We meet on Fridays from 2:00PM to 3:30PM at 384-I including talks and discussions.
The goal of this reading seminar is to understand the recent advances in motivic aspects of Gromov--Witten theory by Katzarkov--Kontsevich--Pantev--Yu, in particular their definitions of Hodge atoms and applications to birational geometry. The invariants are defined through enumerative geometry, that is the quantum connection on the non-archimedean analytic Frobenius manifold defined by genus 0 Gromov--Witten theory.
References
[KKPY] Birational invariants from Hodge structures and quantum multiplication, Katzarkov--Kontsevich--Pantev--Yu
[Iri-B] Quantum cohomology of blowups, Iritani
[Iri-F] Fourier analysis on equivariant quantum cohomology, Iritani
[IK] Quantum cohomology of projective bundles, Iritani--Koto
[Co] Givental's Lagrangian cone and S^1-equivariant Gromov--Witten theory, Coates
[CG] Quantum Riemann--Roch, Lefschetz, and Serre, Coates--Givental
[HYZZ] Decompositions and framings of F-bundles and applications to quantum cohomology, Hinault--Yu--Zhang--Zhang
[Giv] Equivariant Gromov--Witten invariants, Givental
[Clay] Mirror symmetry, Clay monograph
[KKP] Hodge theoretic aspects of mirror symmetry, Katzarkov--Kontsevich--Pantev
Schedule
A1 (1/9) Organization (Jae) [Notes]
High-level structure of the proof, identify the key ingredients and assign talks.
A2 (1/16) Big quantum products and quantum connections ([KKPY] 3.5.1) (Jae)
Gromov--Witten theory, cycle-valued GW invariants, Novikov rings and quantum products, their bulk deformed versions, example computations for the quantum spectrum, formal quantum connections
A3 (1/23) A-model VHS and non-commutative Hodge structures ([KKP] 2, 3.1) (Xinyu) [Notes]
Pure nc-Hodge structures and Z/2-graded variation of (semi-infinite) Hodge structures, quantum connections as VSHS
(1/30) is Miami conference on homological mirror symmetry so we will skip.
A4 (2/6) Example computations of quantum spectrum ([KKPY] 6.1, [Giv]], [Clay] Ch.28) (Sam)
Quantum differential equations, quantum spectrum of semipositive toric lci varieties
A5 (2/13) Non-archimedean A-model F-bundles ([KKPY] 3.1, 3.4, 3.5.2.1) (Ben)
Generalities on F-bundles, Kodaira--Spencer classes (\mu maps) and Euler vector fields, analytic quantum connections on Betti cohomology (for varieties defined over algebraically closed fields of char 0), maximality via divisor axiom of GW theory
A6 (2/20) Basics of motives ([KKPY] 5.1, 3.5.2.2) (Vaughan)
Tannakian philosophy, Weil cohomology theories on Andre motives, Lefschetz characters, Mumford--Tate groups, motivic invariance of quantum products
A7 (2/27) Definition of Hodge Atoms ([KKPY] 5.2, 5.3, 5.4)Â (Cole)
Definition of local G-atoms, description of elementary moves (disjoint unions, blowups, projective bundles) and their compatibility with spectral decomposition, G-atomic F-bundles, numerical invariants of Hodge atoms
A8 (3/6) Atomic decompositions ([KKPY] 4.1, 4.2, 4.3) (???)
Precise statements of spectral decomposition, Iritani blow-up formulas, adaptation to non-archimedean F-bundles
A9 (3/13) Non-rationality of cubic 4-folds ([KKPY] 6.1, 6.2) (???)
Chemical formulae, Non-rationality criterion, Hodge atoms of low-dimensional varieties, the proof of irrationality of very general cubic 4-fold
The remaining talks pertain to structure of quantum cohomology and quantum D-modules for Hamiltonian T-spaces, crucially used for Iritani's proof of the blowup formula. For the core seminar we will discuss the statement of the blowup formula in detail but leave its proof as a black-box. This part could be expanded into a full seminar about "how to compute the quantum spectrum"? and may indeed be separated as a seminar if there is sufficient interest.
B1 Givental formalism
Fundamental solutions of quantum differential equations, J-functions, Givental's overruled Lagrangian cones, Hodge-theoretic mirror symmetry
B2 Applications of Givental's QDE
Using QDEs to compute quantum spectrum for toric complete intersections
B3 Quantum Riemann--Roch and Lefschetz
Twisted Gromov--Witten invariants, reconstruction theorems, quantum spectrum for complete intersections and projective bundles
B4 Equivariant quantum cohomology
Equivariant quantum cohomology, Seidel homomorphisms, shift operators and differential-difference module structures, quantum Kirwan maps, Teleman philosophy
B5 Blowup formula
Wall-and-chamber structure on Kahler moduli space of GIT pre-quotients, deformation to the normal cone, proof of Iritani blow-up formula