Gerstenberger, Andreas (2013): Universal moduli spaces in GromovWitten theory. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
The construction of manifold structures and fundamental classes on the (compactifed) moduli spaces appearing in GromovWitten theory is a longstanding problem. Up until recently, most successful approaches involved the imposition of topological constraints like semipositivity on the underlying symplectic manifold to deal with this situation. One conceptually very appealing approach that removed most of these restrictions is the approach by K. Cieliebak and K. Mohnke via complex hypersurfaces, [CM07]. In contrast to other approaches using abstract perturbation theory, it has the advantage that the objects to be studied still are spaces of holomorphic maps defined on Riemann surfaces. In this thesis this approach is generalised from the case of surfaces of genus 0 dealt with in [CM07] to the general case. In the first section the spaces of Riemann surfaces are introduced, that take the place of the DeligneMumford spaces in order to deal with the fact that the latter are orbifolds. Also, for use in the later parts, the interrelations of these for different numbers of marked points are clarified. After a preparatory section on Sobolev spaces of sections in a fibration, the results presented there are then used, after a short exposition on Hamiltonian perturbations and the associated moduli spaces of perturbed curves, to construct a decomposition of the universal moduli space into smooth Banach manifolds. The focus there lies mainly on the global aspects of the construction, since the local picture, i.e. the actual transversality of the universal CauchyRiemann operator to the zero section, is well understood. Then the compactification of this moduli space in the presence of bubbling is presented and the later construction is motivated and a rough sketch of the basic idea behind it is given. In the last part of the first chapter, the necessary definitions and results are given that are needed to transfer the results on moduli spaces of curves with tangency conditions from [CM07]. There also the necessary restrictions on the almost complex structures and Hamiltonian perturbations from [IP03] are incorporated, that later allow the use of the compactness theorem proved in that reference. In the last part of this thesis, these results are then used to give a definition of a GromovWitten pseudocycle, using an adapted version of the moduli spaces of curves with additional marked points that are mapped to a complex hypersurface from [CM07]. Then a proof that this is welldefined is given, using the compactness theorem from [IP03] to get a description of the boundary and the constructions from the previous parts to cover the boundary by manifolds of the correct dimensions.
Item Type:  Theses (Dissertation, LMU Munich) 

Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 510 Mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date of oral examination:  10. July 2013 
1. Referee:  Cieliebak, Kai 
MD5 Checksum of the PDFfile:  6475670bfba7d5f60d70aa6ad1e8b84a 
Signature of the printed copy:  0001/UMC 21333 
ID Code:  15878 
Deposited On:  22. Jul 2013 09:05 
Last Modified:  23. Oct 2020 13:11 