Mirror Symmetry for Fano Manifolds andRelated Topics
Time: 10-14 December 2018
Venue: Room 127 (Graduate School of Science Bldg no.3), Department ofMathematics, Kyoto University

Mon
TueWedThu
Fri

9:00-10:00
Cho
ViannaKasprzyk
HarderSmirnov

10:30-11:30
ChoViannaSmirnov
HarderSmirnov

13:30-14:30
ViannaCho

15:00-16:00 16:30-17:30
Ikeda
Harder
Ikeda

Kasprzyk
ShamotoKasprzyk
ShamotoTitle:
Cheol-Hyun Cho

1. HMS for toric Fano manifolds
2. Mirror symmetry for orbi-spheres
3. Pairings and Mirror Symmetry
Andrew HarderFibrations and mirror symmetry
Akishi Ikeda
q-stability conditions and q-quadratic diﬀerentials
Alexander Kasprzyk Mirror symmetry for Fano manifolds
Yota Shamoto
Hodge structures on tame compactified
Landau-Ginzburg models
Maxim Smirnov
Quantum cohomology and derived categories ofrational homogeneous spaces
Renato Vianna
Lagrangian fibrations and open Gromov-Witten invariants
See the conference page for more information.
Organizers: Hiroshi Iritani, Yukiko Konishi, Atsushi Takahashi
Acknowledgments: This workshop is supported by Kiban-S 16H06337 (Atsushi Takahashi) and Kiban-C 16K05127 (Hiroshi Iritani).

Cheol-Hyun Cho
Lecture 1. HMS for toric Fano manifolds
We explain how to construct explicit A-infinity functor for toric Fano
manifold which derives homological mirror symmetry between its Fukaya
category and matrix factorization category of its Landau-Ginzburg mirror.
It is based on Floer theoretic construction which uses formal Maurer-Cartan
elements and curved Yoneda embedding. This is a joint work with Hansol
Hong and Siu-Cheong Lau.
Lecture 2. Mirror symmetry for orbi-spheres
We consider a sphere with 3 orbifold points. Its mirror is given by acusp singularity whose closed mirror symmetry has been studied by Rossi,Satake, Takahashi, Ishibashi and Shiraishi. We explain (homological) mirror
symmetry for orbi-spheres using Lagrangian Floer theory. Using an immersed
circle (called Seidel Lagrangian), we define homological mirror functor as well
as Kodaira-Spencer map to prove (homological) mirror symmetry. This isbased on joints works with Lino Amorim, Hansol Hong and Siu-Cheong Lau.
Lecture 3. Pairings and Mirror symmetry
Given a (homological) mirror symmetry between a symplectic manifold
and its Landau-Ginzburg mirrors, we may ask whether mirror symmetry
preserves pairing structures. For a symplectic manifold, Poincare duality
provides pairings for both open and closed theories. A matrix factorization
category has the Kapustin-Li pairing and Jacobian ring has a residue pairing. We introduce what we call, multi-crescent Cardy identity, which is used
to compare these pairings. We will find that interesting conformal factor
arises between these pairings in mirror symmetry. This is a joint work withSangwook Lee and Hyungseok Shin.
Andrew HarderFibrations and mirror symmetry
My first two talks will focus on describing how fibrations on Calabi-Yau
manifolds are related to degenerations under mirror symmetry. In my firsttalk, I will discuss elliptic fibrations on K3 surfaces and how, in Dolgachev
s lattice polarized mirror symmetry, they relate to type II boundary components in the moduli space of mirror K3 surfaces. My second talk will describe
how the same ideas, applied to K3 fibrations on Calabi-Yau threefolds andTyurin degenerations, lead to classification theorems analogous to Kodairas classification of elliptic fibrations, for certain K3 fibered Calabi-Yau threefolds.

