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Dissertations / Theses on the topic 'Moduli spaces of sheaves on surfaces'
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1
Bridgeland, Tom. "Fourier-Mukai transforms for surfaces and moduli spaces of stable sheaves." Thesis, University of Edinburgh, 2002. http://hdl.handle.net/1842/12070.
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In this thesis we study Fourier-Mukai transforms for complex projective surfaces. Extending work of A.I. Bondal and D.O. Orlov, we prove a theorem giving necessary and sufficient conditions for a functor between the derived categories of sheaves on two smooth projective varieties to be an equivalence of categories, and use it to construct examples of Fourier-Mukai transforms for surfaces. In particular we construct new transforms for elliptic surfaces and quotient surfaces. This enables us to identify all pairs of complex projective surfaces having equivalent derived categories of sheaves. We also derive some general properties of Fourier-Mukai transforms, and gives examples of their use. The main applications are to the study of moduli spaces of stable sheaves. In particular we identify many such moduli spaces on elliptic surfaces, generalising results of R. Friedman.
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Scalise, Jacopo Vittorio. "Frames symplectic sheaves on surfaces and their ADHM data." Doctoral thesis, SISSA, 2016. http://hdl.handle.net/20.500.11767/4896.
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This dissertation is centered on the moduli space of what we call framed symplectic sheaves on a surface, compactifying the corresponding moduli space of framed principal SP−bundles.
It contains the construction of the moduli space, which is carried out for every smooth projective surface X with a big and nef framing divisor, and a study of its deformation theory.
We also develop an in-depth analysis of the examples X = P2 and X = Blp (P2 ), showing that the corresponding moduli spaces enjoy an ADHM-type description. In the former case, we prove irreducibility of the space and exhibit a relation with the space of framed ideal instantons on S4 in type C.
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Hoskins, Victoria Amy. "Moduli spaces of complexes of sheaves." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:aedd2719-2a38-41f9-9825-aa8f43fb872c.
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This thesis is on moduli spaces of complexes of sheaves and diagrams of such moduli spaces. The objects in these diagrams are constructed as geometric invariant theory quotients and the points in these quotients correspond to certain equivalence classes of complexes. The morphisms in these diagrams are constructed by taking direct sums with acyclic complexes. We then study the colimit of such a diagram and in particular are interested in studying the images of quasi-isomorphic complexes in the colimit. As part of this thesis we construct categorical quotients of a group action on unstable strata appearing in a stratification associated to a complex projective scheme with a reductive group action linearised by an ample line bundle. We study this stratification for a closed subscheme of a quot scheme parametrising quotient sheaves over a complex projective scheme and relate the Harder-Narasimhan types of unstable sheaves with the unstable strata in the associated stratification. We also study the stratification of a parameter space for complexes with respect to a linearisation determined by certain stability parameters and show that a similar result holds in this case. The objects in these diagrams are indexed by different Harder-Narasimhan types for complexes and are quotients of parameter schemes for complexes of this fixed Harder-Narasimhan type. This quotient is given by a choice of linearisation of the action and so the diagrams depend on these choices. We conjecture that these choices can be made so that for any quasi-isomorphism between complexes representing points in this diagram both complexes are identified in the colimit of this diagram.
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Kool, Martijn. "Moduli spaces of sheaves on toric varieties." Thesis, University of Oxford, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526468.
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Abdellaoui, Gharchia. "Topology of moduli spaces of framed sheaves." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4806.
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Nironi, Fabio. "Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks." Doctoral thesis, SISSA, 2008. http://hdl.handle.net/20.500.11767/4165.
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Sala, Francesco. "Some topics in the geometry of framed sheaves and their moduli spaces." Doctoral thesis, SISSA, 2011. http://hdl.handle.net/20.500.11767/4129.
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This dissertation is primarily concerned with the study of framed sheaves on nonsingular projective varieties and the geometrical properties of the moduli spaces of these objects. In particular, we deal with a generalization to the framed case of known results for (semi)stable torsion free sheaves, such as (relative) Harder-Narasimhan filtration,
Mehta-Ramanathan restriction theorems, Uhlenbeck-Donaldson compactification, Atiyah class and Kodaira-Spencer map. The main motivations for the study of these moduli spaces come from physics, in particular, gauge theory, as we shall explain in the following.
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Sala, Francesco. "Some topics in the geometry of framed sheaves and their moduli spaces." Thesis, Lille 1, 2011. http://www.theses.fr/2011LIL10076/document.
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La thèse est consacrée à l'étude des faisceaux encadrés sur des variétés non-singulières projectives et des propriétés géométriques de leurs espaces de modules. En particulier, on donne une généralisation au cas encadré des résultats connus pour les faisceaux (semi)stables sans torsion non-encadrés, comme l'existence de la filtration de Harder-Narasimhan (relative), théorèmes de restriction de Mehta-Ramanathan, compactification de Donaldson-Uhlenbeck, la définition de la classe d'Atiyah relative et la description de l'application de Kodaira-Spencer via la classe d'Atiyah relative, l'existence d'une structure symplectique holomorphe, dans certains cas, sur les espaces de modules de faisceaux encadrésThe thesis is concerned with the study of framed sheaves on nonsingular projective varieties and the geometrical properties of their moduli spaces. In particular, it deals with a generalization to the framed case of known results for (semi)stable torsion free nonframed sheaves, such as the existence of the (relative) Harder-Narasimhan filtration, Mehta-Ramanathan restriction theorems, Uhlenbeck-Donaldson compactification, the definition of the relative Atiyah class and the description of the Kodaira-Spencer map in terms of the relative Atiyah class, the existence of a symplectic structure, in certain cases, on the moduli spaces of framed sheaves
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Schlüeter, Dirk Christopher. "Universal moduli of parabolic sheaves on stable marked curves." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:b0260f8e-6654-4bec-b670-5e925fd403dd.
