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1
Paterno, Valentina. "Special rationally connected manifolds." Doctoral thesis, Università degli studi di Trento, 2009. https://hdl.handle.net/11572/368764.
Full textAbstract:
We consider smooth complex projective varieties X which are rationally connected by rational curves of degree d with respect to a fixed ample line bundle L on X, and we focus our attention on conic connected manifolds (d=2) and on rationally cubic connected manifolds (d=3).
Conic connected manifolds were studied by Ionescu and Russo; they considered conic connected manifolds embedded in projective space (i.e. L is very ample) and they proved a classification theorem for these manifolds. We show that their classification result holds true assuming just the ampleness of L. Moreover we give a different proof of a theorem due to Kachi and Sato; this result characterizes a special subclass of conic connected manifolds.
As already said before, we study also rationally cubic connected manifolds. We prove that if rationally cubic connected manifolds are covered by “lines†, i.e. by curves of degree 1 with respect to L, then the Picard number of X is equal to or less than 3; moreover we show that if the Picard number is equal to 3 then there is a covering family of “lines†whose numerical class spans a negative extremal ray of the Kleiman-Mori cone of X.
Unfortunately, for rationally cubic connected manifolds which don't admit a covering family of “lines†there isn't an upper bound on the Picard number. However we prove that if we consider rationally cubic connected manifolds which are not covered by “lines†but are Fano then up to a few exceptions in dimension 2 also the Picard number of these manifolds is equal to or less than 3.
In particular, supposing that the dimension of X is greater than 2, we show that either the Picard number is equal to or less than 2 or X is the blow up of projective space along two disjoint subvarieties that are linear subspaces or quadrics.
2
Altavilla, Amedeo. "Quaternionic slice regular functions on domains without real points." Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368416.
Full textAbstract:
In this thesis I've explored the theory of quaternionic slice regular functions. More precisely I've studied some rigidity properties, some differential issues and an application in
complex differential geometry.
This application concerns the constructions and classifications of orthogonal complex structures on open domains of the four dimensional euclidean space.
3
Tasin, Luca. "Birational Maps in the Minimal Model Program." Doctoral thesis, Università degli studi di Trento, 2013. https://hdl.handle.net/11572/368878.
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4
Pernigotti, Letizia. "Geometry of moduli spaces of higher spin curves." Doctoral thesis, Università degli studi di Trento, 2013. https://hdl.handle.net/11572/369303.
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ABSTRACT: Roughly speaking, the moduli space of higher spin curves parametrizes equivalence classes of pairs (C, L) where C is a smooth genus g algebraic curve and L is a line bundle on it whose r-th tensor power is isomorphic to the canonical bundle of the curve. The aim of the talk is to discuss important geometrical properties of these spaces under different points of view: one possible compactification together with the description of the rational Picard group, their birational geometry in some low genus cases and their relation with some special locus inside the classical moduli spaces of curves.
5
Platoni, Irene. "Complete Arcs and Caps in Galois Spaces." Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368555.
Full textAbstract:
Galois spaces, that is affine and projective spaces of dimension N ≥ 2 defined over a finite (Galois) field F_q, are well known to be rich of nice geometric, combinatorial
and group theoretic properties that have also found wide and relevant applications in several branches of Combinatorics, as well as in more practical areas, notably Coding Theory and Cryptography. The systematic study of Galois spaces was initiated in the late 1950’s by the pioneering work of B. Segre [59]. The trilogy [34, 36, 42] covers the general theory of Galois spaces including the study of objects which are linked to linear codes. Typical such objects are plane arcs and their generalizations - especially caps and arcs in higher dimensions - whose code theoretic counterparts are distinguished types of error-correcting and covering linear codes. Their investigation has received a great stimulus from Coding Theory, especially in the last decades; see the survey papers [40, 41]. An important issue in this context is to ask for explicit constructions of small complete arcs and small complete caps. A cap in a Galois space is a set of points no three of which are collinear. A cap is complete if its secants (lines through two points of the set) cover the whole space. An arc in a Galois space of dimension N is a set of points no N+1 of which lying on the same hyperplane. In analogy with caps, an arc which is maximal with respect to set-theoretical inclusion is said to be complete.
Also, arcs coincide with caps in Galois planes. From these geometrical objects, there arise linear codes which turn out to have very good covering properties, provided
that the size of the set is small with respect to the dimension N and the order q of the ambient space.
For the size t(AG(N,q)) of the smallest complete caps in a Galois affine space AG(N,q) of dimension N over F_q, the trivial lower bound is √2q^{N−1/2}. General constructions of complete caps whose size is close to this lower bound are only known for q even and N odd, see [16, 19, 29, 52]. When N is even, complete caps of size of the same order of magnitude as cq^{N/2}, with c a constant independent of q, are known to exist for both the odd and the even order case, see [16, 18, 28, 29, 31] (see also Section 2.2 and the references therein). Whereas, few constructions of small complete arcs in Galois spaces of dimension N>2 are known. In [65, 66, 67], small complete arcs having many points in common with the normal rational curve are investigated (see Section 4.2.3 for comparisons with our results). In this thesis, new infinite families of complete arcs and caps in higher dimensional spaces are constructed from algebraic curves defined over a finite field. In most
cases, no smallest complete caps/arcs were previously known in the literature. Although caps and arcs are rather combinatorial objects, constructions and proofs sometimes heavily rely on concepts and results from Algebraic Geometry in positive characteristic.
6
Cancian, Nicola. "On Semi-isogenous Mixed Surfaces." Doctoral thesis, Università degli studi di Trento, 2017. https://hdl.handle.net/11572/369295.