My final talk will change direction slightly. I will discuss new conjectures,developed with Katzarkov, Pantev, Przyjalkowski and others, about how theweight filtration on a the cohomology log Calabi-Yau variety is related to theperverse Leray filtration on the mirror manifold associated to its aﬃnization
map, and I will describe a collection of examples in which these conjectures
have been verified.
Akishi Ikeda
q-stability conditions and q-quadratic diﬀerentials
For a given triangulated category, we can associate a complex manifold, called the space of Bridgeland stability conditions. By the work ofBridgeland-Smith (BS) and Haiden-Katzarkov-Kontsevich (HKK), for a certain class of triangulated categories associated with marked bordered surfaces, the space of stability conditions on them can be identified with themoduli spaces of quadratic diﬀerentials on Riemann surfaces.
On the recent work with Yu Qiu, we introduced the notion of q-stabilityconditions and showed that the space of q-stability conditions on a certain
double graded triangulated category constructed from a marked borderedsurface can be identified with the moduli space of q-quadratic diﬀerentials(which are multi-valued quadratic diﬀerentials with certain conditions) on thecorresponding Riemann surface. This result looks Calabi-Yau “s” analogue(s is a complex number) of the work of BS for Calabi-Yau 3 categories andalso clarifies the relationship between the works of BS and HKK. In the talk,I will introduce about this work.
Alexander Kasprzyk
Mirror symmetry for Fano manifolds
We will review the approach being developed by Coates, Corti, Kasprzyk,and others to potentially allow for the classification of Fano manifolds via
mirror symmetry. We explain what we can do so far, and what we hope to beable to do in the near future. The focus will be on the combinatorial aspectsof the theory.
Yota Shamoto
Hodge structures on tame compactified Landau-Ginzburg models
A tame compactified Landau-Ginzburg model is a pair (X, f ) of a smoothprojective complex variety X, and a flat projective morphism f from X to theprojective line satisfying some conditions. In this talk, we will consider several kinds of Hodge structures associated with this model. In particular, wediscuss the relation to some conjectures proposed by Katzarkov-KontsevichPantev.

Maxim Smirnov
Quantum cohomology and derived categories of rational homogeneous
spacesAs the title suggests these lectures are devoted to the quantum cohomology and derived categories of coherent sheaves of rational homogeneous
spaces G/P . I will begin by a gentle introduction to quantum cohomology
and derived categories and illustrate both in some simple examples. Then wewill discuss some results on the structure of the small and big quantum cohomology of a rational homogeneous space G/P (e.g. (non-)semisimplicity),and some results the structure of the derived category of coherent sheaves(e.g. existence of full exceptional collections). Finally, we are going to relate both sides via Dubrovin s conjecture and homological mirror symmetry.
These lectures are based on papers joint with (subsets of) J.A. Cruz Morales,
S. Galkin, A. Kuznetsov, A. Mellit, N. Perrin, and some further work inprogress.
Renato Vianna
Lagrangian fibrations and open Gromov-Witten invariants
The first two talks we will address problems of symplectic topology, suchas, what is the space of Lagrangian tori modulo Hamiltonian isotopy in agiven symplectic manifold and what are the shapes of Weinstein neighbourhoods of a given Lagrangian torus. For that, we view Lagrangian tori as fibreof almost toric fibrations (or what we call Gelfand-Cetlin fibrations) and westudy their open Gromov-Witten invariants and their Fukaya algebra. Thiswill be based on recent work with Egor Shelukhin and Dmitry Tonkonog aswell as previous work. As an application of the recent work, we show, under certain assumptions, the unobstructedness of SYZ fibres in a symplectic
Calabi-Yau.
For the third talk we will explore how relate the superpotential of a monotone Lagrangian inside a Donaldson divisor D of X to the superpotentialof its Biran’s lift in X. The relative Gromov-Witten invariants of the pair(X, D) plays a role. Applications include proving existence on infinitely many
monotone Lagrangian tori in symplectic manifolds of arbitrary dimensions,providing relations for relative Gromov Witten invariants for some complete
intersections, and, via the relationship proven by Tonkonog between periodsof the superpotential of a monotone Lagrangian torus and quantum periods,
we recover a formula of Coates-Corti-Galkin-Kasprzyk relating quantum periods of D and X, but in a slightly diﬀerent setting.