Coherent sheaves on Calabi-Yau manifolds, Picard-Fuchs equations and potential functions

Abstract: We investigate the deformation theory of Calabi-Yau threefolds -- simply connected complex projective manifolds with trivial canonical bundle -- together with geometric objects on the Calabi-Yau threefold. The geometric objects in question are curves, surfaces, special coherent sheaves and rank-two vector bundles with a section vanishing in codimension two.

These deformation problems are related to the study of Picard-Fuchs equations. Physicists suggest that a holomorphic potential function the critical locus of which is the space of unobstructed deformations appears as a solution of the Picard-Fuchs equation.
In this thesis, Picard-Fuchs equations for Calabi-Yau threefolds
appearing as complete intersections of codimension two in projective spaces are studied. In addition, Picard-Fuchs equations for pairs of a Calabi-Yau threefold and either a divisor or a curve on the threefold are examined. We construct Picard-Fuchs equations using Griffiths-Dwork reduction.

Based on the work of Jockers and Soroush, we give rigorous mathematical foundations for deriving Picard-Fuchs operators in various cases, in particular for the quintic threefold together with a special divisor.
Furthermore, we initiate a theory of triples consisting of a threefold with two divisors meeting transversally along a curve and set up Picard-Fuchs operators for this situation. The thesis furthermore contains some SINGULAR programmes for explicit calculations

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• Extent: Online-Ressource
• Language: Englisch
• Notes: Universität Freiburg, Dissertation, 2018

Classification and Topics

• Keyword: Komplexe Geometrie ; Stringtheorie ; Hodge-Struktur ; Calabi-Yau-Mannigfaltigkeit

Contributors, Places and Time

• Event: Veröffentlichung
• (where): Freiburg
• (who): Universität
• (when): 2018
• Creator: Peternell, Natalie Kathrin
• Contributor: Scheidegger, Emanuel ; Albert-Ludwigs-Universität Freiburg. Fakultät für Mathematik und Physik

Further information

• DOI: 10.6094/UNIFR/17082
• URN: urn:nbn:de:bsz:25-freidok-170821
• Rights: Kein Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Last update: 25.03.2025, 1:46 PM CET