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The topic of this thesis is the moduli theory of (parabolic) sheaves on stable curves. Using geometric invariant theory (GIT), universal moduli spaces of semistable parabolic sheaves on stable marked curves are constructed: `universal' indicates that these are moduli spaces of pairs where the underlying marked curve may vary as well as the parabolic sheaf (as in the Pandharipande moduli space for pairs of stable curves and torsion-free sheaves without augmentations). As an intermediate step in this construction, we construct moduli spaces of semistable parabolic sheaves on flat families of arbitrary projective schemes (of any dimension or singularity type): this is the technical core of this thesis. These moduli spaces are projective, since they are constructed as GIT quotients of projective parameter spaces. The stability condition for parabolic sheaves depends on a choice of polarisation and is derived from the Hilbert-Mumford criterion. It is not quite the same as traditional stability with respect to parabolic Hilbert polynomials, but it is closely related to it, and the resulting moduli spaces are always compactifications of moduli of slope-stable parabolic sheaves. The construction works over algebraically closed fields of arbitrary characteristic.
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Marque, Nicolas. "Moduli spaces of Willmore immersions." Thesis, Université de Paris (2019-....), 2019. http://www.theses.fr/2019UNIP7127.
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Dans ce travail doctoral, nous commençons par présenter une synthèse du formalisme des immersions faibles de Willmore. A cet effet, nous introduisons les lois de conservation et les exploitons pour retrouver les résultats d'epsilon régularité, ainsi qu'un résultat de régularité faible inédit. Nous présentons ensuite une étude de l'application de Gauss conforme et de ses liens avec la notion de surface de Willmore. Nous en déduisons une loi d'échange de résidus ainsi que d'une caractérisation originale des surfaces étant transformations de surfaces à courbure moyenne constante. Nous appliquons ensuite ces outils aux suites d'immersions de Willmore. Nous montrons tout d'abord qu'elles ne sont pas compactes avec un premier exemple de concentration pour les surfaces de Willmore. Cependant, en se basant sur un résultat d'epsilon régularité demandant un contrôle sur la courbure moyenne, nous montrons une compacité sous un certain plafond d'énergieIn this doctoral work we start by exposing a synthesis of the weak Willmore immersions formalism. To that end, we introduce conservation laws and exploit them to recover the epsilon-regularity theorems, as well as an innovative weak regularity result. We then present a study of the conformal Gauss map and its links with the Willmore surface notion. From this, we deduce an exchange law for residues as well as an original caracterization of surfaces that are conformal transforms of constant mean curvature surfaces. We then apply these tools to sequences of Willmore immersions. We first show that they are not compact wth a first instance of concentration for Willmore surfaces. However, relying upon an epsilon-regularity result based on a small control on the mean curvature, we show compactness below a given threshold
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Pedrini, Mattia. "Moduli spaces of framed sheaves on stacky ALE spaces, deformed partition functions and the AGT conjecture." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4807.
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Pasquinelli, Irene. "Complex hyperbolic lattices and moduli spaces of flat surfaces." Thesis, Durham University, 2018. http://etheses.dur.ac.uk/12863/.
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This work studies the Deligne-Mostow lattices in PU(2,1). These were introduced by Deligne and Mostow in several works, using monodromy of hypergeometric functions. The same lattices were rediscovered by Thurston using a geometric construction, which consists of studying possible configurations of cone points on a sphere of area 1 when the cone angles are prescribed. This space has a complex hyperbolic structure and certain automorphisms of the sphere which swap pairs of cone points, generate a lattice for some choice of initial cone angles (more precisely, the Deligne-Mostow lattices). Among these, we will consider the ones in PU(2,1). We use Thurston's approach to study the metric completion of this space, which is obtained by making pairs of cone points coalesce. Following the works of Parker and Boadi-Parker, we build a polyhedron. Using the Poincaré polyhedron theorem, we prove that the polyhedron we find is indeed a fundamental domain. Moreover, we give presentations for all Deligne-Mostow lattices in PU(2,1), calculate their volumes and show that they are coherent with the known commensurability theorems.
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Rink, Norman Alexander. "Complex geometry of vortices and their moduli spaces." Thesis, University of Cambridge, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.607939.
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Tamplin, L. J. "Cohomology of compactified moduli spaces of bundles over Riemann surfaces." Thesis, University of Oxford, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.393618.
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Brakkee, Emma [Verfasser]. "Moduli spaces of K3 surfaces and cubic fourfolds / Emma Brakkee." Bonn : Universitäts- und Landesbibliothek Bonn, 2019. http://d-nb.info/120001992X/34.
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Bartolini, Gabriel. "On the Branch Loci of Moduli Spaces of Riemann Surfaces." Doctoral thesis, Linköpings universitet, Tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-77449.