Full textAbstract:
Let C be a compact Riemann surface. Let us consider a finite group acting on CxC, having some elements that exchange the factors, and assume that the subgroup of those elements that do not exchange the factors acts freely. We call the quotient a Semi-isogenous Mixed Surface. In this work we investigate these surfaces and we explain how their geometry is encoded in the group. Based on this, we present an algorithm to classify the Semi-isogenous Mixed Surfaces with given geometric genus, irregularity and self-intersection of the canonical class. In particular we give the classification of Semi-isogenous Mixed Surfaces with K^2>0 and holomorphic Euler-Poincaré characteristic equal to 1, where new examples of minimal surfaces of general type appear. Minimality of Semi-isogenous Mixed Surfaces is discussed using two different approaches. The first one involves the study of the bicanonical system of such surfaces: we prove that we can relate the dimension of its first cohomology group to the rank of a linear map that involves only curves. The second approach exploits Hodge index theorem to bound the number of exceptional curves that live on a Semi-isogenous Mixed Surface.
7
Pantaleoni, Andrea <1977>. "Involuzioni di corpi di manici in dimensione 3 ed applicazioni." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/208/1/tesi_Pantaleoni.pdf.
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Pantaleoni, Andrea <1977>. "Involuzioni di corpi di manici in dimensione 3 ed applicazioni." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/208/.
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9
Mandini, Alessia <1979>. "The geometry of the moduli space of polygons in the euclidean space." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/424/1/Tesi_A._Mandini.pdf.
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Mandini, Alessia <1979>. "The geometry of the moduli space of polygons in the euclidean space." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/424/.
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Cerri, Andrea <1978>. "Stability and computation in multidimensional size theory." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/562/1/cerri.pdf.
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Cerri, Andrea <1978>. "Stability and computation in multidimensional size theory." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amsdottorato.unibo.it/562/.
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13
Di, Fabio Barbara <1977>. "Shape from Functions:Enhancing Geometrical-Topological Descriptors." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2009. http://amsdottorato.unibo.it/1961/1/TesiDiFabioPHD.pdf.
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Di, Fabio Barbara <1977>. "Shape from Functions:Enhancing Geometrical-Topological Descriptors." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2009. http://amsdottorato.unibo.it/1961/.
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15
Cavazza, Niccolò <1983>. "Estimating persistent Betti numbers for discrete shape analysis." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2011. http://amsdottorato.unibo.it/3468/1/cavazza_niccolo_tesi.pdf.
Full textAbstract:
Persistent Topology is an innovative way of matching topology and geometry, and it proves to be an effective mathematical tool in shape analysis. In order to express its full potential for applications, it has to interface with the typical environment of Computer Science: It must be possible to deal with a finite sampling of the object of interest, and with combinatorial representations of it.
Following that idea, the main result claims that it is possible to construct a relation between the persistent Betti numbers (PBNs; also called rank invariant) of a compact, Riemannian submanifold X of R^m and the ones of an approximation U of X itself, where U is generated by a ball covering centered in the points of the sampling. Moreover we can state a further result in which, this time, we relate X with a finite simplicial complex S generated, thanks to a particular construction, by the sampling points. To be more precise, strict inequalities hold only in "blind strips'', i.e narrow areas around the discontinuity sets of the PBNs of U (or S). Out of the blind strips, the values of the PBNs of the original object, of the ball covering of it, and of the simplicial complex coincide, respectively.
16
Cavazza, Niccolò <1983>. "Estimating persistent Betti numbers for discrete shape analysis." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2011. http://amsdottorato.unibo.it/3468/.
Full textAbstract:
Persistent Topology is an innovative way of matching topology and geometry, and it proves to be an effective mathematical tool in shape analysis. In order to express its full potential for applications, it has to interface with the typical environment of Computer Science: It must be possible to deal with a finite sampling of the object of interest, and with combinatorial representations of it.
Following that idea, the main result claims that it is possible to construct a relation between the persistent Betti numbers (PBNs; also called rank invariant) of a compact, Riemannian submanifold X of R^m and the ones of an approximation U of X itself, where U is generated by a ball covering centered in the points of the sampling. Moreover we can state a further result in which, this time, we relate X with a finite simplicial complex S generated, thanks to a particular construction, by the sampling points. To be more precise, strict inequalities hold only in "blind strips'', i.e narrow areas around the discontinuity sets of the PBNs of U (or S). Out of the blind strips, the values of the PBNs of the original object, of the ball covering of it, and of the simplicial complex coincide, respectively.
17
Manfredi, Enrico <1986>. "Knots and links in lens spaces." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2014. http://amsdottorato.unibo.it/6265/1/manfredi_enrico_tesi.pdf.
Full textAbstract:
The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk.
In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy.
Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown.
A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described.
A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion.
One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q).
Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift.
The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift.
Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.
18
Manfredi, Enrico <1986>. "Knots and links in lens spaces." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2014. http://amsdottorato.unibo.it/6265/.
Full textAbstract:
The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk.
In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy.
Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown.
A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described.
A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion.
One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q).
Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift.
The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift.
Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.
19
Trozzo, Marco <1986>. "Higgs Bundles and Local Systems on Elliptic Curves." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amsdottorato.unibo.it/7772/1/trozzo_marco_tesi.pdf.
Full textAbstract:
If S is an elliptic curve, the total space X of the cotangent bundle of S is the moduli space of rank one and degree zero Higgs bundles on S and the corresponding character variety Y is C* x C*.
