Dissertations / Theses on the topic 'Kähler metric'

This thesis deals with two different subjects: balanced metrics on complex vector bundles and the diastatic exponential of a symmetric space. Correspondingly we have two main results. In the first one we prove that if a holomorphic vector bundle E over a compact Kähler manifold (M,ω) admits a ω-balanced metric then this metric is unique. In the second one, after defining the diastatic exponential of a real analytic Kähler manifold, we prove that for every point p of an Hermitian symmetric space of noncompact type there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks.

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.

The present thesis consists of three results related to the geometry of rotation invariant Kähler metrics. In the first one, we prove that a 3-codimensional Kähler-Einstein submanifold of the complex projective space with rotation invariant metric is forced to be the product of complex projective spaces. In the second one, we prove that the only stable-projectively induced Ricci-flat Kähler metrics are flat. Finally, we prove as third result that given a Ricciflat radial Kähler metric defined on a complex surface such that the third coefficient of its Tian-Yau-Zelditch expansion vanishes, then it is flat.

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.

Dans cette thèse on s’intéresse à deux types de structures conformes non-dégénérées sur une variété complexe compacte donnée. La première c’est une forme holomorphe symplectique twistée (THS), i.e. une deux-forme holomorphe non-dégénérée à valeurs dans un fibré en droites. Dans le deuxième contexte, il s’agit des métriques localement conformément kähleriennes (LCK). Dans la première partie, on se place sur un variété de type Kähler. Les formes THS généralisent les formes holomorphes symplectiques, dont l’existence équivaut à ce que la variété admet une structure hyperkählerienne, par un théorème de Beauville. On montre un résultat similaire dans le cas twisté, plus précisément: une variété compacte de type kählerien qui admet une structure THS est un quotient fini cyclique d’une variété hyperkählerienne. De plus, on étudie sous quelles conditions une variété localement hyperkählerienne admet une structure THS. Dans la deuxième partie, les variétés sont supposées de type non-kählerien. Nous présentons quelques critères pour l’existence ou non-existence de métriques LCK spéciales, en terme du groupe de biholomorphismes de la variété. En outre, on étudie le problème d’irréductibilité analytique des variétés LCK, ainsi que l’irréductibilité de la connexion de Weyl associée. Dans un troisième temps, nous étudions les variétés LCK toriques, qui peuvent être définies en analogie avec les variétés de Kähler toriques. Nous montrons qu’une variété LCK torique compacte admet une métrique de Vaisman torique, ce qui mène à une classification de ces variétés par le travail de Lerman. Dans la dernière partie, on s’intéresse aux propriétés cohomologiques des variétés d’Oeljeklaus-Toma (OT). Plus précisément, nous calculons leur cohomologie de de Rham et celle twistée. De plus, on démontre qu’il existe au plus une classe de de Rham qui représente la forme de Lee d’une métrique LCK sur un variété OT. Finalement, on détermine toutes les classes de cohomologie twistée des métriques LCK sur ces variétés
In this thesis, we are concerned with two types of non-degenerate conformal structures on a given compact complex manifold. The first structure we are interested in is a twisted holomorphic symplectic (THS) form, i.e. a holomorphic non-degenerate two-form valued in a line bundle. In the second context, we study locally conformally Kähler (LCK) metrics. In the first part, we deal with manifolds of Kähler type. THS forms generalise the well-known holomorphic symplectic forms, the existence of which is equivalent to the manifold admitting a hyperkähler structure, by a theorem of Beauville. We show a similar result in the twisted case, namely: a compact manifold of Kähler type admitting a THS structure is a finite cyclic quotient of a hyperkähler manifold. Moreover, we study under which conditions a locally hyperkähler manifold admits a THS structure. In the second part, manifolds are supposed to be of non-Kähler type. We present a few criteria for the existence or non-existence for special LCK metrics, in terms of the group of biholomorphisms of the manifold. Moreover, we investigate the analytic irreducibility issue for LCK manifolds, as well as the irreducibility of the associated Weyl connection. Thirdly, we study toric LCK manifolds, which can be defined in analogy with toric Kähler manifolds. We show that a compact toric LCK manifold always admits a toric Vaisman metric, which leads to a classification of such manifolds by the work of Lerman. In the last part, we study the cohomological properties of Oeljeklaus-Toma (OT) manifolds. Namely, we compute their de Rham and twisted cohomology. Moreover, we prove that there exists at most one de Rham class which represents the Lee form of an LCK metric on an OT manifold. Finally, we determine all the twisted cohomology classes of LCK metrics on these manifolds