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The spaces of conformally equivalent Riemann surfaces, Mg where g ≥ 1, are not manifolds. However the spaces of the weaker Teichmüller equivalence, Tg are known to be manifolds. The Teichmüller space Tg is the universal covering of Mg and Mg is the quotient space by the action of the modular group. This gives Mg an orbifold structure with a branch locus Bg. The branch loci Bg can be identified with Riemann surfaces admitting non-trivial automorphisms for surfaces of genus g ≥ 3. In this thesis we consider the topological structure of Bg. We study the connectedness of the branch loci in general by considering families of isolated strata and we we establish that connectedness is a phenomenon for low genera. Further, we give the orbifold structure of the branch locus of surfaces of genus 4 and genus 5 in particular, by studying the equisymmetric stratification of the branch locus. Paper 1. In this paper we show that the strata corresponding to actions of order 2 and 3 belong to the same connected component for arbitrary genera. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8. Paper 2. This paper contains a collection of results regarding components of the branch loci, some of them proved in detail in other papers. It is shown that for any integer d if p is a prime such that p > (d + 2)2, there there exist isolated strata of dimension d in the moduli space of Riemann surfaces of genus (d + 1)(p − 1)/2. It is also shown that if we consider Riemann surfaces as Klein surfaces, the branch loci are connected for every genera due to reflections. Paper 3. Here we consider surfaces of genus 4 and 5. Here we study the automorphism groups of Riemann surfaces of genus 4 and 5 up to topological equivalence and determine the complete structure of the equisymmetric stratification of the branch locus. Paper 4. In this paper we establish that the connectedness of the branch loci is a phenomenon for low genera. More precisely we prove that the only genera g where Bg is connected are g = 3, 4, 13, 17, 19, 59.
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Matteini, Tommaso. "Holomorphically symplectic varieties with Prym Lagrangian fibrations." Doctoral thesis, SISSA, 2014. http://hdl.handle.net/20.500.11767/3888.
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The thesis presents a construction of singular holomorphically symplectic varieties as Lagrangian fibrations. They are relative compactified Prym varieties associated to curves on symplectic surfaces with an antisymplectic involution. They are identified with the fixed locus of a symplectic involution on singular moduli spaces of sheaves of dimension 1. An explicit example, giving a singular irreducible symplectic 6-fold without symplectic resolutions, is described for a K3 surface which is the double cover of a cubic surface. In the case of abelian surfaces, a variation of this construction is studied to get irreducible symplectic varieties: relative compactified 0-Prym varieties. A partial classification result is obtained for involutions without fixed points: either the 0-Prym variety is birational to an irreducible symplectic variety of K3[n]-type, or it does not admit symplectic resolutions.
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Camara, Malick. "Tautological rings of moduli spaces of curves." Thesis, Paris 6, 2016. http://www.theses.fr/2016PA066459.
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Les espaces de modules de Riemann répondent au problème de la classification des surfaces de Riemann compactes d'un genre donné. Le sujet de cette thèse est la cohomologie de l'espace des modules des courbes d'un genre donné avec un certain nombre de points marqués. La description de cet anneau a été initiée par D. Mumford puis C. Faber avait proposé une description de l'anneau tautologique des espaces de modules sans points marqués. Une première source de relations provient des relations A. Pixton démontrées par A. Pixton, R. Pandharipande et D. Zvonkine mais on ne sait pas si elles sont complètes. Une autre source de relations utilisée dans ce travail sont les relations de A. Buryak, S. Shadrin et D. Zvonkine. Avant cette thèse, il y avait peu de résultats sur l'anneau tautologique d'espaces de modules de courbes avec un nombre quelconque de points marqués. Cette thèse donne une description complète des l'anneaux tautologiques des espaces de modules de courbes de genres 0, 1, 2, 3 et 4. Un des résultats ayant demandé beaucoup de travail est le groupe de degré 2 de l'anneau tautologique des espaces de modules de courbes lisses de genre 4. Ce groupe demande un travail sur l'annulation de certaines classes tautologiques sur le bord de la compactification de Deligne-Mumford de l'espace des modules en plus d'un astucieux travail numérique. L'espace des modules des courbes réelles de genre 0 et sa théorie de l'intersection sont également étudiés. On peut alors démontrer plusieurs résultats analogues à ceux obtenus dans le cas complexe comme l'équation de la corde. On démontre une formule donnant les nombres d'intersectionThe problem of the moduli spaces of compact Riemann surfaces is the problem of the classification of compact Riemann surfaces of a certain genus. The topic of this thesis is the cohomology of the moduli spaces of curves of a certain genus with marked points and more precisely its subbring called tautological ring. The description of the tautological ring has been initiated by D. Mumford, then C. Faber conjectured a description of the moduli space of curves without marked points. A source of tautological relations are Pixton's relations proven by A. Pixton, R. Pabndharipande and D. Zvonkine. Another source of relations are relations of A. Buryak, S. Shadrin and D. Zvonkine. Before this thesis, there were only few results on the tautological ring of curves with any number of marked points. This thesis gives a complete description of the tautological rings of moduli curves of genera 0, 1, 2, 3 and 4 with any number of marked points. A result which needed a lot of work is the group of degree 2 of the tautological ring of the moudli space of smooth curves of genus 4. We need to work on the vanishing of some tautological classes on the boundary of the Deligne-Mumford compactification of the moduli space of curves and a clever numerical work.The moduli space of real curves of genus 0 and its intersection theory are also studied. Then we can show several results which are analogous to results in the complex case like the string equation. One result of this thesis is a formula giving intersection numbers of products of xi classes.x
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Barros, Ignacio. "K3 surfaces and moduli of holomorphic differentials." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19290.
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In dieser Arbeit behandeln wir die birationale Geometrie verschiedener Modulräume; die Modulräume von Kurven mit einem k-Differential mit vorgeschierbenen Nullen, besser bekannt als Strata von Differenzialen, Moduln von K3 Flächen mit markierten Punkten und Moduln von Kurven. Für bestimmte Geschlechter nennen wir Abschätzungen der Kodaira-Dimension, konstruieren unirationale Parametrisierungen, rationale deckende Kurven und unterschiedliche birationale Modelle.
