Aim: survey what is understood about homological mirror symmetry between derived categories of Fano varieties, and Fukaya-Seidel categories of their mirror Landau-Ginzburg models. We will not aim to give a comprehensive introduction to the theory of Fukaya-Seidel categories or derived categories of coherent sheaves, but rather a qualitative sketch before moving on to the examples of interest. Feel free to suggest topics not already mentioned below!
The group meets on Tuesdays, 10-12, in JCMB 5323.
12/9: Nick Sheridan, Overview (Nick's notes)
24/9: Alexia Corradini, Full exceptional collections (Alexia's notes)
Definition of exceptional collection, and mutation thereof (in triangulated category, see [Rudakov et. al., Helices and Vector Bundles], or in A_\infty category a la [Seidel, Vanishing cycles and mutation, section 5])
Derived category of P^n a la Beilinson
Explicit description of derived category of del Pezzo surface as algebra of modules over a quiver, following section 2 of Auroux-Katzarkov-Orlov, Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves.
1/10: Danil Kozevnikov, Fukaya-Seidel and partially wrapped Fukaya categories (Danil's notes)
A non-technical introduction to the partially wrapped Fukaya category of a stopped Liouville domain a la [Ganatra-Pardon-Shende, Sectorial descent for partially wrapped Fukaya categories].
Fukaya-Seidel category of exact symplectic LG model, and the algebro-geometric source of these [Seidel, More on vanishing cycles and mutation, section 3].
state when these are (expected to be) equivalent [Ganatra-Pardon-Shende, Covariantly functorial wrapped Floer theory on Liouville sectors].
full exceptional collection associated to a choice of vanishing paths, mutation of paths = mutation of exceptional collection [Seidel, Vanishing cycles and mutation].
in general, may have non-isolated singular locus, or orbifolding: expect a semi-orthogonal decomposition.
8/10: Parth Shimpi, First examples of HMS (Parth's notes)
P^1 (points; ample line bundles)
P^2 (exceptional collection; see notes from previous reading group)
del Pezzo surfaces following Auroux-Katzarkov-Orlov
state HMS for smooth Fano toric varieties without proof? Note this gives existence of full exceptional collection for DCoh(smooth Fano toric variety) associated to vanishing paths.
15/10: Jeff Hicks, Abouzaid-Auroux-Katzarkov mirror construction (Jeff's notes)
22/10: NO MEETING
29/10: (*different location: JCMB 5326*) Augustinas Jacovskis, Derived categories of Fano threefolds (Augustinas' notes)
Overview of Fano 3-folds; what their s.o.d. look like [Kuznetsov, Derived categories of Fano threefolds]; for Picard rank one, [Bayer-Lahoz-Macri-Stellari, Stability conditions on Kuznetsov components, p. 24].
5/11: Sukjoo Lee, Mirrors to Fano threefolds (Sukjoo's notes)
Qualitative description of mirrors to Fano threefolds, with selected examples, following [Cheltsov-Przyjalkowski, Katzarkov-Kontsevich-Pantev conjecture for Fano threefolds] and references therein. See also [Doran-Harder-Katzarkov-Ovcharenko-Przyjalkowski, Modularity of Landau-Ginzburg models].
19/11: Ilaria Di Dedda, Fano hypersurfaces in projective spaces, their Abouzaid-Auroux-Katzarkov/Hori-Vafa mirrors (Ilaria's notes). Another helpful reference: arXiv:2410.14678.
26/11: Bogdan Simeonov, Cubic fourfold (Bogdan's notes).
5/12 (note special day): Victor Przyjalkowski, Fibers of Landau--Ginzburg models, invariants of Fano varieties, and some categories (Victor's notes).
10/12: Giulia Gugiatti, Quartic double solid (Giulia's notes).
?: Kuznetsov components looking like DCoh(curve)
[Cannizzo, Categorical mirror symmetry on cohomology for a complex genus 2 curve]
Fano 3-folds where Kuznetsov component looks like derived category of a curve, e.g. intersection of two quadrics in P^5 [Kapustin-Katzarkov-Orlov-Yotov, Homological mirror symmetry for varieties of general type, section 7].
?: different Fano 3-folds with the same Kuznetsov component [Kapustin-Katzarkov-Orlov-Yotov, Homological mirror symmetry for varieties of general type, remark 6.3].
?: deformations of LG models, and moduli of mirror Fano varieties.