La géométrie symplectique et la géométrie complexe sont intimement liées, en particulier par les techniques asymptotiquement holomorphes de Donaldson et Auroux d'une part et par les travaux d’Eliashberget et Cieliebak sur la pseudoconvexité d'autre part. Les travaux présentés dans cette thèse sont motivés par ces deux liens. On donne d’abord la caractérisation symplectique suivante des constantes de Seshadri. Dans une variété complexe, la constante de Seshadri d’une classe de Kähler entière en un point est la borne supérieure des capacités de boules standard admettant, pour une certaine forme de Kähler dans cette classe, un plongement holomorphe et iso-Kähler de codimension 0 centré en ce point. Ce critère était connu de Eckl en 2014 ; on en donne une preuve différente. La deuxième partie est motivée par la question suivante de Donaldson : <<Toute sphère lagrangienne d'une variété projective complexe est-elle un cycle évanescent d'une déformation complexe vers une variété à singularité conique ?>> D'une part, on présente toute sous-variété lagrangienne close d’une variété symplectique/kählérienne close dont les périodes relatives sont entières comme lieu des minima d’une exhaustion <<convexe>> définie sur le complémentaire d'une section hyperplane symplectique/complexe. Dans le cadre kählérien, <<convexe>> signifie strictement plurisousharmonique tandis que dans le cadre symplectique, cela signifie de Lyapounov pour un champ de Liouville. D'autre part, on montre que toute sphère lagrangienne d'un domaine de Stein qui est le lieu des minima d’une fonction <<convexe>> est un cycle évanescent d'une déformation complexe sur le disque vers un domaine à singularité conique. / Symplectic geometry and complex geometry are closely related, in particular by Donaldson and Auroux’s asymptotically holomorphic techniques and by Eliashberg and Cieliebak’s work on pseudoconvexity. The work presented in this thesis is motivated by these two connections. We first give the following symplectic characterisation of Seshadri constants. In a complex manifold, the Seshadri constant of an integral Kähler class at a point is the upper bound on the capacities of standard balls admitting, for some Kähler form in this class, a codimension 0 holomorphic and iso-Kähler embedding centered at this point. This criterion was known by Eckl in 2014; we give a different proof of it. The second part is motivated by Donaldon’s following question: ‘Is every Lagrangian sphere of a complex projective manifold a vanishing cycle of a complex deformation to a variety with a conical singularity?’ On the one hand, we present every closed Lagrangian submanifold of a closed symplectic/Kähler manifold whose relative periods are integers as the lowest level set of a ‘convex’ exhaustion defined on the complement of a symplectic/complex hyperplane section. In the Kähler setting ‘complex’ means strictly plurisubharmonic while in the symplectic setting it refers to the existence of a Liouville pseudogradient. On the other hand, we prove that any Lagrangian sphere of a Stein domain which is the lowest level-set of a ‘convex’ function is a vanishing cycle of some complex deformation over the disc to a variety with a conical singularity.

<p>For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.</p>

For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.

Some problems concerning holomorphic continuation of the class of bounded holo­morphic functions from bounded domains in Cn that are domains of holomorphy are solved. A bounded domain of holomorphy Ω in C2 with nonschlicht H°°-envelope of holomorphy is constructed and it is shown that there is a point in D for which Glea­son’s Problem for H°°(Ω) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C2 for which the volume of the H°°-envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves. If H°°(Ω) is the Banach algebra of bounded holomorphic functions on a bounded domain Ω in Cn and if p is a point in Ω, then the following problem is known as Gleason’s Problem for Hoo(Ω) : Is the maximal ideal in H°°(Ω) consisting of functions vanishing at p generated by (z1 -p1) , ... ,   (zn - pn) ? A sufficient condition for solving Gleason’s Problem for 77°° (Ω) for all points in Ω is given. In particular, this condition is fulfilled by a convex domain Ω with Lip1+e boundary (0 &lt; e &lt; 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C2. One of the ideas in the methods of proof is integration along specific polygonal lines. Certain properties of some open sets defined by global plurisubharmonic func­tions in Cn are studied. More precisely, the sets Du = {z e Cn : u(z) &lt; 0} and Eh = {{z,w) e Cn X C : h(z,w) &lt; 1} are considered where ti is a plurisubharmonic function of minimal growth and h≠0 is a non-negative homogeneous plurisubharmonic function. (That is, the functions u and h belong to the classes L(Cn) and H+(Cn x C) respectively.) It is examined how the fact that Eh and the connected components of Du are H°°-domains of holomorphy is related to the structure of the set of disconti­nuity points of the global defining functions and to polynomial convexity. Moreover it is studied whether these notions are preserved under a certain bijective mapping from L(Cn) to H+(Cn x C). Two counterexamples are given which show that polynomial convexity is not preserved under this bijection. It is also proved, for example, that if Du is bounded and if the set of discontinuity points of u is pluripolar then Du is of type H°°. A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions. / digitalisering@umu.se

