic exponential of a symmetric space. Correspondingly we have two main results. In the first one we prove that if a holomorphic vector bundle E over a compact Kähler manifold (M,ω) admits a ω-balanced metric then this metric is unique. In the second one, after defining the diastatic exponential of a real analytic Kähler manifold, we
prove that for every point p of an Hermitian symmetric space of noncompact type there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks.
17
Kuschkowitz, Mark [Verfasser]. "Equivariant Vector Bundles and Rigid Cohomology on Drinfeld's Upper Half Space over a Finite Field / Mark Kuschkowitz." Wuppertal : Universitätsbibliothek Wuppertal, 2016. http://d-nb.info/112004460X/34.
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Arcara, Daniele. "Moduli spaces of vector bundles on curves." 2003. http://purl.galileo.usg.edu/uga%5Fetd/arcara%5Fdaniele%5F200305%5Fphd.
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Reede, Fabian. "Moduli spaces of bundles over two-dimensional orders." Doctoral thesis, 2013. http://hdl.handle.net/11858/00-1735-0000-001A-7778-7.
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Wir studieren Moduln über Maximalordnungen auf glatten projektiven Flächen und ihre Modulräume. Wir untersuchen null- und zweidimensionale Modulräume auf K3 und abelschen Flächen für unverzweigte Ordnungen, den sogenannten Azumaya Algebren. Weiterhin untersuchen wir Modulräume für spezielle verzweigte Ordnungen auf der projektiven Ebene. Wir beweisen das diese Räume immer glatt sind. Mit Hilfe dieses Ergebnisses studieren wir die Deformationstheorie der betrachteten Moduln. Im letzten Kapitel konstruieren wir explizite Ordnungen und berechnen einige Modulräume.
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Nevins, Thomas A. "Moduli spaces of framed sheaves on ruled surfaces /." 2000. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:9965126.
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Iena, Oleksandr [Verfasser]. "Modification of Simpson moduli spaces of 1-dimensional sheaves by vector bundles : an experimental example / Oleksandr Iena." 2009. http://d-nb.info/994346085/34.
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VANZO, DAVIDE. "Instanton bundles and their moduli spaces." Doctoral thesis, 2017. http://hdl.handle.net/2158/1079371.
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This work studies the moduli space of instanton bundles on P^{2n+1} focusing its attention on certain kind of families: Rao-Skiti and 't Hooft. It proves that these two components are linked in their moduli space.
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Krepski, Derek. "Pre-quantization of the Moduli Space of Flat G-bundles." Thesis, 2009. http://hdl.handle.net/1807/19047.
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This thesis studies the pre-quantization of quasi-Hamiltonian group actions from a
cohomological viewpoint. The compatibility of pre-quantization with symplectic reduction
and the fusion product are established, and are used to understand the necessary and sufficient conditions for the pre-quantization of M(G,S), the moduli space of
at flat G-bundles over a closed surface S.
For a simply connected, compact, simple Lie group G, M(G,S) is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this thesis determines the obstruction, namely a certain 3-dimensional cohomology class, that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are
determined explicitly for all non-simply connected, compact, simple Lie groups G. Partial results are obtained for the case of a surface S with marked points.
Also, it is shown that via the bijective correspondence between quasi-Hamiltonian
group actions and Hamiltonian loop group actions, the corresponding notions of prequantization coincide.
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Baird, Thomas John. "The moduli space of flat G-bundles over a nonorientable surface." 2008. http://link.library.utoronto.ca/eir/EIRdetail.cfm?Resources__ID=742554&T=F.
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Keshari, Dinesh Kumar. "Infinitely Divisible Metrics, Curvature Inequalities And Curvature Formulae." Thesis, 2012. http://etd.iisc.ernet.in/handle/2005/2332.
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The curvature of a contraction T in the Cowen-Douglas class is bounded above by the
curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples
of operators in the Cowen-Douglas class.
Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ).
Clearly, finding relationships amongs the complex geometric invariants inherent in the
short exact sequence
0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0
is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy
c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )).
We obtain a refinement of this formula:
trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).