In this thesis we study the relationship between the existence of extremal Kähler metrics and stability. We introduce a space of symplectic potentials for toric manifolds, which we show gives metrics with mixed Poincaré type and cone angle singularities. We show uniqueness and that existence implies stability for extremal metrics arising from these potentials. For quadrilaterals, we give a computable criterion for stability in certain cases, giving a definite log-stable region for generic quadrilaterals. We use computational tools to find new examples of stable and unstable toric manifolds. For Poincaré type manifolds with an S1-action, we prove a version of the LeBrun-Simanca openness theorem and Arezzo-Pacard blow-up theorem.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.
Includes bibliographical references (p. 185-200).
This Thesis is concerned with the study of the geometry and structure of the space of Kihler metrics representing a fixed cohomology class on a compact Kähler manifold. The first part of the Thesis is concerned with a problem of geometric quantization: Can the geometry of the infinite-dimensional space of Kähler metrics be approximated in terms of the geometry of the finite-dimensional spaces of FubiniStudy Bergman metrics sitting inside it? We restrict to toric varieties and prove the following result: Given a compact Riemannian manifold with boundary and a smooth map from its boundary into the space of toric Kähler metrics there exists a harmonic map from the manifold with these boundary values and, up to the first two derivatives, it is the limit of harmonic maps from the Riemannian manifold into the spaces of Bergman metrics. This generalizes previous work of Song-Zelditch on geodesics in the space of toric Kähler metrics. In the second part of the Thesis we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamical systems on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow as well as another flow on the space of Kähler metrics. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. As an application, we address several questions in Kähler geometry related to canonical metrics, energy functionals, the Moser-Trudinger-Onofri inequality, Nadel-type multiplier ideal sheaves, and the structure of the space of Kähler metrics.
by Yanir Akiva Rubinstein.
Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.
Includes bibliographical references (p. 61-64).
In this thesis, we derive the asymptotic expansion of the Kiihler-Einstein metrics on certain quasi-projective varieties, which can be compactified by adding a divisor with simple normal crossings. The weighted Cheng-Yau Hilder spaces and the log-filtrations based on the bounded geometry are introduced to characterize the asymptotics. We first develop the analysis of the Monge-Ampere operators on these weighted spaces. We construct a family of linear elliptic operators which can be viewed as certain conjugacies of the specially linearized Monge-Ampbre operators. We derive a theorem of Fredholm alternative for such elliptic operators by the Schauder theory and Yau's generalized maximum principle. Together these results derive the isomorphism theorems of the Monge-Ampbre operators, which imply that the Monge-Ampere operators preserve the log-filtration of the Cheng-Yau Holder ring. Next, by choosing a canonical metric on the submanifold, we construct an initial Kidhler metric on the quasi-projective manifold such that the unique solution of the Monge-Ampere equation belongs to the weighted -1 Cheng-Yau Hölder ring. Moreover, we generalize the Fefferman's operator to act on the volume forms and obtain an iteration formula.
(cont.) Finally, with the aid of the isomorphism theorems and the iteration formula we derive the desired asymptotics from the initial metric. Furthermore, we prove that the obtained asymptotics is canonical in the sense that it is independent of the extensions of the canonical metric on the submanifold.
by Damin Wu.
Ph.D.

