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Aspects of enumerative and categorical algebraic geometry
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- Author / Creator
- Chidambaram, Nitin Kumar
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In this thesis, we study some aspects of algebraic geometry that have had a
significant influx of ideas from physics. The first part focuses on the Eynard-
Orantin topological recursion and its variants as a theory of enumerative ge-
ometry. We investigate the conjectural relationship between the topological
recursion and quantum curves in the case of elliptic curves. We show that the
perturbative wave-function is not the solution to a quantum curve, while the
non-perturbative one is (up to certain order in ħ).
We define the formalism of Higher Airy Structures (HAS), which quantized
higher order Lagrangians in a symplectic vector space. By showing that the
Bouchard-Eynard topological recursion is a HAS, we find a necessary and
sufficient condition on the spectral curve for producing symmetric correlators
ω g,n . We construct numerous other examples, some of which are (sometimes
conjecturally) related to FJRW theory or open (r-spin) intersection theory.
In this thesis, we study some aspects of algebraic geometry that have had a significant influx of ideas from physics. The first part focuses on the Eynard-Orantin topological recursion and its variants as a theory of enumerative geometry. We investigate the conjectural relationship between the topological recursion and quantum curves in the case of elliptic curves. We show that the perturbative wave-function is not the solution to a quantum curve, while the non-perturbative one is (up to certain order in $ \hbar $).We define the formalism of Higher Airy Structures (HAS), which quantized higher order Lagrangians in a symplectic vector space. By showing that the Bouchard-Eynard topological recursion is a HAS, we find a necessary and sufficient condition on the spectral curve for producing symmetric correlators $ \omega_{g,n} $. We construct numerous other examples, some of which are (sometimes conjecturally) related to FJRW theory or open ($ r $-spin) intersection theory.
In the second part of this thesis, we focus on the study of derived categories and its connections to birational geometry; in particular, we are interested in a conjecture of Bondal and Orlov about flops. Using the presentation of a flop as a variation of geometric invariant theory (VGIT) problem, Ballard, Diemer and Favero proposed a Fourier-Mukai kernel and conjectured that it induces a derived equivalence. We verify this conjecture in the case of Grassmann flops first. Then, we tackle the case of singular VGIT problems using semi-free commutative dg-algebra resolutions, and prove that we obtain derived equivalences of dg-schemes, under some conditions.
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- Subjects / Keywords
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- Graduation date
- Fall 2020
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- Type of Item
- Thesis
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- Degree
- Doctor of Philosophy
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- License
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