Calabi-Yau hypersurfaces and complete intersections in toric varieties

  • Author / Creator
    Novoseltsev, Andrey Y
  • In this thesis we report on several projects that stemmed out from an attempt to obtain an example for the last class in Doran-Morgan classification of ``variations of Hodge structure which can underlie families of Calabi-Yau threefolds over the thrice-punctured sphere with $b^3 = 4$, or equivalently $h^{2,1} = 1$''. First, a framework for toric varieties and their Calabi-Yau subvarieties has been implemented in the free open-source mathematical software system Sage. While there are other software packages, both commercial and free, for toric geometry, Sage has the advantage of smooth integration of numerous libraries for other related objects such as graphs, symbolic expressions, fast linear algebra, arbitrary precision and exact arithmetic, etc., combined with a powerful yet simple interface. We hope that our framework will prove useful both in research and teaching. Next, closed-form combinatorial expressions were obtained for Hodge numbers $h^{p,1}(X)$ of Calabi-Yau nef complete intersections of two hypersurfaces in toric varieties. Such formulas were long known for anticanonical hypersurfaces, while for nef complete intersections one had to use a highly-recursive generating function, whose actual computation requires significant resources. Our result provides a more efficient way to compute Hodge numbers of given Calabi-Yau varieties and can potentially be exploited to search for complete intersections with prescribed Hodge numbers. Finally, we have used torically induced fibrations by $M$-polarized K3-surfaces to construct an explicit geometric transition between an anticanonical hypersurface and a nef complete intersection through a singular subfamily of hypersurfaces. While we have concentrated on varieties inspired by the aforementioned Doran-Morgan classification, similar techniques may be used for (partial) desingularization of other singular subfamilies of generically smooth hypersurfaces.

  • Subjects / Keywords
  • Graduation date
  • Type of Item
  • Degree
    Doctor of Philosophy
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  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
  • Institution
    University of Alberta
  • Degree level
  • Department
    • Department of Mathematical and Statistical Sciences
  • Supervisor / co-supervisor and their department(s)
    • Doran, Charles (Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Buckingham, Paul (Mathematical and Statistical Sciences)
    • Page, Don (Physics)
    • Bouchard, Vincent (Mathematical and Statistical Sciences)
    • Cox, David (Mathematics, Amherst College, MA, USA)
    • Lewis, James (Mathematical and Statistical Sciences)