Dans la veine du programme d'Erlangen de Klein, travaux d'E. Cartan, M. Gromov, et d'autres, ce travail se trouve à cheval, entre la géométrie et les actions de groupes. Le thème global serait de comprendre les groupes d'isométries des variétés pseudo-riemanniennes. Plus précisément, suivant une "conjecture vague" de Gromov, classifier les variétés pseudo-riemanniennes dont le groupe d'isométries agit non-proprement, i.e. que son action ne préserve pas de métrique riemannienne auxiliaire?Plusieurs travaux ont été accomplis dans le cas des métriques lorentziennes (i.e. de signature (- +...+)). En revanche, le cas pseudo-riemannien général semble hors de portée.Les structures Hermite-Lorentz se trouvent entre le cas lorentzien et le premier cas pseudo-riemannien général, i.e. de signature (- - +…+). De plus, elle se définit sur des variétés complexes, et promet une extra-rigidité. Plus précisément, une structure Hermite-Lorentz sur une variété complexe consiste en une métrique pseudo-riemannienne de signature (- - +…+) qui est hermitienne au sens qu'elle est invariante par la structure presque complexe. Par analogie au cas hermitien classique, on définit naturellement une notion de métrique Kähler-Lorentz.Comme exemple, on a l'espace de Minkowski complexe ; dans un certain sens, on a un temps de dimension 1 complexe (du point de vue réel, le temps est 2-dimensionnel). On a également l'espace de Sitter et anti de Sitter complexes. Ils ont une courbure holomorphe constante, et généralisent dans ce sens les espaces projectifs et hyperboliques complexes.Cette thèse porte sur les variétés Hermite-Lorentz homogènes. En plus des exemples cités, il y a deux autres espaces symétriques, qui peuvent naturellement jouer le rôle de complexification des espaces de Sitter et anti de Sitter réels.Le résultat principal de la thèse est un théorème de rigidité de ces espaces symétriques : tout espace Hermite-Lorentz homogène à isotropie irréductible est l'un des cinq espaces symétriques précédents. D'autres résultats concernent le cas où l'on remplace l'hypothèse d'irréductibilité par le fait que le groupe d'isométries soit semi-simple
In the vein of Klein's Erlangen program, the research works of E. Cartan, M.Gromov and others, this work straddles between geometry and group actions. The overall theme is to understand the isometry groups of pseudo-Riemannian manifolds. Precisely, following a "vague conjecture" of Gromov, our aim is to classify Pseudo-Riemannian manifolds whose isometry group act’s not properly, i.e that it’s action does not preserve any auxiliary Riemannian metric. Several studies have been made in the case of the Lorentzian metrics (i.e of signature (- + .. +)). However, general pseudo-Riemannian case seems out of reach. The Hermite-Lorentz structures are between the Lorentzian case and the former general pseudo-Riemannian, i.e of signature (- -+ ... +). In addition, it’s defined on complex manifolds, and promises an extra-rigidity. More specifically, a Hermite-Lorentz structure on a complex manifold is a pseudo-Riemannian metric of signature (- -+ ... +), which is Hermitian in the sense that it’s invariant under the almost complex structure. By analogy with the classical Hermitian case, we naturally define a notion of Kähler-Lorentz metric. We cite as example the complex Minkowski space in where, in a sense, we have a one-dimensional complex time (the real point of view, the time is two-dimensional). We cite also the de Sitter and Anti de Sitter complex spaces. They have a constant holomorphic curvature, and generalize in this direction the projective and complex hyperbolic spaces.This thesis focuses on the Hermite-Lorentz homogeneous spaces. In addition with given examples, two other symmetric spaces can naturally play the role of complexification of the de Sitter and anti de Sitter real spaces.The main result of the thesis is a rigidity theorem of these symmetric spaces: any space Hermite-Lorentz isotropy irreducible homogeneous is one of the five previous symmetric spaces. Other results concern the case where we replace the irreducible hypothesis by the fact that the isometry group is semisimple

Cette thèse traite des solitons de Kähler-Ricci qui sont des généralisations naturelles des métriques de Kähler-Einstein. Elle est divisée en deux parties. La première étudie la décomposition solitonique de l’espace des champs de vecteurs holomorphes dans le cas des variétés toriques. La seconde partie étudie de manière analytique les variétés horosphériques en redémontrant par la méthode de la continuité l’existence de solitons de Kähler-Ricci sur ces variétés et en calculant après la borne supérieure de Ricci
This thesis deal with Kähler-Ricci solitons which are natural generalizations of Kähler-Einstein metrics. It is divided into two parts. The first one studies the solitonic decomposition of the space of holomorphic vector spaces in the case of toric manifold. The second one studies is an analytic way the existence of horospherical Kähler-Ricci solitons on those manifolds and then computes the greatest Ricci lower bound