In Kapitel 1 führen wir die zu untersuchenden Objekte ein und geben einen kurzen Überblick ihrer wichtigsten Eigenschaften und offenen Problemen. In Kapitel 2 konstruieren wir einen Hilfsmodulraum, der als Brücke zwischen bestimmten finiten Quotienten von Mgn für kleines g und den Moduln der polarisierten K3 Flächen vom Geschlecht 11 dient. Wir entwickeln die Deformationstheorie, die nötig ist, um die Eigenschaften und die oben genannten Modulräume zu erforschen.
In Kapitel 3 bedienen wir uns dieser Werkzeuge, um birationale Modelle für Moduln polarisierter K3 Flächen vom Geschlecht 11 mit markierten Punkten zu konstruieren. Diese nutzen wir, um Resultate über die Kodaira-Dimension herzuleiten. Wir beweisen, dass der Modulraum von polarisierten K3 Flächen vom Geschlecht 11 mit n markierten Punkten unirational ist, falls n<=6, und uniruled, falls n<=7. Wir beweisen auch, dass die Kodaira-Dimension von Modulraum von polarisierten K3 Flächen vom Geschlecht 11 mit n markierten Punkten nicht-negativ ist für n>= 9. Im letzten Kapitel gehen wir noch auf die fehlenden Fälle der Kodaira-Klassifizierung von Mgnbar ein.
Schliesslich behandeln wir in Kapitel 4 die birationale Geometrie mit Blick auf die Strata von holomorphen und quadratischen Differentialen. Wir zeigen, dass die Strata holomorpher und quadratischer Differentiale von niedrigem Geschlecht uniruled sind, indem wir rationale Kurven mit pencils auf K3 und del Pezzo Flächen konstruieren. Durch das Beschränken des Geschlechts 3<= g<=6 bilden wir projektive Bündel über rationale Varietäten, die die holomorphe Strata mit maximaler Länge g-1 dominieren. Also zeigen wir auch, dass diese Strata unirational sind.In this thesis we investigate the birational geometry of various moduli spaces; moduli spaces of curves together with a k-differential of prescribed vanishing, best known as strata of differentials, moduli spaces of K3 surfaces with marked points, and moduli spaces of curves. For particular genera, we give estimates for the Kodaira dimension, construct unirational parameterizations, rational covering curves, and different birational models. In Chapter 1 we introduce the objects of study and give a broad brush stroke about their most important known features and open problems. In Chapter 2 we construct an auxiliary moduli space that serves as a bridge between certain finite quotients of Mgn for small g and the moduli space of polarized K3 surfaces of genus eleven. We develop the deformation theory necessary to study properties of the mentioned moduli space. In Chapter 3 we use this machinery to construct birational models for the moduli spaces of polarized K3 surfaces of genus eleven with marked points and we use this to conclude results about the Kodaira dimension. We prove that the moduli space of polarized K3 surfaces of genus eleven with n marked points is unirational when n<= 6 and uniruled when n<=7. We also prove that the moduli space of polarized K3 surfaces of genus eleven with n marked points has non-negative Kodaira dimension for n>= 9. In the final section, we make a connection with some of the missing cases in the Kodaira classification of Mgnbar. Finally, in Chapter 4 we address the question concerning the birational geometry of strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus 3<= g<=6 we construct projective bundles over rational varieties that dominate the holomorphic strata with length at most g-1, hence showing in addition, these strata are unirational.
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Fortuna, Mauro [Verfasser]. "Some moduli spaces of curves and surfaces : topology and Kodaira dimension / Mauro Fortuna." Hannover : Gottfried Wilhelm Leibniz Universität Hannover, 2021. http://d-nb.info/1229615636/34.
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Bartolini, Gabriel. "On the Branch Loci of Moduli Spaces of Riemann Surfaces of Low Genera." Licentiate thesis, Linköping : Department of Mathematics, Linköping University, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-51519.
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Karadogan, Gulay. "The Moduli Of Surfaces Admitting Genus Two Fibrations Over Elliptic Curves." Phd thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/12606084/index.pdf.
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In this thesis, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes H(1,X(d),n) of morphisms of degree n from elliptic curves to the modular curve X(d), d≤
3. Ultimately, we show that the moduli spaces, considered, are fiber spaces over the affine line A¹
with fibers determined by the components of H (1,X(d),n).
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Stein, Luba [Verfasser]. "On the Hilbert uniformization of moduli spaces of flat G-bundles over Riemann surfaces / Luba Stein." Bonn : Universitäts- und Landesbibliothek Bonn, 2014. http://d-nb.info/1047145499/34.
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Ehrenfried, Ralf. "Die Homologie der Modulräume berandeter Riemannscher Flächen von kleinem Geschlecht." Bonn : [s.n.], 1998. http://catalog.hathitrust.org/api/volumes/oclc/41464656.html.
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Goluboff, Justin Ross. "Genus Six Curves, K3 Surfaces, and Stable Pairs:." Thesis, Boston College, 2020. http://hdl.handle.net/2345/bc-ir:108715.
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Thesis advisor: Maksym FedorchukA general smooth curve of genus six lies on a quintic del Pezzo surface. In [AK11], Artebani and Kondō construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this dissertation, we construct a smooth Deligne-Mumford stack P₀ parametrizing certain stable surface-curve pairs which essentially resolves this map. Moreover, we give an explicit description of pairs in P₀ containing special curves
Thesis (PhD) — Boston College, 2020
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics
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Ling, Songbo [Verfasser], and Fabrizio [Akademischer Betreuer] Catanese. "Classification and Moduli Spaces of Surfaces of General Type with Pg = q =1 / Songbo Ling ; Betreuer: Fabrizio Catanese." Bayreuth : Universität Bayreuth, 2017. http://d-nb.info/113794448X/34.
