We will meet on Wednesdays 4-5:00pm in MR13 (except on Mar 11, when the Geometry and Topology Colloquim takes place), for a 1 hour talk followed by a question and discussion session.
The goal of this seminar to the understand the recent work of Katzarkov, Kontsevich, Pantev and Yu on certain new birationality results, such as the irrationality of a general cubic fourfold, obtained from enumerative geometry. Specifically, we will study their construction of a new birational invariant called "Hodge atom", which is defined via a non-archimedean version of the Frobenius manifold associated to the Gromov-Witten theory of an algebraic variety.
If you have questions, suggestions or want to sign up for a talk, please feel free to email Ailsa Keating (amk50@cam.ac.uk) or Zihong Chen (zc431@cam.ac.uk).
Main references:
Birational invariants from Hodge structures and quantum multiplication, Katzarkov-Kontsevich-Pantev-Yu
Hodge theoretic aspects of mirror symmetry, Katzarkov-Kontsevich-Pantev
Quantum cohomology of blowups, Iritani
Quantum cohomology of projective bundles, Iritani-Koto
Decompositions and framings of F-bundles and applications to quantum cohomology, Hinault-Yu-Zhang-Zhang
Schedule
Week 1 (Feb 18) Zihong : Overview and organization (notes).
Week 2 (Feb 25) Maxim: Quantum cohomology and quantum connection ([KKPY] 3.5): definition and axioms of cycle-valued Gromov-Witten classes and the (bulk-deformed) quantum cohomology; example computations; the (big) quantum connection.
Week 3 (Mar 4) Alexia: The A-model F-bundle ([KKPY] 3.1, 3.4, 3.5) : definition of (formal/non-archimedean) F-bundles; maximality and the Euler vector field; non-archimedean convergence of the Gromov-Witten potential.
(Mar 11, no talk due to Geometry and Topology Colloquim)
Week 4 (Mar 18) Zihong: Spectral decomposition of F-bundles ([KKPY] 4.1, [HYZZ] 3.1, 3.2, 3.3): the Frobenius theorem for PDEs over non-archimedean fields; spectral decomposition of maximal F-bundles and non-archimedean convergence.
Week 5 (April 29?) Xuanchun: Hodge atoms ([KKPY] 4.2, 4.3, 5.3, 5.4): transformation of spectral decompositions of the A-model F-bundles under blowups and taking projective bundle; atoms and atomic F-bundles.
(May 6: tentative date for Geometry and Topology Colloquim)
Week 6 (May 13?) Ivan: Irrationality of general cubic 4-folds ([KKPY] 6.1, 6.2): Givental's fundamental solution; computations of Hodge atoms in low dimensions; irrationality of a very general cubic 4-fold.
Other: rationality over non-algebraically-closed fields, atoms and homological mirror symmetry....