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.

Schedule
01/18
Organizational meeting
01/25
No seminar (Miami conference)
02/01
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
02/08
Shaoyun Bai
Quantum Riemann-Roch
Reference: [CG]
02/15
Melissa Liu
Shift operators
References: [I1]
02/22
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]
02/29
Konstantin Aleshkin
Quantum cohomology of projective bundles II: Fourier aspects
TBA
Reference: [IK]
03/07
Konstantin Aleshkin
Quantum cohomology of blowups I
TBA
Reference: [I]
03/14
No Seminar (spring break)
03/21
Sam Dehority
Quantum cohomology of blowups II
TBA
Reference: [I]
03/28
Sam Dehority
Quantum cohomology of blowups III
TBA
Reference: [I]