Habermann, Matthew;
(2021)
Homological mirror symmetry for invertible curve singularities.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
The central theme of this thesis is homological mirror symmetry for curve singularities defined by invertible polynomials. The main results are contained in Chapters 3, 4, and 5. Chapter 3 is based on joint work with Jack Smith, and establishes homological Berglund--Hübsch mirror symmetry for invertible polynomials in two variables by matching generating collections on both sides. Along the way, we show that the category of graded matrix factorisations has a tilting object, confirming a conjecture of Lekili and Ueda in the case of curves. In Chapter 4, we build on the results of Chapter 3 to establish a derived equivalence between the Fukaya category of the Milnor fibre and the derived category of perfect complexes on the proposed mirror. The strategy of proof builds on that of Lekili and Ueda, and uses a moduli of $A_\infty$-structures argument. A key step in the proof of this result is to reconstruct the Milnor fibres by a gluing procedure. In Chapter 5, we prove homological mirror symmetry for a framework which generalises that of invertible curve singularities. Namely, the B--model is taken to be a chain or ring of weighted projective lines joined nodally such that each irreducible component is allowed to have non-trivial generic stabiliser, and the A--model is built using the gluing construction of Chapter 4. As a special case, this completely resolves a conjecture of Lekili and Ueda on invertible polynomials in complex dimension one. This also re-establishes the results of Chapter 4 by different methods and generalises the results of Lekili and Polishchuk. As a corollary, we prove some derives equivalences between categories of sheaves on the B--models by studying when the corresponding A--models are graded symplectomorphic.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Homological mirror symmetry for invertible curve singularities |
| Event: | UCL (University College London) |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2021. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10136637 |
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