Comparison of non-Archimedean and logarithmic mirror constructions via the Frobenius structure theorem: https://arxiv.org/abs/2204.00940 (To appear in Journal of the LMS)
Motivated by comparing two recent mirror constructions for a log Calabi-Yau pair (X,D), one by Gross and Siebert, and the other by Keel and Yu, I compare curve counts used in constructing the candidate mirror algebras. The result is that under some mild assumptions on D, the log Gromov-Witten invariants used to define the Gross-Siebert intrinsic mirror constructions are the same as the naive counts of rational curves used to define the Keel-Yu mirror.
Birational invariance in punctured log Gromov-Witten theory: https://arxiv.org/abs/2210.06079 (To appear in Algebraic Geometry)
In the same spirit of the investigation carried out by Abramovich and Wise for ordinary log Gromov-Witten invariants, I investigate how moduli spaces of punctured log maps behave under log étale modifications to the boundary for a log smooth pair (X,D). These are birational morphisms which admit combinatorial descriptions in a manner analogous to toric blowups in toric geometry. It turns out that the stack M(X') of punctured log maps to the modification X' of a target X has an explicit description in terms of the stack M(X) and the corresponding tropical moduli problems.
Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten theory of root stacks: https://arxiv.org/abs/2403.05376 (submitted)
Using the orbifold/log Gromov-Witten correspondence of Battistella, Nabijou, and Ranganathan, combined with the birational invariance in punctured log Gromov-Witten theory established in the paper above, I relate the log invariants used to define the intrinsic mirror algebra of Gross and Siebert with a class of orbifold invariants defined by Tseng and You. In particular, this relation gives another proof of associativity of the intrinsic mirror algebra, as well as proof of the (weak) Frobenius structure conjecture. Along the way, I extend the study above of the behaviour of log Gromov-Witten invariants under log étale morphisms to include root stacks and ramified base change.
Quantum periods, toric degenerations and intrinsic mirror symmetry : https://arxiv.org/abs/2501.01408
Quantum periods are an important class of Gromov-Witten invariant for a Fano variety X, playing a central role in the Fano search program. We demonstrate how intrinisc mirror symmetry naturally yield Landau-Ginzburg superpotentials whose classical periods recover the quantum periods of a given Fano variety. Moreover, we explore how these superpotentials in various settings featuring cluster varieties encode toric degenerations of the Fano variety. Finally, we show that the quantum periods are enough invariants to determine all log Gromov-Witten invariants involved in the definition of the miror algebra associated X and a smooth anticanonical divisor D.
Birational invariance in log Gromov-Witten theory and double ramification cycles with target (In preperation)
Higher genus log-orbifold correspondence (joint with Robert Crumplin, in preperation)