Studying categorical aspects of the Landau-Ginzburg B-model using variations of geometric invariant theory

Malter, Aimeric ORCID: 0000-0001-7377-1103 (2023). Studying categorical aspects of the Landau-Ginzburg B-model using variations of geometric invariant theory. University of Birmingham. Ph.D.

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Abstract

In this thesis we use variations of geometric invariant theory to study the derived categories of coherent sheaves associated to complete intersections in toric varieties. In the context of mirror symmetry, a given Calabi-Yau variety may not have a unique mirror associated to it. Finding relations between derived categories associated to distinct mirror constructions leads to unification results under Homological Mirror Symmetry. In particular, in this thesis we prove the equivalence of two constructions to a complete intersection of cubics in ℙ\(^{5}\), one due to Batyrev and Borisov, the other due to Libgober and Teitelbaum. The proof relies on methods of partial compactifications and variations of geometric invariant theory, and is the first of its kind to relate derived categories for complete intersections and not hypersurfaces.

Singular complete intersections present an obstacle when applying these methods to a wider context, and we do not obtain equivalences of derived categories in general. Partial compactifications and variations of geometric invariant theory however remain a strong tool in studying the derived categories of singular complete intersections. In this thesis, we give a framework in which we can use these methods to obtain crepant categorical resolutions. We illustrate this framework by giving a family of examples which directly generalises the mirror construction by Libgober and Teitelbaum, then categorically resolving the derived categories of coherent sheaves by the derived categories of coherent sheaves associated to a family of Batyrev-Borisov mirrors.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):