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Fakultät für Mathematik, Informatik und Statistik - Digitale Hochschulschriften der LMU - Teil 01/02 Podcast
Universal moduli spaces in Gromov-Witten theory

Universal moduli spaces in Gromov-Witten theory

Fakultät für Mathematik, Informatik und Statistik - Digitale Hochschulschriften der LMU - Teil 01/02 vor 13 Jahren

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The construction of manifold structures and fundamental classes on
the (compactifed) moduli spaces appearing in Gromov-Witten theory
is a long-standing problem. Up until recently, most successful
approaches involved the imposition of topological constraints like
semi-positivity 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
Deligne-Mumford 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 Cauchy-Riemann 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
Gromov-Witten 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
well-defined 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.