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Rodado, A. Armando J. "Weierstrass points and canonical cell decompositions of the moduli and teichmüller spaces of riemann surfaces of genus two /." Connect to thesis, 2007. http://eprints.unimelb.edu.au/archive/00003539.
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Boes, Felix Jonathan [Verfasser]. "On moduli spaces of Riemann surfaces : new generators in their unstable homology and the homotopy type of their harmonic compactification / Felix Jonathan Boes." Bonn : Universitäts- und Landesbibliothek Bonn, 2018. http://d-nb.info/1170778070/34.
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Cruz, Juan Antonio Pacheco. "Espaços de Moduli de complexos quadráticos e de suas superfícies singulares." Universidade Federal de Juiz de Fora (UFJF), 2015. https://repositorio.ufjf.br/jspui/handle/ufjf/4696.
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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Um complexo de retas quadrático, ou simplesmente um complexo quadrático, é um conjunto de retas do espaço projetivo Pn (n = 3, no nosso caso) que satisfazem uma equação quadrática. Um complexo quadrático também pode ser considerado como um feixe de quádricas e portanto tem um símbolo de Segre bem definido. Sabe-se que as retas de um dado complexo, passando por um ponto p ∈P3, formam em geral um cone quadrático. Os pontos nos quais esses cones são a união de dois planos formam uma superfície em P3, chamada Superfície Singular do complexo. O objetivo desse trabalho é, fixado um símbolo de Segre, construir o espaço de Moduli dos complexos quadráticos, o espaço de Moduli das superfícies singulares desses complexos e então estudar a relação entre esse espaços.
A quadratic line complex, or a quadratic complex, is by definition a set of lines in a projective space Pn (n = 3, in our case) which satisfy a given quadratic equation. A quadratic complex can also be considered as a pencil of quadrics. Hence, it has a well defined Segre symbol. It is a classical fact that lines of a given complex through any point p ∈P3 form in general a quadratic cone. The points such that theses cones break up into two planes form a surface, the Singular Surface of the complex. The objective of this work is, for a fixed Segre symbol, to construct the Moduli space of quadratic complex, the Moduli space of corresponding singular surfaces and to study the relation between them.
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Cartier, Sébastien. "Surfaces des espaces homogènes de dimension 3." Phd thesis, Université Paris-Est, 2011. http://tel.archives-ouvertes.fr/tel-00672332.
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Ce mémoire porte sur l'étude des surfaces minimales et de courbure moyenne constante dans les espaces homogènes de dimension 3. Nous établissons les formules de Sym-Bobenko pour les surfaces de courbure moyenne constante 1/2 de H^2xR et minimales du groupe de Heisenberg, et donnons des exemples de construction de telles immersions par la méthode DPW. Nous montrons également que des propriétés de symétrie passent aux correspondances de type surfaces sœurs et cousines, ce qui entraîne l'existence de graphes entiers de courbure moyenne constante 1/2 à bout vertical dans H^2xR qui ne sont pas de révolution. Nous reprenons ensuite l'étude des bouts verticaux d'immersions de courbure moyenne constante 1/2 dans H^2xR. Nous munissons une famille de graphes entiers d'une structure de variété lisse et en déduisons un analogue pour H^2xR d'un théorème de A. E. Treibergs pour l'espace de Minkowski. Nous nous intéressons également aux déformations des anneaux de révolution. Une conséquence directe est l'existence d'anneaux immergés qui ne sont pas de révolution. Nous construisons notamment des anneaux dont les bouts n'ont pas le même axe. Enfin, nous décrivons les invariants de Nœther correspondant aux isométries des espaces homogènes pour les surfaces minimales et de courbure moyenne constante. Nous utilisons le formalisme de la géométrie de contact qui permet l'écriture de formules explicites en toute généralité, et nous étudions l'évolution des formes de Nœther sous l'action des isométries des espaces homogènes. Nous calculons ces invariants dans le cas des anneaux déformés de H^2xR, et dans celui des anneaux horizontaux du groupe de Heisenberg
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Morzadec, Thomas. "Compactification géométrique de l'espace de modules des structures de demi-translation sur une surface." Thesis, Université Paris-Saclay (ComUE), 2015. http://www.theses.fr/2015SACLS225/document.