Submitted by Erika Demachki (erikademachki@gmail.com) on 2017-03-17T21:10:56Z No. of bitstreams: 2 Dissertação - Laena Furtado Borges - 2017.pdf: 2090414 bytes, checksum: afc3416e502ab5aedc5390b7986a9fcf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5)
Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-20T13:53:29Z (GMT) No. of bitstreams: 2 Dissertação - Laena Furtado Borges - 2017.pdf: 2090414 bytes, checksum: afc3416e502ab5aedc5390b7986a9fcf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5)
Made available in DSpace on 2017-03-20T13:53:29Z (GMT). No. of bitstreams: 2 Dissertação - Laena Furtado Borges - 2017.pdf: 2090414 bytes, checksum: afc3416e502ab5aedc5390b7986a9fcf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-03
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this work, we will present some concepts of quasi-Einstein metrics. From this, we will enunciate and demonstrate rigidity results for quasi-Einstein metrics until we have enough material to demonstrate a stiffness result for quasi-Einstein metrics of dimension two. Finally, we will give some concepts of Kähler metrics, prove a theorem and finally demonstrate a corollary that connects the main theorem of our work with Kähler metrics.
Nesse trabalho, apresentaremos alguns conceitos de métricas quasi-Einstein. A partir disso, enunciaremos e demonstraremos resultados de rigidez para métricas quasi-Einstein, até que tenhamos material suficiente para a demonstração de um resultado de rigidez para métricas quasi-Einstein em dimensão dois. Por fim, daremos alguns conceitos de métricas kähler, provaremos um teorema e por fim demonstraremos um corolário que conecta o teorema principal do nosso trabalho com as métricas Kähler.

Le résultat principal de cette thèse est l'obtention d'une condition nécessaire et suffisante pour l'existence d'une métrique de Kähler-Einstein sur une compactification bi-équivariante lisse et Fano d'un groupe complexe réductif connexe. Ces variétés comprennent les variétés toriques et les compactifications magnifiques de groupes semisimples adjoints.Dans la première partie de ce travail sont développés les outils nécessaires à l'étude de l'existence de métriques de Kähler-Einstein sur ces variétés. Nous calculons en particulier la Hessienne complexe d'une fonction $Ktimes K$-invariante sur la complexification d'un groupe compact $K$. Nous associonségalement, à toute métrique invariante à courbure positive sur un fibré linéarisé ample sur une compactification de groupe, une fonction convexe dont le comportement asymptotique est prescrit. Ceci est utilisé une première fois pour obtenir une formule pour l'invariant alpha d'un fibré en droite ample sur une compactification de groupe Fano. Cette formule est obtenue par le calcul des seuils log canoniques des métriques hermitiennes invariantes à courbure positive, et induit, dans le cas particulier des variétés toriques, un résultat obtenu auparavant, figurant dans l'article par ailleurs inclus en appendice de la thèse.Nous prouvons ensuite le résultat principal en obtenant des estimées $C^0$ le long de la méthode de continuité, en se ramenant à une équation de Monge-Ampèreréelle sur un cône. La condition obtenue est que le barycentre du polytope associé à la compactification de groupe, par rapport à la mesure de Duistermaat-Heckman, doit être dans une zone particulière du polytope. Cette condition peut être vérifiée sur les exemples, donne de nouveaux exemples de variétés deKähler-Einstein Fano, et donne aussi un exemple qui n'admet aucun soliton de Kähler-Ricci. Nous calculons de plus la plus grande borne inférieure de Ricci lorsqu'il n'y a pas de métrique de Kähler-Einstein
The main result of this work is a necessary and sufficient condition for the existence of a Kähler-Einstein metric on a smooth and Fano bi-equivariant compactification of a complex connected reductive group. Examples of such varieties include wonderful compactifications of adjoint semisimple groups.The tools needed to study the existence of Kähler-Einstein metrics on these varieties are developed in the first part of the work, including a computation of the complex Hessian of a $Ktimes K$-invariant function on the complexification of a compact group $K$. Another step is to associate to any non-negatively curved invariant hermitian metric on an ample linearized line bundle on a group compactification a convex function with prescribed asymptotic behavior. This is used a first time to derive a formula for the alpha invariantof an ample line bundle on a Fano group compactification. This formula is obtained through the computation of the log canonical thresholds of any non-negatively curved invariant hermitian metric, and gives the sameresult, for toric manifolds, as the one we obtained before, in an article that is included in this thesis as an appendix.Then we prove the main result by obtaining $C^0$ estimates along the continuity method, using the tools developed to reduce to a real Monge-Ampère equation on a cone. The condition obtained is that the barycenter of the polytope associated to the group compactification, with respect to the Duistermaat-Heckman measure, lies in a certain zone in the polytope. This condition can be checked on examples, gives new examples of Fano Kähler-Einstein manifolds, and also gives an example that admits no Kähler-Ricci solitons. We also compute the greatest Ricci lower bound when there are no Kähler-Einstein metrics