The punctual Hilbert scheme X^[n] of X can be identified with the moduli space of stable marked Higgs bundles on S and there is a natural isomorphism of graded vector spaces between the rational cohomology groups of the Hilbert schemes of X and Y that exchanges the perverse Leray filtration on X^[n] with the halved weight filtration on Y^[n].
We prove that there is a diffeomorphism between the Hilbert schemes that induces the given isomorphism in cohomology. We also give a complete description of Higgs bundles corresponding to subschemes of length n ≤ 3. Moreover, we discuss a conjecture by Simpson on the compactification of the moduli space of Higgs bundles and on the dual boundary complex of the character variety, proving a result going in the direction of Simpson’s conjecture.
20
Ruffoni, Lorenzo <1989>. "The Geometry of Branched Complex Projective Structures on Surfaces." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amsdottorato.unibo.it/7860/1/ruffoni_lorenzo_tesi.pdf.
Full textAbstract:
We study the geometry of deformations of structures locally modelled on the Riemann sphere, up to branched covers, focusing on structures with quasi-Fuchsian holonomy and on structures which admit holomorphically trivial deformations. Applications to Riemann-Hilbert problems are discussed.
21
Savini, Alessio <1990>. "Numerical invariants and volume rigidity for hyperbolic lattices." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amsdottorato.unibo.it/8464/1/alessio_savini_tesi.pdf.
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We prove a generalization of Mostow-Prasad rigidity by showing that the volume function on the PO(m,1)-character variety of a non-uniform real hyperbolic lattice of PO(p,1) stays away from its maximum outside a suitable analytic neighborhood of the class of the discrete and faithful representation, when m>=p>=3. The same for non-uniform complex and quaternionic hyperbolic lattices for m>=p>=2.
When G is a non-uniform lattice of PSL(2,C) without torsion we define the omega-Borel invariant for representations into SL(n,C_om) and we discuss its properties.
22
Felisetti, Camilla <1990>. "Two applications of the decomposition theorem to moduli spaces." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amsdottorato.unibo.it/8681/1/felisetti_camilla_tesi.pdf.
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The decomposition theorem is a statement about the (derived) direct image of the intersection cohomology by an algebraic projective map. The decomposition theorem and more generally the theory of perverse sheaves have found many interesting applications, especially in representation theory. Usually a lot of work is needed to apply it in concrete situations, to identify the various summands. This thesis proposes two applications of the decomposition theorem.
In the first we consider the moduli space of Higgs bundles of rank 2 and degree 0 over a curve of genus 2. Applying the decomposition theorem, we are able to compute the weight polynomial of the intersection cohomology of this moduli space.
The second result contained in this thesis is concerned with the general problem of determining the support of a map, and therefore in line with the ”support theorem” by Ngo.
We consider families C ! B of integral curves with at worst planar singularities, and the relative ”nested” Hilbert scheme C^[m,m+1]. Applying the technique of higher discriminants, recently developed by Migliorini and Shende, we prove that in this case there are no supports other than the whole base B of the family. Along the way we investigate smoothness properties of C[m,m+1], which may be of interest on their own.
23
Grossi, Annalisa <1992>. "Automorphisms of O'Grady's sixfolds." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amsdottorato.unibo.it/9441/1/Tesi%20Dottorato.pdf.
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We study automorphisms of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to the O’Grady’s sixfold. We classify non-symplectic and symplectic automorphisms using lattice theoretic criterions related to the lattice structure of the second integral cohomology. Moreover we introduce the concept of induced automorphisms. There are two birational models for O'Grady's sixfolds, the first one introduced by O'Grady, which is the resolution of singularities of the Albanese fiber of a moduli space of sheaves on an abelian surface, the second one which concerns in the quotient of an Hilbert cube by a symplectic involution. We find criterions to know when an automorphism is induced with respect to these two different models, i.e. it comes from an automorphism of the abelian surface or of the Hilbert cube.
24
Quercioli, Nicola <1992>. "On the topological theory of Group Equivariant Non-Expansive Operators." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amsdottorato.unibo.it/9770/1/Tesi-Dottorato-Quercioli.pdf.
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In this thesis we aim to provide a general topological and geometrical framework for group equivariance in the machine learning context. A crucial part of this framework is a synergy between persistent homology and the theory of group actions. In our approach, instead of focusing on data, we focus on suitable operators defined on the functions that represent the data. In particular, we define group equivariant non-expansive operators (GENEOs), which are maps between function spaces endowed with the actions of groups of transformations. We investigate the topological, geometric and metric properties of the space of GENEOs. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of GENEOs and proving some results about our model. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show some new methods to build different classes of GENEOs in order to populate and approximate the space of GENEOs. Moreover, we define a suitable Riemannian structure on manifolds of GENEOs making available the use of gradient descent methods.
25
Mella, Alessandro <1992>. "Non-topological persistence for data analysis and machine learning." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amsdottorato.unibo.it/9809/1/Thesis_reviewed.pdf.
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This thesis main objective is to study possible applications of the generalisation of persistence theory introduced in [1], [2]. This generalisation extends the notion of persistence to a wider categorical setting, avoiding constructing secondary structures as topological spaces.
The first field analysed is graph theory. At first, we studied which classical graph theory invariants could be used as rank function. Another aspect analysed in this thesis is the extension of the study of connectivity in graphs from a persistence viewpoint started in [1] to oriented graphs. Moreover, we studied how different orientation of the same underlying graph can change the distribution of cornerpoints in persistence diagrams, both in deterministic and random graphs.