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L'objectif de la thèse est de construire une compactification géométrique de l'espace des structures de demi-translation sur une surface S compacte, connexe, orientable, de genre au moins égal à 2. Il s’inscrit dans le très large thème d’étude des déformations de structures géométriques sur les surfaces. Une structure de demi-translation sur S est une métrique localement euclidienne (de courbure constante nulle) sur S, avec des singularités coniques d'angles k pi, avec k un entier et k>2, telle que l'holonomie de tout lacet lisse de S, disjoint des singularités, est Id ou -Id.Je définis l'ensemble des structures mixtes sur S, qui sont des structures arborescentes (au sens de Drutu-Sapir), équivariantes par le groupe fondamentalde S et CAT(0), obtenues par recollement de pièces par des arêtes, éventuellement réduites à des points, telles que l'espace obtenu par écrasement des pièces est un arbre réel simplicial (la plupart des arêtes ont une longueur non nulle), et les pièces sont ou bien des arbres réels, ou bien des revêtements universels de sous-surfaces (ouvertes) de S, munies de structures de demi-translation. Je munis l'espace Mix(Sigma) des (classes d'isométries équivariantes par le groupe fondamental de S) de structures mixtes sur S d'une topologie géométrique naturelle, appelée topologie de Gromov équivariante. Je montre alors, par des techniques d'ultralimites à la Gromov, que l'espace Flat(S) des (classes d'isotopie de) structures de demi-translation sur S, identifié à l’ensemble des structures de demi-translation équivariantes par le groupe fondamental de S sur le revêtement universel de S, est un ouvert dense de Mix(S), et que le projectifié PMix(S), muni de la topologie quotient, est compact. Le projectifié PMix(S) est donc une compactification du projectifié PFlat(S) de l'espace Flat(S) (qui s'identifie à l'espace des structure de demi-translation d'aire 1 sur S)The goal of this thesis is to build a geometric compactification of the space of half-translation structures on a connected, compact surface S, with genus at least 2. It is a part of the wide thema of study of the deformations of metric structures on surfaces.A half-translation structure on S is a locally euclidean metric (with null constant curvature) on S, with conical singularities of angles k pi, with k an integer and k>2, such that the holonomy of every smooth curve of S, disjoint from the singularities, is contained in Id or -Id.I define the set of mixed structures on S, which are tree-graded spaces (in the sense of Drutu-Sapir), equivariant by the fundamental group of S and CAT(0), obtained by gluing some pieces by some edges, possibly reduced to a point, such that the space obtained by replacing the pieces by some points is a simplicialtree (most edges have a positive length), and the pieces are either some trees or some universal covers of (open) subsurfaces of S endowed with a half-translation structures. I endow the space Mix(S) of (classes of isometry equivariant by the fundamental group of S of) mixed structures on S with a natural geometric topology, called the Gromov equivariant topology. I show, by techniques using ultralimits "à la Gromov", that the space Flat(S) of (isotopy classes of) half-translation structures on S, identified with the set of half-translation structures on the universal cover of S which are equivariant for the fundamental group of S, is a dense and open subset of Mix(S), and the projectified space PMix(S) is compact. The projectified space PMix(S) is then a compactification of the projectified space PFlat(S) (which identifies with the space of half-translations structures of area 1 on S
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Zúñiga, Javier. "Semistable Graph Homology." Pontificia Universidad Católica del Perú, 2016. http://repositorio.pucp.edu.pe/index/handle/123456789/96300.
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Using the orbicell decomposition of the Deligne-Mumford compactification of the moduli space of Riemann surfaces studied before by the author, a chain complex based on semistable ribbon graphs is constructed which is an extension of the Konsevich’s graph homology.En este trabajo mediante la descomposicion orbicelular de la compacticacion de Deligne-Mumford del espacio de moduli de supercies de Riemann (estudiada antes por el autor) construimos un complejo basado en grafos de cinta semiestables, lo cual constituye una extension de la homologa de grafos de Kontsevich.
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Finski, Siarhei. "On some problems of holomorphic analytic torsion." Thesis, Sorbonne Paris Cité, 2019. https://theses.md.univ-paris-diderot.fr/FINSKI_Siarhei_va.pdf.
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Le but de cette thèse est d'étudier la torsion analytique dans deux contextes différents. Dans le premier contexte, on étudie l'asymptotique de la torsion analytique, quand un fibré vectoriel holomorphe hermitien est tordué par une puissance croissant du fibré en droites positif. Dans le deuxième contexte, on généralise la théorie de la torsion analytique pour des surfaces de Riemann avec des pointes hyperboliques. Motivé par des singularités de la métrique complète de courbure scalaire constante -1 sur des surfaces de Riemann stables épointées, on demande que la métrique sur la surface de Riemann soit lisse seulement en dehors d'un nombre fini des points au voisinage auxquelles elle peut avoir des singularités comme la métrique de Poincaré sur un disque épointé. On fixe un fibré vectoriel holomorphe hermitien qui peut avoir au pire des singularités logarithmiques au voisinage des points marqués. Pour ces données, en renormalisant la trace de l'opérateur de la chaleur, on construit la torsion analytique et on étudie ces propriétésIn the first context, we study the asymptotics of the analytic torsion, when a Hermitian holomorphic vector bundle is twisted by an increasing power of a positive line bundle. In the second context, we generalize the theory of analytic torsion for surfaces with hyperbolic cusps. Motivated by singularities appearing in complete metrics of constant scalar curvature -1 on stable Riemann surfaces, we suppose that the metric on the surface is smooth outside a finite number points in the neighborhood of which it can to have singularities like Poincaré metric has on a punctured disc. We fix a Hermitian holomorphic vector bundle which has at worst logarithmic singularities in the neighborhood of the marked points. For these data, by renormalizing the trace of the heat operator, we construct the analytic torsion and study its properties. Then we study the properties of the analytic torsion in family setting: we prove the curvature theorem, we study the behavior of the analytic torsion when the cusps are created by degeneration and we give some applications to the moduli spaces of pointed curves
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Sauvaget, Adrien. "Théorie de l’intersection sur les espaces de différentielles holomorphes et méromorphes." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066460/document.
