NCMS · A new approach in curve counting theories and mirror symmetry

A new approach in curve counting theories and mirror symmetry

[2204.00509] Relative quantum cohomology under birational transformations (arxiv.org). In my work, I studied how relative quantum cohomology, defined using orbifold Gromov—Witten theory, varies understand birational transformations. This property is known as the main difference between orbifold and logarithmic Gromov—Witten invariants. Instead of considering single invariants, I considered a generating function and studied the property on the structural level. I have been revising and improving this article, and now it is under review in Advances in Mathematics. [2112.12891] Degenerations, fibrations and higher rank Landau-Ginzburg models (arxiv.org). This is joint work with C. Doran and J. Kostiuk. We generalize the Doran—Harder—Thompson conjecture beyond the Calabi—Yau setting and to more general and more complicated degenerations. We have been revising and improving this article, and now it is under review in Advances in Mathematics. My paper (titled:” The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map”). Given a smooth log Calabi--Yau pair (X,D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau--Ginzburg potential and show that it is a generating function of two-point relative Gromov--Witten invariants of (X,D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem to compute two-point relative invariants. When D is nef, we compute the proper Landau--Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of Grafnitz—Ruddat--Zaslow that the proper Landau--Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period. I further generalize the result to consider the theta functions for more general pairs (simple normal crossing pairs): arXiv:2403.17077. We introduce a new type of orbifold invariants for snc pairs, called mid-age invariants, and use these invariants to define orbifold invariants associated with the broken line type. Then, we define the orbifold theta functions as generating functions of orbifold invariants with mid-ages. We show that these orbifold theta functions are well-defined and satisfy the multiplication rule. In joint work with Yu Wang (arXiv:2403.17200), we are able to generalize the local-log-orbifold correspondence in an unexpectedly different direction. Given a smooth projective variety X and a smooth nef divisor D, we identify genus zero relative Gromov--Witten invariants of (X,D) with (n+1) relative markings with genus zero relative/orbifold Gromov--Witten invariants of a P1-bundle with n relative markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov--Witten invariants with genus zero absolute Gromov--Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov--Witten invariants.

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