The other application field analysed is image processing. We adapted the notion of steady and ranging sets to the category of sets and used them to define activation and deactivation rules for each pixel. These notions allowed us to define a filter capable of enhancing the signal of pixels close to a border. This filter has proven to be stable under salt and pepper noise perturbation.
At last, we used this filter to define a novel pooling layer for convolutional neural networks. In the experimental part, we compared the proposed layer with other state-of-the-art layers. The results show how the proposed layer outperform the other layers in term of accuracy. Moreover, by concatenating the proposed and the Max pooling, it is possible to improve accuracy further.
[1] Bergomi, M.G., Ferri, M., Vertechi, P., Zuffi, L. (2020), Beyond topological persistence: Starting from networks, arXiv.
[2] Bergomi, M. G., & Vertechi, P. (2020). Rank-based persistence. Theory and Applications of Categories, 35, 228-260.
26
Prieto, Montañez Yulieth Katterin <1993>. "Automorphisms on algebraic varieties: K3 surfaces, hyperkähler manifolds, and applications on Ulrich bundles." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2022. http://amsdottorato.unibo.it/10149/1/PhD_tesis.pdf.
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One of the main tools to study the geometry of complex algebraic varieties is the group of automorphisms. The first part of this thesis concerns the study of symplectic automorphisms of finite order on K3 surfaces, and birational symplectic maps of finite order on projective hyperkähler manifolds which are deformation equivalent to the Hilbert scheme of K3 surfaces. In the second part of this thesis, the automorphism groups of rational homogeneous spaces are used to study Ulrich bundles in smooth projective varieties.
27
Sarti, Filippo <1993>. "Numerical invariants for measurable cocycles." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2022. http://amsdottorato.unibo.it/10160/2/tesi.pdf.
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The theory of numerical invariants for representations can be generalized to measurable cocycles. This provides a natural notion of maximality for cocycles associated to complex hyperbolic lattices with values in groups of Hermitian type. Among maximal cocycles, the class of Zariski dense ones turns out to have a rigid behavior.
An alternative implementation of numerical invariants can be given by using equivariant maps at the level of boundaries and by exploiting the Burger-Monod approach to bounded cohomology.
Due to their crucial role in this theory, we prove existence results in two different contexts.
Precisely, we construct boundary maps for non-elementary cocycles into the isometry group of CAT(0)-spaces of finite telescopic dimension and for Zariski dense cocycles into simple Lie groups.
Then we approach numerical invariants.
Our first goal is to study cocycles from complex hyperbolic lattices into the Hermitian group SU(p,q).
Following the theory recently developed by Moraschini and Savini, we define the Toledo invariant by using the pullback along cocycles, also by involving boundary maps.
For cocycles Γ × X → SU(p,q) with 1
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Portioli, Marco <1992>. "The Hitchin map for one-nodal base curves of compact type." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2022. http://amsdottorato.unibo.it/10455/1/portioli_marco_tesi.pdf.
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Studying moduli spaces of semistable Higgs bundles (E, \phi) of rank n on a smooth curve C, a key role is played by the spectral curve X (Hitchin), because an important result by Beauville-Narasimhan-Ramanan allows us to study isomorphism classes of such Higgs bundles in terms of isomorphism classes of rank-1 torsion-free sheaves on X. This way, the generic fibre of the Hitchin map, which associates to any semistable Higgs bundle the coefficients of the characteristic polynomial of \phi, is isomorphic to the Jacobian of X. Focusing on rank-2 Higgs data, this construction was extended by Barik to the case in which the curve C is reducible, one-nodal, having two smooth components. Such curve is called of compact type because its Picard group is compact.
In this work, we describe and clarify the main points of the construction by Barik and we give examples, especially concerning generic fibres of the Hitchin map. Referring to Hausel-Pauly, we consider the case of SL(2,C)-Higgs bundles on a smooth base curve, which are such that the generic fibre of the Hitchin map is a subvariety of the Jacobian of X, the Prym variety. We recall the description of special loci, called endoscopic loci, such that the associated Prym variety is not connected. Then, letting G be an affine reductive group having underlying Lie algebra so(4,C), we consider G-Higgs bundles on a smooth base curve. Starting from the construction by Bradlow-Schaposnik, we discuss the associated endoscopic loci.
By adapting these studies to a one-nodal base curve of compact type, we describe the fibre of the SL(2,C)-Hitchin map and of the G-Hitchin map, together with endoscopic loci.
In the Appendix, we give an interpretation of generic spectral curves in terms of families of double covers.
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Paterno, Valentina. "Special rationally connected manifolds." Doctoral thesis, University of Trento, 2009. http://eprints-phd.biblio.unitn.it/159/1/tesi_PATERNO.pdf.
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We consider smooth complex projective varieties X which are rationally connected by rational curves of degree d with respect to a fixed ample line bundle L on X, and we focus our attention on conic connected manifolds (d=2) and on rationally cubic connected manifolds (d=3).
Conic connected manifolds were studied by Ionescu and Russo; they considered conic connected manifolds embedded in projective space (i.e. L is very ample) and they proved a classification theorem for these manifolds. We show that their classification result holds true assuming just the ampleness of L. Moreover we give a different proof of a theorem due to Kachi and Sato; this result characterizes a special subclass of conic connected manifolds.
As already said before, we study also rationally cubic connected manifolds. We prove that if rationally cubic connected manifolds are covered by “lines”, i.e. by curves of degree 1 with respect to L, then the Picard number of X is equal to or less than 3; moreover we show that if the Picard number is equal to 3 then there is a covering family of “lines” whose numerical class spans a negative extremal ray of the Kleiman-Mori cone of X.