Dans cette thèse nous étudions des questions de stabilité géométrique pour des variétés kähleriennes à courbure scalaire constante (cscK) avec classe de cohomologie transcendante. En tant que point de départ, nous introduisons des notions généralisées de K-stabilité, étendant une image classique introduite par G. Tian et S. Donaldson dans le cadre des variétés polarisées. Contrairement à la théorie classique, ce formalisme nous permet de traiter des questions de stabilité pour des variétés kähleriennes compactes non projectives ainsi que des variétés projectives munis de polarisations non rationnelles. Dans une première partie, nous étudions les rayons sous-géodésiques associés aux configurations tests dites cohomologiques, objets introduitent dans cette thèse. Nous établissons ainsi des formules fondamentales pour la pente asymptotique d'une famille de fonctionnelles d'énergie, le long de ces rayons géodésiques. Ceci est lié au couplage de Deligne en géométrie algébrique, et ce formalise permet en particulier de comprendre le comportement asymptotique d'un grand nombre de fonctionnelles d'énergie classiques en géométrie kählerienne, y compris la fonctionnelle d'Aubin-Mabuchi et la K-énergie. En particulier, ceci fournit une approche pluripotentielle naturelle pour étudier le comportement asymptotique des fonctionnelles d'énergie dans la théorie de K-stabilité. En s'appuyant sur cette première partie, nous démontrons ensuite un certain nombre de résultats de stabilité pour les variétés cscK. Tout d'abord, nous prouvons que les variétés cscK sont K-semistables dans notre sens généralisé, prolongeant ainsi un résultat dû à Donaldson dans le cadre projectif. En supposant que le groupe d'automorphisme est discret, nous montrons en outre que la K-stabilité est une condition nécessaire pour l'existence des métriques cscK sur des variétés kähleriennes compactes. Plus précisément, nous prouvons que la coercivité de la K-énergie implique la K-stabilité uniforme, ainsi généralisant des résultats de Mabuchi, Stoppa, Berman, Dervan et Boucksom-Hisamoto-Jonsson pour des variétés polarisées. Cela donne une preuve nouvelle et plus générale d'une direction de la conjecture Yau-Tian-Donaldson dans ce contexte. L'autre direction (suffisance de K-stabilité) est considérée comme l'un des problèmes ouverts les plus importants en géométrie kählerienne. Nous donnons enfin des résultats partiels dans le cas des variétés kähleriennes compactes qui admettent des champs de vecteurs holomorphes non triviaux. Nous discutons également autour des perspectives et applications de notre théorie de K-stabilité pour les variétés kähleriennes avec classe transcendante, notamment à l'étude des lieux de stabilité dans le cône de Kähler
In this thesis we are interested in questions of geometric stability for constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class. As a starting point we develop generalized notions of K-stability, extending a classical picture for polarized manifolds due to G. Tian, S. Donaldson, and others, to the setting of arbitrary compact Kähler manifolds. We refer to these notions as cohomological K-stability. By contrast to the classical theory, this formalism allows us to treat stability questions for non-projective compact Kähler manifolds as well as projective manifolds endowed with non-rational polarizations. As a first main result and a fundamental tool in this thesis, we study subgeodesic rays associated to test configurations in our generalized sense, and establish formulas for the asymptotic slope of a certain family of energy functionals along these rays. This is related to the Deligne pairing construction in algebraic geometry, and covers many of the classical energy functionals in Kähler geometry (including Aubin's J-functional and the Mabuchi K-energy functional). In particular, this yields a natural potential-theoretic aproach to energy functional asymptotics in the theory of K-stability. Building on this foundation we establish a number of stability results for cscK manifolds: First, we show that cscK manifolds are K-semistable in our generalized sense, extending a result due to S. Donaldson in the projective setting. Assuming that the automorphism group is discrete we further show that K-stability is a necessary condition for existence of constant scalar curvature Kähler metrics on compact Kähler manifolds. More precisely, we prove that coercivity of the Mabuchi functional implies uniform K-stability, generalizing results of T. Mabuchi, J. Stoppa, R. Berman, R. Dervan as well as S. Boucksom, T. Hisamoto and M. Jonsson for polarized manifolds. This gives a new and more general proof of one direction of the Yau-Tian-Donaldson conjecture in this setting. The other direction (sufficiency of K-stability) is considered to be one of the most important open problems in Kähler geometry. We finally give some partial results in the case of compact Kähler manifolds admitting non-trivial holomorphic vector fields, discuss some further perspectives and applications of the theory of K-stability for compact Kähler manifolds with transcendental cohomology class, and ask some questions related to stability loci in the Kähler cone

We analyse the holomorphic curvature of Kähler metrics on generalised flag manifolds with respect to the question of strict positivity. The main results are twofold: Firstly, we show that most generalised flag manifolds with second betti number smaller than 3 have positive holomorphic curvature for any Kähler metric. Secondly, using fairly different techniques we obtain that every generalised flag manifold of rank four or less has positive holomorphic curvature with respect to the Kähler-Einstein metric.

