Aspects of enumerative and categorical algebraic geometry

Abstract

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.