Unfortunately, for rationally cubic connected manifolds which don't admit a covering family of “lines” there isn't an upper bound on the Picard number. However we prove that if we consider rationally cubic connected manifolds which are not covered by “lines” but are Fano then up to a few exceptions in dimension 2 also the Picard number of these manifolds is equal to or less than 3.
In particular, supposing that the dimension of X is greater than 2, we show that either the Picard number is equal to or less than 2 or X is the blow up of projective space along two disjoint subvarieties that are linear subspaces or quadrics.
30
Frapporti, Davide. "Mixed quasi-étale surfaces and new surfaces of general type." Doctoral thesis, University of Trento, 2012. http://eprints-phd.biblio.unitn.it/686/1/PhDThesis.pdf.
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In this thesis we define and study the mixed quasi-étale surfaces. In particularwe classify all the mixed quasi-étale surfaces whose minimal resolution of the singularities is a regular surface with p_g=0 and K^2>0.
It is a well known fact that each Riemann surface with p_g=0 is isomorphic to P^1.
At the end of XIX century M. Noether conjectured that an analogous statement holds for the surfaces: in modern words, he conjectured that every smooth projective surface with p_g=q=0 be rational.
The first counterexample to this conjecture is due to F. Enriques (1869).
He constructed the so called Enriques surfaces.
The Enriques-Kodaira classification divides compact complex surfaces in four main classes according to their Kodaira dimension k: -oo, 0, 1, 2. A surface is said to be of general type if k=2.
Nowadays this class is much less understood than the other three. The Enriques surfaces have k=0.
The first examples of surfaces of general type with p_g=0 have been constructed in the 30's by L. Campedelli e L. Godeaux.
The idea of Godeaux to construct surfaces was to consider the quotient of simpler surfaces by the free action of a finite group.
In this spirit, Beauville proposed a simple construction of surfaces of general type, considering the quotient of a product of two curves C_1 and C_2 by the free action of a finite group G. Moreover he gave an explicit example with p_g=q=0 considering the quotient of two Fermat curves of degree 5 in P^2.
There is no hope at the moment to achieve a classification of the whole class of the surfaces of general type. Since for a surface in this class the Euler characteristic of the structure sheaf \chi is strictly positive, one could hope that a classification of the boundary case \chi=1 is more affordable.
Some progresses in this direction have been done in the last years through the work of many authors, but this (a priori small) case has proved to be very challenging, and we are still very far from a classification of it. At the same time, this class of surfaces, and in particular the subclass of the surfaces with p_g=0 contains some of the most interesting surfaces of general type.
If S is a surface of general type with \chi=1, which means p_g=q, then p_g = q < 5, and if p_g=q=4, then S is birational to the product of curves of genus 2.
The surfaces with p_g = q = 3 are completely classified.
The cases p_g = q < 3 are still far from being classified.
Generalizing the Beauville example, we can consider the quotient (C_1 x C_2)/G, where the C_i are Riemann surfaces of genus at least two, and G is a finite group.
There are two cases: the mixed case where the action of G exchanges the two factors (and then C_1 = C_2); and the unmixed case where G acts diagonally.
Many authors studied the surfaces birational to a quotient of a product of two curves, mainly in the case of surfaces of general type with \chi=1.
In all these works the authors work either in the unmixed case or in the mixed case under the assumption that the group acts freely.
The main purpose of this thesis is to extend the results and the strategies of the above mentioned papers in the non free mixed case.
Let C be a Riemann surface of genus at least 2, let G be a finite group that acts on C x C with a mixed action, i.e. there exists an element in G that exchanges the two factors. Let G^0 be the index two subgroup of the elements that do not exchange the factors. We say that X=(C x C)/G is a mixed quasi-étale surface if the quotient map C x C -> (C x C)/G has finite branch locus.
We present an algorithm to construct regular surfaces as the minimal resolution of the singularities of mixed quasi-étale surfaces.
We give a complete classification of the regular surfaces with p_g=0 and K^2>0 that arise in this way.
Moreover we show a way to compute the fundamental group of these surfaces and we apply it to the surfaces we construct.
Some of our construction are more interesting than others. We have constructed two numerical Campedelli surfaces (K^2 = 2) with topological fundamental group Z/4Z. Two of our constructions realize surfaces whose topological type was not present in the literature before.
We also have three examples of Q-homology projective planes, and two of them realize new examples of Q-homology projective planes.
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Altavilla, Amedeo. "Quaternionic slice regular functions on domains without real points." Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1089/2/Thesis.pdf.
Full textAbstract:
In this thesis I've explored the theory of quaternionic slice regular functions. More precisely I've studied some rigidity properties, some differential issues and an application in
complex differential geometry.
This application concerns the constructions and classifications of orthogonal complex structures on open domains of the four dimensional euclidean space.
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Tasin, Luca. "Birational Maps in the Minimal Model Program." Doctoral thesis, University of Trento, 2013. http://eprints-phd.biblio.unitn.it/1110/1/Thesis.pdf.
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Pernigotti, Letizia. "Geometry of moduli spaces of higher spin curves." Doctoral thesis, University of Trento, 2013. http://eprints-phd.biblio.unitn.it/1141/1/TesiDottoratoPernigotti.pdf.
Full textAbstract:
ABSTRACT: Roughly speaking, the moduli space of higher spin curves parametrizes equivalence classes of pairs (C, L) where C is a smooth genus g algebraic curve and L is a line bundle on it whose r-th tensor power is isomorphic to the canonical bundle of the curve. The aim of the talk is to discuss important geometrical properties of these spaces under different points of view: one possible compactification together with the description of the rational Picard group, their birational geometry in some low genus cases and their relation with some special locus inside the classical moduli spaces of curves.