This thesis examines orbifold versions of three results concerning the existence of canonical metrics in the Kahler setting. The first of these is Yau's solution to Calabi's conjecture, which demonstrates the existence of a Kahler metric with prescribed Ricci form on a compact Kahler manifold. The second is a variant of Yau's solution in a certain non-compact setting, namely, the setting in which the Kahler manifold is assumed to be asymptotic to a cone. The final result is one due to Uhlenbeck and Yau which asserts the existence of Kahler-Einstein metrics on stable vector bundles over compact Kahler manifolds.

In this thesis we investigate two approaches to the problem of existence of metrics of constant scalar curvature in a fixed Kähler class. In the first part, we examine the equation for constant scalar curvature under the assumption of toric symmetry, thus reducing the problem to a fourth order nonlinear degenerate elliptic equation for a convex function defined in a polytope in ℝ^n. We obtain partial results on this equation using an associated Monge-Ampère equation to determine the boundary behavior of the solution. In the second part, we consider the asymptotics of certain energy functionals and their relation to stability and the existence of minimizers. We derive explicit formulas for their asymptotic slopes, which allows one to determine whether or not (X,L) is stable, and in some cases rule out the existence of a canonical metric.

This thesis consists of two parts. In the first part, we introduce coupled Kähler- Einstein and Hermitian-Yang-Mills equations. It is shown that these equations have an interpretation in terms of a moment map. We identify a Futaki-type invariant as an obstruction to the existence of solutions of these equations. We also prove a Matsushima- Lichnerowicz-type theorem as another obstruction. Using the Calabi ansatz, we produce nontrivial examples of solutions of these equations on some projective bundles. Another class of nontrivial examples is produced using deformation. In the second part, we prove a priori estimates for vortex-type equations. We then apply these a priori estimates in some situations. One important application is the existence and uniqueness result concerning solutions of the Calabi-Yang-Mills equations. We recover a priori estimates of the J-vortex equation and the Monge-Ampère vortex equation. We establish a corre- spondence result between Gieseker stability and the existence of almost Hermitian-Yang- Mills metric in a particular case. We also investigate the Kählerity of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations.

Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2017.
En esta tesis estudiamos las estructuras localmente conformes Kähler (LCK) y localmente conformes simplécticas (LCS) invariantes a izquierda en grupos de Lie, o equivalentemente tales estructuras en álgebras de Lie. Luego se buscan retículos (subgrupos discretos co-compactos) en dichos grupos. De esta manera obtenemos estructuras LCK o LCS en las solvariedades compactas (cociente de un grupo de Lie por un retículo). Específicamente estudiamos las estructuras LCK en solvariedades con estructuras complejas abelianas. Luego describimos explícitamente la estructura de las álgebras de Lie que admiten estructuras de Vaisman. También determinamos los grupos de Lie casi abelianos que admiten estructuras LCK o LCS y además analizamos la existencia de retículos en ellos. Finalmente desarrollamos un método para construir de manera sistemática ejemplos de álgebras de Lie equipadas con estructuras LCK o LCS a partir de un álgebra de Lie que ya admite tales estructuras y una representación compatible.
In this thesis we study left invariant locally conformal Kähler (LCK) structures and locally conformal symplectic structures (LCS) on Lie groups, or equivalently such structures on Lie algebras. Then we analize the existence of lattices (co-compact discrete subgroups) on these Lie groups. Therefore, we obtain LCK or LCS structures on compact solvmanifolds (quotients of a Lie group by a lattice). Specifically we study LCK structures on solvmanifold where the complex structure is abelian. Then we describe the structure of a Lie algebra admitting a Vaisman structure. On the other hand we determine the almost abelian Lie groups equipped with a LCK or LCS structures, and we also analize the existence of lattices on these groups. Finally we construct a method to produce examples of Lie algebras admitting LCK or LCS structures beginning with a Lie algebra with these structures and a compatible representation.