Mirror Symmetry for Fano
Manifolds and Related Topics
(previous workshops 2014,
2015,
2016,
2017)
Venue: Room 127 (conference room), Department of Mathematics (Graduate School
of Science Bldg no.3), Kyoto Univeristy
(map, direction)
Time: 1014 December 2018
Invited Speakers (each speaker will give 23 lectures)
CheolHyun Cho
Andrew Harder
Akishi Ikeda
Alexander Kasprzyk
Yota Shamoto
Maxim Smirnov
Renato Vianna
Schedule: (pdf)

9:0010:00 
10:3011:30 
13:3014:30 
15:0016:00 
16:3017:30 
Mon 
Cho 1 
Cho 2 
Vianna 1 
Ikeda 1 

Tue 
Vianna 2 
Vianna 3 
Cho 3 
Harder 1 
Ikeda 2 
Wed 
Kasprzyk 1 
Smirnov 1 



Thu 
Harder 2 
Harder 3 
Kasprzyk 2 
Kasprzyk 3 

Fri 
Smirnov 2 
Smirnov 3 
Shamoto 1 
Shamoto 2 

Title and Abstract:
CheolHyun Cho
Lecture 1. HMS for toric Fano
manifolds
We explain how to
construct explicit Ainfinity functor for toric Fano manifold which
derives homological mirror symmetry between its Fukaya
category and matrix factorization category of its LandauGinzburg
mirror. It is based on Floer theoretic construction
which uses formal MaurerCartan elements and curved Yoneda embedding. This is a joint work with Hansol Hong and SiuCheong Lau.
Lecture 2. Mirror symmetry for orbispheres
We consider a sphere with 3 orbifold points. Its mirror is given by a cusp singularity
whose closed mirror
symmetry has been studied by Rossi, Satake,
Takahashi, Ishibashi and Shiraishi. We explain (homological) mirror symmetry
for orbispheres using Lagrangian
Floer theory. Using an immersed circle (called Seidel
Lagrangian), we define homological mirror functor as well as KodairaSpencer
map to prove (homological) mirror symmetry. This is based on joints works with
Lino Amorim, Hansol Hong
and SiuCheong Lau.
Lecture 3. Pairings and Mirror symmetry
Given a (homological) mirror symmetry
between a symplectic manifold and its LandauGinzburg 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 KapustinLi pairing and Jacobian ring has a
residue pairing. We introduce what we call, multicrescent 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 with Sangwook Lee and Hyungseok Shin.
Andrew Harder
Fibrations and mirror symmetry
My first two talks will focus on describing
how fibrations on CalabiYau manifolds are related to
degenerations under mirror symmetry. In my first talk, 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 CalabiYau threefolds and Tyurin
degenerations, lead to classification theorems analogous to Kodaira’s
classification of elliptic fibrations, for certain K3
fibered CalabiYau threefolds.
My final talk will change direction
slightly. I will discuss new conjectures, developed with Katzarkov,
Pantev, Przyjalkowski and
others, about how the weight filtration on a the cohomology
log CalabiYau variety is related to the perverse 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
qstability conditions and qquadratic
differentials
For a given triangulated category, we can
associate a complex manifold, called the space of Bridgeland
stability conditions. By the work of BridgelandSmith
(BS) and HaidenKatzarkovKontsevich
(HKK), for a certain class of triangulated categories associated with marked
bordered surfaces, the space of stability conditions on them can be identified
with the moduli spaces of quadratic differentials on Riemann surfaces.
On the recent work with Yu Qiu, we introduced the notion of qstability conditions and
showed that the space of qstability conditions on a certain double graded
triangulated category constructed from a marked bordered surface can be
identified with the moduli space of qquadratic differentials (which are
multivalued quadratic differentials with certain conditions) on the
corresponding Riemann surface. This result looks CalabiYau “s” analogue (s is
a complex number) of the work of BS for CalabiYau 3 categories and also
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 be able to do in the near future. The focus will be on the
combinatorial aspects of the theory.
Yota Shamoto
Hodge
structures on tame compactified LandauGinzburg models
A tame compactified
LandauGinzburg model is a pair (X, f) of a smooth
projective complex variety X, and a flat projective morphism f from X to the
projective line satisfying some conditions. In this talk, we will consider
several kinds of Hodge structures associated with this model. In particular, we
discuss the relation to some conjectures proposed by KatzarkovKontsevichPantev.
Maxim Smirnov
Quantum
cohomology and derived categories of rational
homogeneous spaces
As 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 we will
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 in progress.
Renato Vianna (pdf)
Lagrangian fibrations
and open GromovWitten invariants
The first two talks we will address
problems of symplectic topology, such as, what is the
space of Lagrangian tori modulo Hamiltonian isotopy
in a given symplectic manifold and what are the
shapes of Weinstein neighbourhoods of a given Lagrangian torus. For that, we view Lagrangian
tori as fibre of almost toric
fibrations (or what we call GelfandCetlin
fibrations) and we study their open GromovWitten invariants and their Fukaya
algebra. This will be based on recent work with Egor Shelukhin and Dmitry Tonkonog as
well as previous work. As an application of the recent work, we show, under
certain assumptions, the unobstructedness of SYZ fibres in a symplectic
CalabiYau.
For the third talk we will explore how
relate the superpotential of a monotone Lagrangian inside a Donaldson divisor D of X to the superpotential of its Biran's
lift in X. The relative GromovWitten 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 periods of the superpotential
of a monotone Lagrangian torus and quantum periods,
we recover a formula of CoatesCortiGalkinKasprzyk relating quantum periods of D and X, but in
a slightly different setting.
Organizers: Hiroshi
Iritani, Yukiko Konishi, Atsushi Takahashi
Acknowledgements: This workshop is supported by KibanS 16H06337 (Atsushi
Takahashi) and KibanC 16K05127(Hiroshi Iritani).