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Platoni, Irene. "Complete Arcs and Caps in Galois Spaces." Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1364/1/tesi.pdf.
Full textAbstract:
Galois spaces, that is affine and projective spaces of dimension N ≥ 2 defined over a finite (Galois) field F_q, are well known to be rich of nice geometric, combinatorial
and group theoretic properties that have also found wide and relevant applications in several branches of Combinatorics, as well as in more practical areas, notably Coding Theory and Cryptography. The systematic study of Galois spaces was initiated in the late 1950’s by the pioneering work of B. Segre [59]. The trilogy [34, 36, 42] covers the general theory of Galois spaces including the study of objects which are linked to linear codes. Typical such objects are plane arcs and their generalizations - especially caps and arcs in higher dimensions - whose code theoretic counterparts are distinguished types of error-correcting and covering linear codes. Their investigation has received a great stimulus from Coding Theory, especially in the last decades; see the survey papers [40, 41]. An important issue in this context is to ask for explicit constructions of small complete arcs and small complete caps. A cap in a Galois space is a set of points no three of which are collinear. A cap is complete if its secants (lines through two points of the set) cover the whole space. An arc in a Galois space of dimension N is a set of points no N+1 of which lying on the same hyperplane. In analogy with caps, an arc which is maximal with respect to set-theoretical inclusion is said to be complete.
Also, arcs coincide with caps in Galois planes. From these geometrical objects, there arise linear codes which turn out to have very good covering properties, provided
that the size of the set is small with respect to the dimension N and the order q of the ambient space.
For the size t(AG(N,q)) of the smallest complete caps in a Galois affine space AG(N,q) of dimension N over F_q, the trivial lower bound is √2q^{N−1/2}. General constructions of complete caps whose size is close to this lower bound are only known for q even and N odd, see [16, 19, 29, 52]. When N is even, complete caps of size of the same order of magnitude as cq^{N/2}, with c a constant independent of q, are known to exist for both the odd and the even order case, see [16, 18, 28, 29, 31] (see also Section 2.2 and the references therein). Whereas, few constructions of small complete arcs in Galois spaces of dimension N>2 are known. In [65, 66, 67], small complete arcs having many points in common with the normal rational curve are investigated (see Section 4.2.3 for comparisons with our results). In this thesis, new infinite families of complete arcs and caps in higher dimensional spaces are constructed from algebraic curves defined over a finite field. In most
cases, no smallest complete caps/arcs were previously known in the literature. Although caps and arcs are rather combinatorial objects, constructions and proofs sometimes heavily rely on concepts and results from Algebraic Geometry in positive characteristic.
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Cancian, Nicola. "On Semi-isogenous Mixed Surfaces." Doctoral thesis, University of Trento, 2017. http://eprints-phd.biblio.unitn.it/2605/2/tesi.pdf.
Full textAbstract:
Let C be a compact Riemann surface. Let us consider a finite group acting on CxC, having some elements that exchange the factors, and assume that the subgroup of those elements that do not exchange the factors acts freely. We call the quotient a Semi-isogenous Mixed Surface. In this work we investigate these surfaces and we explain how their geometry is encoded in the group. Based on this, we present an algorithm to classify the Semi-isogenous Mixed Surfaces with given geometric genus, irregularity and self-intersection of the canonical class. In particular we give the classification of Semi-isogenous Mixed Surfaces with K^2>0 and holomorphic Euler-Poincaré characteristic equal to 1, where new examples of minimal surfaces of general type appear. Minimality of Semi-isogenous Mixed Surfaces is discussed using two different approaches. The first one involves the study of the bicanonical system of such surfaces: we prove that we can relate the dimension of its first cohomology group to the rank of a linear map that involves only curves. The second approach exploits Hodge index theorem to bound the number of exceptional curves that live on a Semi-isogenous Mixed Surface.
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Tironi, A. L. "Smooth complex projective manifolds with reducible hyperplane sections of special type." Doctoral thesis, Università degli Studi di Milano, 2006. http://hdl.handle.net/2434/56513.
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Dalvit, Ester. "New proposals for the popularization of braid theory." Doctoral thesis, Università degli studi di Trento, 2011. https://hdl.handle.net/11572/368720.
Full textAbstract:
Braid theory is a very active research field. Braids can be studied from variuos points of view and have applications in different fields, e.g. in mathematical physics and in biology.
In this thesis we provide a formal introduction to some topics in the mathematical theory of braids and two possible approaches to this field at a popular level: a movie and a workshop.
The scientific movie addressed to a non-specialist audience has been realized using the free ray-tracer POV-Ray. It is divided into four parts, each of which has a length of about 15 minutes. The content ranges from the introduction of basic concepts to deep results.
The workshop activity is based on the action of braids on loops and aims to invite and lead the audience to a mathematical formalization of the principal concepts involved: braids, curves and group actions.
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GALATI, CONCETTINA. "Number of moduli of families of plane curves with nodes and cusps." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2006. http://hdl.handle.net/2108/210.
Full textAbstract:
In my Ph.D.-thesis I computed the number of moduli of certain families of plane curves with nodes and cusps. Let Σn k,d ⊂ P(H0(P2,OP2(n))) := PN, with N = n(n+3)2 , be the closure, in the Zariski’s topology, of the locally closed set of reduced and
irreducible plane curves of degree n with k cusps and d nodes. We recall that, if k = 0,
the varieties Vn,g = Σn0,d are called the Severi varieties of irreducible plane curves of degree n and geometric genus g = n−1
2 − d. Let Σ⊂ Σn
k,d be an irreducible component of Σn
k,d and let g = n−1
2 −d−k be the geometric genus of the plane curve corresponding to the general point of Σ. It is naturally defined a rational map
ΠΣ : Σ Mg,
sending the general point [Γ] ∈ Σ to the isomorphism class of the normalization of the
plane curve Γ corresponding to the point [Γ]. We set number of moduli of Σ := dim(ΠΣ(Σ)).
