Ricci-flat Complex Geometry and its Applications

Autor :Marco Kühnel
Herkunft :OvGU Magdeburg, Fakultät für Mathematik
Datum :03.02.2010
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Kurzfassung :In chapter two we have a look at divisors and metrics invariant under some big symmetry group. This will give us a class of examples of complements of singular divisors with complete, Ricci-flat metrics. The Kähler condition will prove to be an extra condition on the symmetry group, as explained later in chapter three. In this singular case, the Kähler cone proves to be rather big, so the question of complete, Ricci-flat Kähler metrics in every Kähler class is vivid and we obtain a confirming answer for the highly symmetric case in chapter three. Contrary to this, a result of the author and Bert Koehler shows triviality of the Kähler cone can be achieved, if the divisor D is smooth and X fulfils some conditions of topological simplicity. In this case any complete, Ricci-flat Kähler metric differs from any given initial metric only by a Kähler potential. So we look for a solution to a complex Monge-Amp\'ere equation A bounded solution to this problem has been found by Tian and Yau for an appropriate initial metric. In terms of volume growth of geodesic balls there is a difference between the symmetric and the smooth case. In the symmetric case the polynomial growth rate can jump according to the jump of multiplicities of the divisor, but is always integer whereas in the smooth case it is rational non-integer. So the asymptotics of known complete, Ricci-flat Kähler metrics differ depending on whether D is smooth or not. In chapter four we inquire further into the asymptotics of the metric proved to exist by Tian and Yau. The results are the joint work of the author with Bert Koehler. We prove that the initial metric and the Ricci-flat solution metric differ only by any given negative power of the radial coordinate when approaching D. This result is expected to have analytic applications also for tackling the singular case. A similar result has been obtained at the same time by Santoro, independently and with a different technique. Finally, we obtain an extendability result for automorphisms using the existence result of Tian and Yau, much in the spirit of Schumacher's extension result in the Ricci-negative case. Chapter 5 is dedicated to the study of curvature preserving deformations of holomorphic vector bundles. Independently of the intrinsic interest in vector bundles, this could prove to be useful for the study of deformations of Ricci-flat open manifolds. The obstruction for this problem is proved to be a certain cohomology group, trivial in the Kähler case. Examples of non-Kähller manifolds are studied and criteria given for the property of a given vector bundle to admit only trivial deformations as curvature preserving deformations- In the last chapter we apply compact Ricci-flatness in order to construct octic hypersurfaces in projective 3-space with many nodes. The maximal number of nodes constructed here is 128.
Schlagwörter :Ricci-flat, complex, Monge-Ampere, Calabi-Yau, asymptotics
Rechte :Dieser Text ist urheberrechtlich geschützt
Größe :III, 106 S.
Erstellt am :04.03.2010 - 07:15:12
Letzte Änderung :22.04.2010 - 09:01:52
MyCoRe ID :HALCoRe_document_00007956
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