Patrick Lei

Enumerative invariants and birational transformations (Spring 2024)
About this seminar

The goal of this seminar is to understand the behavior of enumerative invariants under birational transformations. In the case of the Gromov-Witten theory of smooth projective varieties, the weak factorization theorem (see this paper) tells us that it suffices to consider blowups along smooth centers. In this case, a correspondence between quantum D-modules was proven by Iritani [I], which has applications in the announced proof by Katzarkov-Kontsevich-Pantev-Yu of the irrationality of the cubic fourfold. In our journey to understanding this work, we will learn about many important tools in enumerative geometry. First, we will need to understand some foundational tools like the Givental formalism and shift operators:

We will then study the technically easier case of projective bundles [IK] (recall that the blowup replaces the center with the projectivization of its normal bundle) to understand the techniques before heading to the main result.

In the remaining part of the semester, we will explore some related work, for example on the crepant transformation conjecture.

Organizational meeting
No seminar (Miami conference)
Patrick Lei
Givental formalism
I will explain how to package Gromov-Witten invariants into structures that are amenable to systematic study.
References: [G], Coates's thesis, Dubrovin
Shaoyun Bai
Quantum Riemann-Roch
Reference: [CG]
Melissa Liu
Shift operators
Reference: [I1]
Che Shen
Quantum cohomology of projective bundles I -- a mirror theorem
We will follow [IK] to talk about a mirror theorem for projective bundles. More precisely, the I-function of a projective bundle can be constructed from the J-function of the underlying vector bundle by exploiting Givental's Lagrangian cone formalism and the quantum Riemann-Roch theorem of Coates-Givental.
Reference: [IK]
Konstantin Aleshkin
Quantum cohomology of projective bundles II: Fourier aspects
Reference: [IK]
Konstantin Aleshkin
Quantum cohomology of blowups I
Reference: [I]
No Seminar (spring break)
Sam Dehority
Quantum cohomology of blowups II
Reference: [I]
Sam Dehority
Quantum cohomology of blowups III
Reference: [I]
Patrick Lei
Generalities on orbifold cohomology and toric DM stacks
I will explain various technicalities in Gromov-Witten theory for Deligne-Mumford stacks and how to construct toric Deligne-Mumford stacks from (extended) stacky fans.
References: [CIJ], Abramovich-Graber-Vistoli, Tseng, Borisov-Chen-Smith, Jiang
Davis Lazowski
Crepant transformation conjecture for toric complete intersections I
Reference: [CIJ]
Davis Lazowski
Crepant transformation conjecture for toric complete intersections II
Reference: [CIJ]
Konstantin Aleshkin
Wall-crossing for Grassmannian flops
I will talk about a couple of papers that just appeared on arXiv. Two groups of people independently generalize CIJ wall-crossing to a simple non-abelian quotient using abelianization ideas. I plan to talk about the results and the role that CIJ's proof of the crepant transformation conjecture plays in these constructions.
References: Lutz-Shafi-Webb, Priddis-Shoemaker-Wen