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University of Michigan

1. Slapar, Marko. Real surfaces in complex surfaces.

Degree: PhD, Pure Sciences, 2003, University of Michigan

URL: http://hdl.handle.net/2027.42/123705 

Let S ⊂ X be a real compact surface, Cinfinity embedded in a complex surface X. The problem of existence of a regular Stein neighborhood basis of S in X has so far not been well understood. In a generic position, there are only finitely many complex points on X, which can be classified as either elliptic or hyperbolic. By a result of Bishop, the nonexistence of elliptic complex points on S is a necessary condition for the existence of a Stein neighborhood basis of S. We show that an embedded surface S, without elliptic complex points, and with the extra condition of flatness at hyperbolic complex points, has a regular Stein neighborhood basis in X. A connection between complex points and unions of totally real planes is then explored to prove a similar result for certain polynomially convex unions of totally real planes. Using these results, together with the global theory of complex points on embedded real surfaces, we give some new examples of totally real surfaces and Stein domains inside complex elliptic surfaces. Advisors/Committee Members: Fornaess, John Erik (advisor).

Subjects/Keywords: Complex Surfaces; Pseudoconvex Domains; Real Surfaces; Stein Manifold

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Slapar, M. (2003). Real surfaces in complex surfaces. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/123705

Chicago Manual of Style (16th Edition):

Slapar, Marko. “Real surfaces in complex surfaces.” 2003. Doctoral Dissertation, University of Michigan. Accessed March 08, 2021. http://hdl.handle.net/2027.42/123705.

MLA Handbook (7th Edition):

Slapar, Marko. “Real surfaces in complex surfaces.” 2003. Web. 08 Mar 2021.

Vancouver:

Slapar M. Real surfaces in complex surfaces. [Internet] [Doctoral dissertation]. University of Michigan; 2003. [cited 2021 Mar 08]. Available from: http://hdl.handle.net/2027.42/123705.

Council of Science Editors:

Slapar M. Real surfaces in complex surfaces. [Doctoral Dissertation]. University of Michigan; 2003. Available from: http://hdl.handle.net/2027.42/123705


University of Michigan

2. Mihailescu, Eugen. Periodic points and hyperbolicity in higher dimensional complex dynamics.

Degree: PhD, Pure Sciences, 1999, University of Michigan

URL: http://hdl.handle.net/2027.42/131726 

Higher dimensional complex dynamics has experienced a tremendous growth in the past decade and much of this has been inspired by its connections with other fields. In Chapter II we study the periodic points that can appear for a torus action on a Kobayashi hyperbolic Stein manifold. Here we prove that under certain restrictions, there exist only a finite number of periodic points of all periods. Then we study the fixed points for an S1 action on a Stein manifold; multidimensional residue theory is used to prove the existence of at most one fixed point in many cases. We also give counterexamples of manifolds with actions having an arbitrary number of fixed points. In Chapter III, we generalize an important Corollary from Chapter II to manifolds having any finite number of generators for their second cohomology group. There are also given several applications to Siegel domains for holomorphic endomorphisms of P2; a new class of such Siegel domains is constructed. In Chapter IV we study the opposite case, hyperbolic maps on P2, which are proved to have no Siegel domains. We show that the set of ws-hyperbolic maps without cycles is open, which further provides a large class of examples of ws-hyperbolic: maps constructed from simpler ones. We also conjecture that the interior of the set K- of points with bounded backward iterates, is empty, at least for certain classes of strongly hyperbolic maps on P 2. In the final section we prove that a repellor with empty interior for a special class of functions, will have Lebesgue measure zero. Advisors/Committee Members: Fornaess, John E. (advisor).

Subjects/Keywords: Complex; Dimensional; Dynamics; Higher; Hyperbolicity; Periodic Points; Siegel Domains; Stein Manifold

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Mihailescu, E. (1999). Periodic points and hyperbolicity in higher dimensional complex dynamics. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/131726

Chicago Manual of Style (16th Edition):

Mihailescu, Eugen. “Periodic points and hyperbolicity in higher dimensional complex dynamics.” 1999. Doctoral Dissertation, University of Michigan. Accessed March 08, 2021. http://hdl.handle.net/2027.42/131726.

MLA Handbook (7th Edition):

Mihailescu, Eugen. “Periodic points and hyperbolicity in higher dimensional complex dynamics.” 1999. Web. 08 Mar 2021.

Vancouver:

Mihailescu E. Periodic points and hyperbolicity in higher dimensional complex dynamics. [Internet] [Doctoral dissertation]. University of Michigan; 1999. [cited 2021 Mar 08]. Available from: http://hdl.handle.net/2027.42/131726.

Council of Science Editors:

Mihailescu E. Periodic points and hyperbolicity in higher dimensional complex dynamics. [Doctoral Dissertation]. University of Michigan; 1999. Available from: http://hdl.handle.net/2027.42/131726

3. Vérine, Alexandre. Quelques propriétés symplectiques des variétés Kählériennes : Some symplectic properties of Kähler manifolds.

Degree: Docteur es, Mathématiques, 2018, Lyon

URL: http://www.theses.fr/2018LYSEN038 

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…

Advisors/Committee Members: Giroux, Emmanuel (thesis director).

Subjects/Keywords: Fonctions plurisousharmoniques; Domaines de Stein; Cycles évanescents; Cobordismes de Weinstein; Sections hyperplanes; Constantes de Seshadri; Variétés symplectiques; Plurisubharmonic functions; Vanishing cycles; Stein domains; Weinstein cobordisms; Hyperplane sections; Seshadri constants; Symplectic manifolds

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Vérine, A. (2018). Quelques propriétés symplectiques des variétés Kählériennes : Some symplectic properties of Kähler manifolds. (Doctoral Dissertation). Lyon. Retrieved from http://www.theses.fr/2018LYSEN038

Chicago Manual of Style (16th Edition):

Vérine, Alexandre. “Quelques propriétés symplectiques des variétés Kählériennes : Some symplectic properties of Kähler manifolds.” 2018. Doctoral Dissertation, Lyon. Accessed March 08, 2021. http://www.theses.fr/2018LYSEN038.

MLA Handbook (7th Edition):

Vérine, Alexandre. “Quelques propriétés symplectiques des variétés Kählériennes : Some symplectic properties of Kähler manifolds.” 2018. Web. 08 Mar 2021.

Vancouver:

Vérine A. Quelques propriétés symplectiques des variétés Kählériennes : Some symplectic properties of Kähler manifolds. [Internet] [Doctoral dissertation]. Lyon; 2018. [cited 2021 Mar 08]. Available from: http://www.theses.fr/2018LYSEN038.

Council of Science Editors:

Vérine A. Quelques propriétés symplectiques des variétés Kählériennes : Some symplectic properties of Kähler manifolds. [Doctoral Dissertation]. Lyon; 2018. Available from: http://www.theses.fr/2018LYSEN038

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