Aspects of enumerative and categorical algebraic geometry

• Author / Creator

  Chidambaram, Nitin Kumar

• 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.

• Subjects / Keywords

  • airy structures
  • derived categories
  • algebraic geometry
  • topological recursion
  • flops

• Graduation date

  Fall 2020

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-s5ky-km66

• License

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