Cette thèse porte sur la théorie du pluripotentiel et des problèmes d'équidistribution. Elle consiste en 4 chapitres. Le premier chapitre se consarce à l'étude de la régularité de la solution de l'équation de Monge-Ampère complexe sur une variété kahlérienne compacte X. Plus précisement, à l'aide des outils de la géométrie Cauchy-Riemann, on montre que la dernière équation possède une (unique) solution holdérienne pour une large classe géométrique de mesures de probabilités supportées par des sous-variétés réelles de X. Dans le chapitre 2, on étudie l'intersection des courants positifs fermés de grand bidegré. On y prouve que le produit extérieur de deux courants positifs fermés dont l'un possède un superpotentiel continu est positif fermé. Ceci généralise un résultat classique pour les courants de bidegré (1,1). Les deux chapitres suivants sont des applications de la théorie du pluripotentiel à des problèmes d'équidistribution. Dans le chapitre 3, on donne une vitesse explicite de convergence pour l'équidistribution des points de Fekete dans un compact K de l'espace euclidien à bord lisse par morceaux vers la mesure d'équilibre de K. Ici, les points de Fekete sont des bons points dans le problème d'interpolation d'une fonction continue sur K par des polynômes. Un tel contrôle de vitesse est crucial en pratique qu'on utilise les points de Fekete. La thèse se termine par le chapitre 4 où on prouve un analogue de la loi de Weyl pour les résonances d'un opérateur de Schodinger générique sur l'espace euclidien de dimension impair. Les résonances sont des objets centraux dans l'étude des opérateurs de Schrodinger. Elles jouent un rôle similaire à celui des valeurs propres dans le cadre compact. / This thesis concerns the pluripotential theory and equidistribution problems. It consists of 4 chapters. The first chapter is dedicated to the study of the regularity of the solution of the complexe Monge-Ampère equation on a compact Kahler manifold X. More precisely, using tools from the Cauchy-Riemann geometry, we prove that the last equation possesses a unique Holder continuous solution for a large geometric class of probability measures supported on real submanifolds of X. In the chapter 2, we study the intersecton of positive closed currents of higher bidegree. We prove there that the wedge product of two such currents one of which has a continuous superpotential est closed and positive. This property generalises a classical result for currents of bidegree (1,1). The next two chapters are applications of the pluripotential theory to equidistribution problems. In the chapter 3, we give an explicit speed of convergence for the equidistribution of Fekete's points in a compact subset K of the Euclidean space with piecewise smooth boundary toward the equilibrium measure of K. Here, the Fekete's points are good points for the interpolation problem of continuous functions by polynomials on K. A such control of speed is crucial in practice when ones use Fekete's points. The thesis is ended by the chapter 4 where we prove an analogue of Weyl's law for the resonances of a generic Schrodinger operator on an Euclidean space of odd dimension. The resonances are central objects in the research of Schrodinger operators. They play a similar role to that of eigenvalues in the compact setting.

Cette thèse étudie les équidistributions de zéros de sections holomorphesaléatoires de fibrés en droites pour les mesures modérées. Elle consiste en deuxparties.Dans la première partie, nous construisons une famille étendue de mesuressingulières modérées sur des espaces projectifs. Ces mesures sont générées pardes fonctions quasi-plurisousharmoniques avec les potentiels höldériens.Le deuxième partie traite une propriété d' équidistribution dans un contextegénéral. Nous établissons un théorème d'équidistribution dans le cas dequelques fibrés en droites gros munis de métriques singulières. Une vitesse deconvergence précise pour l'équidistribution est obtenue. / This thesis investigates the equidistributions of zeros of random holomorphic sections of line bundles for moderate measures. It consists of two parts. In the first part, we construct a large family of singular moderate measures on projective spaces. These measures are generated by quasi-plurisubharmonic functions with Holder potentials.The second part deals with an equidistribution property in general settings. We establish an equidistribution theorem in the case of several big line bundles endowed with singular metrics. A precise convergence speed for the equidistribution is obtained.