Primitive Forms, Mirror Symmetry and Related Topics 2014
Date: 24 Dec 2014 – 26 Dec 2014
Place: Room 127 (Building no.3), Department of Mathematics, Kyoto University
10:30 – 11:30 11:45 – 12:45
14:30 – 15:30 15:45 – 16:45 17:00 – 18:00
24 OhtaSandalunch Kajiura
Sugiyama
Ono
9:30 – 10:30
25 K. Saito
26 Ueda11:00 – 12:00
14:00 – 15:00 15:15 – 16:15 16:30 – 17:30
Horilunch Yoshinaga
Si LiMilanovTodalunch Bondal
UeharaM. Saito

24 December
10:30 – 11:30 Hiroshi Ohta
11:45 – 12:45 Fumihiko Sanda14:30 – 15:30 Hiroshige Kajiura
15:45 – 16:45 Satoshi Sugiyama
17:00 – 18:00 Kaoru Ono

Trace map in cyclic A∞ category
Fukaya categories and blowups
More on A∞ enhancements of triangulated
categories
On the Fukaya-Seidel categories of surface Lefschetz
fibrations
Idempotents in quantum cohomology andnon-dispaceable Lagrangian submanifolds

25 December
9:30 – 10:30 Kyoji Saito
11:00 – 12:00
14:00 – 15:00
15:15 – 16:15
16:30 – 17:30

Coherency of direct images of de-Rham complex
by open holomorphic map
Kentaro Hori
Grade restriction rule in Rødland model
Masahiko Yoshinaga Primitive forms and the Kostant-Kirillov form
Si Li
Mirror theorem between Landau-Ginzburg models
Todor Milanov
Mirror symmetry and the global ancestor potential

26 December
9:30 – 10:30 Kazushi Ueda
11:00 – 12:00 Yukinobu Toda
14:00 – 15:00 Alexey Bondal
15:15 – 16:15 Hokuto Uehara
16:30 – 17:30 Morihiko Saito

On a certain generalization of triangle singularities
Non-commutative width and Gopakumar-Vafa
invariants
The structure of the null-category for birational
morphisms of reldim 1Autoequivalences of derived categories of elliptic
surfaces with non-zero Kodaira dimensions.
On the structure of Brieskorn lattices.

This workshop is supported by JSPS Grant-in-Aid for Challenging Exploratory Research 26610008 and Grant-in-Aid for Young Scientists (A) 20684003 (Takahashi).
Organizer: Hiroshi Iritani, Yukiko Konishi, Astushi Takahashi
12