If k < 3n, then
(1) dim(ΠΣ(Σ)) ≤ min(dim(Mg), dim(Mg) + ρ − k),
where ρ := ρ(2, g, n) = 3n − 2g − 6 is the Brill-Neother number of the linear series of
degree n and dimension 2 on a smooth curve of genus g. We say that Σ has the expected number of moduli if the equality holds in (1). By classical Brill-Neother theory when ρ is positive and by a well know result of Sernesi when ρ ≤ 0, we have that Σn0,d, (which is irreducible), has the expected number of moduli for every d ≤ n−1
2 . Working out the main ideas and techniques that Sernesi uses in [1], under the hypothesis k > 0, in my Ph.D.-thesis I find sufficient conditions in order that an irreducible component Σ ⊂ Σn
k,d has the expected number of moduli. If Σ verifies these properties, then ρ ≤ 0. By using
induction on the degree n and on the genus g of the general curve of the family, I prove
that, if ρ ≤ 0 and k ≤ 6, then there exists at least one irreducible component of Σn
k,d
with expected number of moduli equal to 3g−3+ρ−k. By using this result and a result of Eisembud and Harris, from which it follows that, if ρ is positive enough and k ≤ 3
then dim(ΠΣ(Σ)) = 3g − 3, I prove that Σn1,d (which is irreducible) has the expected number of moduli for every d ≤ n−1
2 , i.e. for every ρ. I am extending this result to the case k ≤ 3. Finally, I consider the case of irreducible sextics with six cusps. It is
classically know that Σ66,0 contains at least two irreducible components Σ1 and Σ2. The general point of Σ1 parametrizes a sextic with six cusps on a conic, whereas the general element of Σ2 corresponds to a sextic with six cusps not on a conic. I prove that Σ1 and Σ2 have expected number of moduli. I don’t still know example of irreducible complete families of plane curves with nodes and cusps having number of moduli smaller that the expected. Finally, in the first sections of my thesis, following essentially Zariski’s papers, I introduce classical techniques used to study and describe the geometry of a family of
plane curves with assigned singularities. Then, I briefly resume the more modern results by Wahl on families of plane curves with nodes and cusps. I also give some applications of Horikawa deformation theory to the study of deformations of plane curves. Finally, I devoted a section of my thesis to the versal deformation family of plane curve singularity.
In particular, by using the results of [3] and [2] and a simple argument of projective geometry, I proved that in the equigeneric locus of the ´etale versal deformation space B of an ordinary plane curve singularity there are only points corresponding to a plane curve with only ordinary multiple points. I mean that this result is known, but I haven’t found in literature a proof of this.
References
[1] E. Sernesi:On the existence of certain families of curves, Invent. math. vol. 75, (1984).
[2] A. Morelli:Un’osservazione sulle singolarita’ delle trasformate birazionali di una curva algebrica,
Rend. Acc. Sci. Napoli, serie 4 vol. 29 (1962), p.59-64.
[3] A. Franchetta: Osservazioni sui punti doppi delle superfici algebriche, Rend. Acc. dei Lincei, gennaio 1946.
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ALTOMANI, ANDREA. "Orbits of real forms in complex flag manifolds." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2008. http://hdl.handle.net/2108/722.
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FLAMINI, FLAMINIO. "Families of nodal curves on smooth projective surfaces." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2000. http://hdl.handle.net/2108/74808.
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DI, NEZZA ELEONORA. "Geometry of complex Monge-Ampère equations on compact Kähler manifolds." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2014. http://hdl.handle.net/2108/202161.
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EL, EMAM CHRISTIAN. "Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space." Doctoral thesis, Università degli studi di Pavia, 2020. http://hdl.handle.net/11571/1361034.
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SPELTA, IRENE. "On Shimura Subvarieties of the Torelli Locus and Ramified Prym Maps." Doctoral thesis, Università degli studi di Pavia, 2020. http://hdl.handle.net/11571/1367134.
Full textAbstract:
The purpose of the thesis is two-fold: to investigate the existence of totally geodesic subvarieties of the moduli space of principally polarized abelian
varieties, Ag, contained in the Jacobian locus and to study the geometry of certain positive dimensional fibres of some ramified Prym maps. Totally
geodesic subvarieties constitute a useful tool to study the extrinsic geometry of the Jacobian locus inside Ag and they are involved in the rather famous
Coleman-Oort conjecture. Furthermore, they motivate our interest in Prym maps. Indeed it turns out that certain positive dimensional fibres represent
a good place to look for totally geodesic subvarieties.
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TAMBASCO, SALVATORE. "On the volume of Fano K-moduli spaces." Doctoral thesis, Università degli studi di Pavia, 2020. http://hdl.handle.net/11571/1371999.