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Nous construisons l'espace des différentielles stables : un espace des modules de différentielles méromorphes avec des pôles d'ordres fixés. Cet espace est un cône au dessus de l'espace Mg,n des courbes stables. Si l'ensemble de poles est vide, il s'agit du fibré de Hodge. Nous introduisons l'anneau tautologique du projectivisé de l'espace des différentielles stables par analogie avec Mg,n. L'espace des différentielles stables est stratifié en fonction des ordres des zéros de la différentielle. Nous montrons que la classe de cohomologie Poincaré-duale de chaque strate est tautologique et peut être calculée explicitement, ce qui constitue le résultat principal de la thèse. Nous appliquons ces résultats pour calculer des nombres de Hurwitz et pour prouver plusieurs identités dans le groupe de Picard des strates. Ensuite, nous nous intéressons aux espaces des modules des différentielles d'ordre supérieur. Une courbe munie d'une k-différentielle holomorphe possède un revêtement naturel de groupe de Galois Z/kZ. Le fibré de Hodge sur la courbe revêtante se décompose en une somme directe de sous-fibrés en fonction du car- actère de Z/kZ. Nous calculons la première classe de Chern de chacun de ces sous-fibrés. Un dernier chapitre sera consacré à l'exposé des liens conjecturaux entre les classes des strates de différentielles, les espaces de courbes r-spin et les cycles de double ramificationWe construct the space of stable differentials: a moduli space of meromorphic differentials with poles of fixed order. This space is a cone over the moduli space Mg,n of stable curves. If the set of poles is empty, then this cone is the Hodge bundle. We introduce the tautological ring of the projectivized space of stable differentials by analogy with Mg,n. The space of stable differentials is stratified according to the orders of zeros of the differential. We show that the Poincaré-dual cohomology classes of these strata are tautological and can be explicitly computed, this constitutes the main result of this thesis. We apply this result to compute Hurwitz numbers and to show several identities in the Picard group of the strata. Then, we interest ourselves to moduli spaces of differentials of superior order. A curve endowed with a k-differential carry a natural ramified covering of Galois group Z/kZ. The Hodge bundle over the covering curve is decomposed into a direct sum of sub-vector bundles according to the character of Z/kZ. We compute the first Chern class of each of these sub-bundles. A last chapter will be dedicated to the presentation of conjectural relations between classes of strata of differentials, moduli of r-spin structures and double ramification cycles
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Cattaneo, Alberto. "Non-symplectic automorphisms of irreducible holomorphic symplectic manifolds." Thesis, Poitiers, 2018. http://www.theses.fr/2018POIT2322/document.
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Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1.Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectiqueWe study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution
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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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Zowislok, Markus [Verfasser]. "On moduli spaces of semistable sheaves on K3 surfaces / vorgelegt von Markus Zowislok." 2010. http://d-nb.info/1003549594/34.
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Chang, Chi-Kang, and 張繼剛. "Desingularized moduli spaces of torsion-free semistable sheaves on a K3 surface." Thesis, 2018. http://ndltd.ncl.edu.tw/handle/fufjab.
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碩士國立臺灣大學
數學研究所
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Abstract The aim of this article is to study Kieran G. O’Grady’s paper "Desingularized moduli spaces of sheaves on a K3" in 1998, where the author constructs the moduli space of rank two torsion-free semistable sheaves on a non-singular K3 surface with c1 = 0 and c2 = c a even number not less then 4. This moduli space is denoted by Mc, which is a G.I.T. quotient from the Quot-scheme and is singular. By using Kirwan’s method of successive blow ups of the strictly semistable loci with reductive stabilizer, one can obtain a desingularization Mcc of Mc. What’s surprising is that when c = 4, there is a Mori extremal divisorial contraction of Mc4 so that the outcome is a hyperk¨ahler manifold Mf4. Moreover, the natural map from Mf4 to M4 is a morphism and hence a simplectic desingularization of M4. The hyperk¨ahler manifold Mf4 is not birational/deformation equivalence to another two typical constructions of HK manifolds: the Hilbert schemes of points and Kummer varieties. Key words: moduli space of sheaves, semistable sheaves, geometric invariant theory, symplectic resolution, hyperk¨ahler variety.
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Petro, Matthew. "Moduli spaces of Riemann surfaces." 2008. http://www.library.wisc.edu/databases/connect/dissertations.html.
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Baranovsky, Vladimir. "Moduli of sheaves on surfaces and action of the oscillator algebra /." 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:9965058.
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Maslovaric, Marcel. "Variational Geometric Invariant Theory and Moduli of Quiver Sheaves." Doctoral thesis, 2018. http://hdl.handle.net/11858/00-1735-0000-002E-E430-8.
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HARDER, ANDREW. "Moduli Spaces of K3 Surfaces with Large Picard Number." Thesis, 2011. http://hdl.handle.net/1974/6646.
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Morrison has constructed a geometric relationship between K3 surfaces with large Picard number and abelian surfaces. In particular, this establishes that the period spaces of certain families of lattice polarized K3 surfaces (which are closely related to the moduli spaces of lattice polarized K3 surfaces) and lattice polarized abelian surfaces are identical. Therefore, we may study the moduli spaces of such K3 surfaces via the period spaces of abelian surfaces.
In this thesis, we will answer the following question: from the moduli space of abelian surfaces with endomorphism structure (either a Shimura curve or a Hilbert modular surface), there is a natural map into the moduli space of abelian surfaces, and hence into the period space of abelian surfaces. What sort of relationship exists between the moduli spaces of abelian surfaces with endomorphism structure and the moduli space of lattice polarized K3 surfaces? We will show that in many cases, the endomorphism ring of an abelian surface is just a subring of the Clifford algebra associated to the N\'eron-Severi lattice of the abelian surface. Furthermore, we establish a precise relationship between the moduli spaces of rank 18 polarized K3 surfaces and Hilbert modular surfaces, and between the moduli spaces of rank 19 polarized K3 surfaces and Shimura curves.