Abstracts
Hiroshi Ohta
Title: Trace map in cyclic A∞ category
Abstract: Based on my joint work with Fukaya, Oh, Ono, I will introduce the trace
map in cyclic A∞ category as one-loop partition function, and discuss some aspect of
the universal deformation of the potential function in Lagrangian Floer theory for toriccase.
Fumihiko Sanda
Title: Fukaya categories and blowups
Abstruct: In this talk, we study about the behavior of Fukaya categories under blowups
by using SYZ mirror symmetry. We also explain how to extract information about
quantum cohomology from Fukaya categories.
Hiroshige Kajiura
Title: More on A∞ enhancements of triangulated categories.
Abstract: In this talk we concentrate on triangulated categories generated by exceptional collections and discuss various classes of such triangulated categories whose Ainﬁnity enhancements are unique or nonunique up to A-inﬁnity equivalences. We alsomention geometric examples corresponding to some of these triangulated categories.
Satoshi Sugiyama
Title: On the Fukaya-Seidel categories of surface Lefschetz fibrations
Abstract: In this talk, we study about the Fukaya-Seidel category of exact Lefschetz
ﬁcrations that the dimension of the ﬁbre is two and possible application to low dimension
topology. In such a case, the derived Fukaya-Seidel category is independent of the choiceof the exact symplectic structure and can be computed in combinatorial way. I will
introduce some computational examples and some problems.
Kaoru Ono
Title: Idempotents in quantum cohomology and non-dispaceable Lagrangian submanifolds
Abstract: In a joint work with Fukaya, Oh and Ohta, we gave a criterion for a Lagrangian submanifold to be heavy/superheavy with respect to a partial symplectic
quasi-state associated with an idempotent of quantum cohomology. It ﬁts quite wellwith a generation criterion of Fukaya category, which we obtained with Abouzaid. In
this talk, I would like to explain our criterion. In particular, I would like to consider thefollowing example. The quantum cohomology of the symplectic blow-up of a symplectically aspherical manifold (X, ω) contains a ﬁeld as a direct summand. The unit of theﬁeld is an idempotent of quantum cohomology. If the size of the symplectic blow-up isless than a constant C(X, ω), we can ﬁnd a superheavy Lagrangian torus with respect
to the symplectic quasi state associated with this idemponent.3Kyoji Saito
Title: Coherency of direct images of de-Rham complex by open holomorphic map.
Abstract: There are two formulations of Coherence Theorem for proper morphisms.
One: an algebraic proof for projective morphisms by Grothendieck, and the other: ancomplex analytic proof for proper holomorphic morphisms by Grauert. The latter uses
functional analysis of the Fréchet spaces of holomorphic functions on Stein-manifolds,
but was somehow forgotten. However, the advantage is that its technique is applicable
to non-proper morphisms. I’ll explain how the technique is used to show a cohenencyof certain relatice de-Rham cohomology groups for some Stein morphisms based on an
unpublished old result of the speaker.
Kentaro Hori
Title: Grade Restriction Rule in Rødland Model
Abstract: The talk is about D-branes in “Rødland model”, a gauged linear sigma
model with two geometric phases — Pfaﬃan phase and Grassmannian phase, as wellas in another model (“Hosono-Takagi model”) with the similar feature. The ”grade
restriction rule” for each window between the two phases is presented. This yields
the rule of D-brane transport through the windows and in particular the monodromy
around the singular points. The hemisphere partition function is used to ﬁnd the rule.
Based on a joint work with Richard Eager, Johanna Knapp and Mauricio Romo.
Masahiko Yoshinaga
Title: Primitive forms and the Kostant-Kirillov form.
Abstract: The Weyl group quotient of the Cartan subalgebra for a simple Lie algebraof type ADE has the ﬂat structures. There are two ways to describe them. One is by
the general framework of primitive forms (based on the fact that the Kostant-Kirillov
form can be considered as a primitive form). The other is by the invariant-theoretic
method. In this talk, we will develop an invariant-theoretic way of computing GaussManin diﬀerentiation of the Kostant-Kirillov form, and then discussing the relationshipbetween the above two constructions.
Si Li
Title: Mirror Theorem between Landau-Ginzburg models.
Abstract: In this talk, I will prove the Landau-Ginzburg mirror symmetry conjecture
for general quasi-homogenous singularities, i.e., the FJRW theory (LG A-model) of
such polynomials is equivalent to the Saito-Givental theory (LG B-model) of the mirror
polynomial. This is joint work with Weiqiang He, Rachel Webb and Yefeng Shen.

4Todor Milanov
Title: Mirror symmetry and the global ancestor potential.
Abstract: I’m planning to talk about a joint work in progress with H. Iritani, Y.
Ruan, and Y. Shen. Using Givental’s higher genus reconstruction and Saito’s theoryof primitive forms we deﬁne a global total ancestor potential as a function on the total
space of a certain line bundle with values in a Fock space. I will discuss also theapplications of this construction to mirror symmetry.
Kazushi Ueda
Title: On a certain generalization of triangle singularities
Abstract: Triangle singularities are Fuchsian singularities associated with von Dyck
groups, which are index two subgroups of Schwarz triangle groups. Hypersurface triangle singularities are classiﬁed by Dolgachev, and give 14 exceptional unimodal singularities classiﬁed by Arnold. In the talk, we will discuss a joint work with Kenji Hashimotoand Hwayoung Lee on a higher-dimensional generalization of triangle singularities. Inparticular, we show that there are only ﬁnitely many hypersurface singularities of thistype in each dimension, and give a complete list in dimension 3.
Yukinobu Toda
Title: Non-commutative width and Gopakumar-Vafa invariants.
Abstract: I show that the non-commutative widths for ﬂopping curves on smooth 3-folds
introduced by Donovan-Wemyss are described by Katz’s genus zero Gopakumar-Vafa
invariants.
Alexey Bondal
Title: The structure of the null-category for birational morphisms of reldim 1.
Abstract: We will give an algebraic description for the null-category of a birational
morphism between smooth surfaces.
Hokuto Uehara
Title: Autoequivalences of derived categories of elliptic surfaces with non-zero Kodairadimensions.
Abstract: We study the group of autoequivalences of the derived categories of coherent
sheaves on smooth projective elliptic surfaces with non-zero Kodaira dimensions. We
ﬁnd a description of it when each reducible ﬁber is a cycle of (−2)-curves.
Morihiko Saito
Title: On the structure of Brieskorn lattices.
Abstract: We plan to explain the structure of Brieskorn lattices related to primitive
forms including the formal setting recently studied by C. Li, S. Li, and K. Saito.