Full textAbstract:
In questa tesi abbiamo calcolato il volume CM, ossia il grado del fibrato in rette CM disceso sul K-spazio dei moduli Fano, delle del Pezzo di grado quattro per qualunque dimensione. Inoltre, abbiamo calcolato il volume CM del K-spazio dei moduli degli arrangiamenti in iperpiani in dimensione 1 e 2. Infine, abbiamo correlato tali volumi al volume Weil-Petersson estendendo la nozione di metrica di Weil-Petersson anche nel caso log.In this thesis we compute the CM volume, that is the degree of the descended CM line bundle, of the Fano K-moduli space of Quartic del Pezzo in any dimension, and of the K-moduli space of the log Fano hyperplane arrangements of dimension one and two. Furthermore, we relate these volumes to the Weil-Petersson volumes by extending the notion of Weil-Petersson metric in the log case
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NOCERA, Guglielmo. "A study of the spherical Hecke category via derived algebraic geometry." Doctoral thesis, Scuola Normale Superiore, 2022. https://hdl.handle.net/11384/125742.
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Monge, Maurizio. "A constructive theory for extensions of p-adic fields." Doctoral thesis, Scuola Normale Superiore, 2012. http://hdl.handle.net/11384/85667.
Full textAbstract:
The subject of this thesis in the study of nite extensions of p-adic fields, in different aspects. Via the study of the Galois module of p-th power classes
L =(L )p of a general Galois extension L=K of degree p, it is possible to deduce and classify the extensions of degree p2 of a p-adic field. We exhibit formulae
counting how many times a certain group appears as Galois group of the normal
closure, generalizing previous results.
In general degree we give a synthetic formula counting isomorphism classes
of extensions of fixed degree. The formula is obtained via Krasner formula and
a simple group-theoretic Lemma allowing to reduce the problem to counting
cyclic extensions, which can be done easily via local class eld theory.
When K is an unrami ed extension of Qp we study the problem of giving
necessary and su cient conditions on the coe cients of an Eisenstein polynomial
for it to have a prescribed group as Galois group of the splitting field. The
techniques introduced allow to recover very easily Lbekkouri's result on cyclic
extensions of degree p2, and to give a complete description of the Galois group,
with its rami cation filtration, for splitting fields of Eisenstein polynomials of
degree p2 which are a general p-extension. We then show how the same methods
can be used to characterize Eisenstein polynomials defining a cyclic extension
of degree p3.
We then study Eisenstein polynomials in general, describing a family of
special reduced polynomials which provide almost unique generators of totally
rami ed extensions, and a reduction algorithm. The number of special polynomials
generating a fixed extension L=K is always smaller than the number of
conjugates of L over K, so that each Galois extension is generated by exactly
one special polynomial. We give an algorithm to recover all special polynomials
generating one extension, and a criterion that allows to detect when the extension
generated by an Eisenstein polynomial is different from a fixed extension
whose special generators are all given, the criterion does not only depend only
on the usual distance on the set of Eisenstein polynomials defined by Krasner
and others. An algorithm to construct the special polynomial generating an
abelian class eld is given, provided a suitable description of a candidate norm
subgroup of K x.
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Ruggiero, Matteo. "The valuative tree, rigid germs and Kato varieties." Doctoral thesis, Scuola Normale Superiore, 2011. http://hdl.handle.net/11384/85671.
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CARACENI, ALESSANDRA. "The geometry of large outerplanar and half-planar maps." Doctoral thesis, Scuola Normale Superiore, 2015. http://hdl.handle.net/11384/109626.
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MANCA, BENEDETTO. "DPW potentials for compact symmetric CMC surfaces in the 3-sphere." Doctoral thesis, Università degli Studi di Cagliari, 2019. http://hdl.handle.net/11584/271388.
Full textAbstract:
Minimal and CMC immersions of a compact surface M in the 3-sphere can be studied via their associated family of flat SL(2, C)-connections on a rank 2 holomorphic vector bundle E over M. However, for surfaces of genus greater than 2 it is a difficult task to describe family of holomorphic flat connections. It is easier to consider a related family of meromorphic flat connections and then reconstruct the associated family of the immersion from it.
The aim of this thesis is to show that it is possible to define a family of meromorphic flat connections on a class of CMC surfaces in S3, from which it is possible to reconstruct the immersion.
We consider CMC surfaces M in the 3-sphere having a group of symmetries which is finite and such that the quotient of the surface by the group is the Riemann sphere.
We show that the surfaces constructed by Lawson in 1970 and by Karcher, Pinkall and Sterling in 1988, belong to this class of surfaces.
We define a holomorphic vector bundle V over the Riemann sphere, equipped with a parabolic structure. We consider a family of logarithmic flat connections on V and we show that such family of logarithmic connections has a prescribed asymptotic.
The main theorem of the thesis shows that the family of logarithmic connections on V can be used to define a DPW potential on the CMC surface satisfying the necessary properties to reconstruct the immersion of the surface M into the 3-sphere via loop group factorisation.
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CANNAS, AGHEDU FRANCESCO. "Quantizations of Kähler metrics on blow-ups." Doctoral thesis, Università degli Studi di Cagliari, 2021. http://hdl.handle.net/11584/309588.
Full textAbstract:
The thesis consists of three main results related to Kähler metrics on blow-ups. In the first one, we prove that the blow-up C ̃^2 of C^2 at the origin endowed with the Burns–Simanca metric g_BS admits a regular quantization. We use this fact to prove that all coefficients in the Tian-Yau-Catlin-Zelditch expansion for the Burns–Simanca metric vanish and that a dense subset of (C ̃^2,g_BS) admits a Berezin quantization. In the second one, we prove that the generalized Simanca metric on the blow-up C ̃^n of C^n at the origin is projectively induced but not balanced for any integer n>=3. Finally, we prove as third result that any positive integer multiple of the Eguchi–Hanson metric, defined on a dense subset of C ̃^2/Z_2, is not balanced.