Finally, we will calculate the moduli space of E_8^2 + <4>-polarized K3 surfaces as a family of elliptic K3 surfaces in Weierstrass form and use this new family to find families of rank 18 and 19 polarized K3 surfaces which are related to abelian surfaces with real multiplication or quaternionic multipliction via the Shioda-Inose construction.Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-08-12 14:38:04.131
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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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"Survey on the canonical metrics on the Teichmüller spaces and the moduli spaces of Riemann surfaces." 2010. http://library.cuhk.edu.hk/record=b5896649.
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Chan, Kin Wai."September 2010."
Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.
Includes bibliographical references (leaves 103-106).
Abstracts in English and Chinese.
Chapter 1 --- Introduction --- p.8
Chapter 2 --- Background Knowledge --- p.13
Chapter 2.1 --- Results from Riemann Surface Theory and Quasicon- formal Mappings --- p.13
Chapter 2.1.1 --- Riemann Surfaces and the Uniformization The- orem --- p.13
Chapter 2.1.2 --- Fuchsian Groups --- p.15
Chapter 2.1.3 --- Quasiconformal Mappings and the Beltrami Equation --- p.17
Chapter 2.1.4 --- Holomorphic Quadratic Differentials --- p.20
Chapter 2.1.5 --- Nodal Riemann Surfaces --- p.21
Chapter 2.2 --- Teichmuller Theory --- p.24
Chapter 2.2.1 --- Teichmiiller Spaces --- p.24
Chapter 2.2.2 --- Teichmuller's Distance --- p.26
Chapter 2.2.3 --- The Bers Embedding --- p.26
Chapter 2.2.4 --- Teichmuller Modular Groups and Moduli Spaces of Riemann Surfaces --- p.27
Chapter 2.2.5 --- Infinitesimal Theory of Teichmiiller Spaces --- p.28
Chapter 2.2.6 --- Boundary of Moduli Spaces of Riemann Sur- faces --- p.29
Chapter 2.3 --- Schwarz-Yau Lemma --- p.30
Chapter 3 --- Classical Canonical Metrics on the Teichnmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.31
Chapter 3.1 --- Finsler Metrics and Bergman Metric --- p.31
Chapter 3.1.1 --- Definitions and Properties of the Metrics --- p.32
Chapter 3.1.2 --- Equivalences of the Metrics --- p.33
Chapter 3.2 --- Weil-Petersson Metric --- p.36
Chapter 3.2.1 --- Definition and Properties of the Weil-Petersson Metric --- p.36
Chapter 3.2.2 --- Results about Harmonic Lifts --- p.37
Chapter 3.2.3 --- Curvature Formula for the Weil-Petersson Met- ric --- p.41
Chapter 4 --- Kahler Metrics on the Teichmiiller Spaces and the Moduli Spaces of Riemann Surfaces --- p.42
Chapter 4.1 --- McMullen Metric --- p.42
Chapter 4.1.1 --- Definition of the McMullen Metric --- p.42
Chapter 4.1.2 --- Properties of the McMullen Metric --- p.43
Chapter 4.1.3 --- Equivalence of the McMullen Metric and the Teichmuller Metric --- p.45
Chapter 4.2 --- Kahler-Einstein Metric --- p.50
Chapter 4.2.1 --- Existence of the Kahler-Einstein Metric --- p.50
Chapter 4.2.2 --- A Conjecture of Yau --- p.50
Chapter 4.3 --- Ricci Metric --- p.51
Chapter 4.3.1 --- Definition of the Ricci Metric --- p.51
Chapter 4.3.2 --- Curvature Formula of the Ricci Metric --- p.53
Chapter 4.4 --- The Asymptotic Behavior of the Ricci Metric --- p.61
Chapter 4.4.1 --- Estimates on the Asymptotics of the Ricci Metric --- p.61
Chapter 4.4.2 --- Estimates on the Curvature of the Ricci Metric --- p.83
Chapter 4.5 --- Perturbed Ricci Metric --- p.92
Chapter 4.5.1 --- Definition and the Curvature Formula of the Perturbed Ricci Metric --- p.92
Chapter 4.5.2 --- Estimates on the Curvature of the Perturbed Ricci Metric --- p.93
Chapter 4.5.3 --- Equivalence of the Perturbed Ricci Metric and the Ricci Metric --- p.96
Chapter 5 --- Equivalence of the Kahler Metrics on the Teichmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.98
Chapter 5.1 --- Equivalence of the Ricci Metric and the Kahler-Einstein Metric --- p.98
Chapter 5.2 --- Equivalence of the Ricci Metric and the McMullen Metric --- p.99
Bibliography --- p.103
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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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Amaris, Armando Jose Rodado. "Weierstrass points and canonical cell decompositions of the moduli and Teichmuller Spaces of Riemann surfaces of genus two." 2007. http://repository.unimelb.edu.au/10187/2259.
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A genus-two Riemann surface admits a canonical decomposition into Dirichlet polygons determined by its six Weierstrass points. All possible associated graphs are determined explicitly from circle packing problems, solved by systems of linear inequalities whose solutions determine a finite 6-dimensional polyhedral complex in 12-dimensional space. The 6-dimensional Moduli Space of genus-two Riemann surfaces inherits a canonical explicit decomposition into Euclidean polyhedra, giving new natural coordinates for the Teichmuller Space of all possible constant curvature geometries on a marked genus-two surface.
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Ferreira, Susana Raquel Carvalho. "Schottky principal G-bundles over compact Riemann surfaces." Doctoral thesis, 2014. http://hdl.handle.net/10362/13333.
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