About TIMG

The main purpose of Trends in Modern Geometry is to intensively discuss recent remarkable progress in the field of geometry, especially complex/Kaehler geometry, symplectic geometry, and Riemannian geometry. Another goal of this symposium is to give guidance for beginning graduate students, and especially to motivate them to further study these areas of geometry.

Archives: TIMG 2014

Speakers

Miguel Abreu (IST, Portugal)

Xiuxiong Chen (Stony Brook)

Ronan J. Conlon (UQAM, Montreal)

Michael Entov (Technion)

Mikio Furuta (Tokyo)

Jun-Muk Hwang (KIAS)

Hiroshi Iritani (Kyoto)

Chi Li (Stony Brook)

Andre Neves (Imperial College)

Yong-Geun Oh (POSTECH)

Schedule

The symposium will be held in University of Tokyo on Monday and Tuesday, and in Nasu on Thursday and Friday.

Title and Abstract

(Occasionally Added)

**Title: "Kaehler-Sasaki metrics on toric symplectic cones" **

In this talk I will present an explicit construction of scalar-flat toric
Kaehler-Sasaki metrics on toric symplectic cones, using action-angle
coordinates and symplectic potentials. This gives rise to constant scalar
curvature Sasaki metrics on the associated smooth toric contact manifolds.
I will also discuss several examples, including generalizations of the
Sasaki-Einstein metrics on \(S^2 \times S^3\) contructed in 2004 by the
mathematical-physicists J. Gauntlett, D. Martelli, J. Sparks and D. Waldram.
This is joint work with J. Mourão, J. P. Nunes and R. Sena-Dias.

**Title: "On a "new continuity path” for constant scalar curvature Kähler metrics" **

In this talk, we will discuss a "new continuity path” which links the solution of cscK metrics
with a solution of certain 2nd fully nonlinear equation.
This is largely an expository talk where we explain various aspects of geometric and analysis
centered around this "new path”.

**Title: "Asymptotically Conical Calabi-Yau manifolds" **

Asymptotically Conical (AC) Calabi-Yau manifolds are Ricci-
flat Kähler manifolds that are modelled on a Ricci-flat Kähler cone at
infinity. I will describe a method to determine every AC Calabi-Yau
manifold modelled on some given Ricci-flat Kähler cone. I will then
present some examples. This is joint work with Hans-Joachim Hein (
University of Maryland).

**Title: "Unobstructed symplectic packing for tori" **

I will discuss why any finite collection of disjoint (not
necessarily equal) standard symplectic balls admits a symplectic
embedding to an even-dimensional torus \(T\) equipped with a Kähler form, as
long as their total volume is less than the volume of \(T\). The proof uses
a number of powerful rigidity results from complex geometry. If the
cohomology class of the Kähler form on \(T\) is not proportional to a
rational one, a similar claim holds also for symplectic embeddings of
any number of equal polydisks. The proof of the latter result involves
Ratner's orbit closure theorem.

This is a joint work with M.Verbitsky.

**Title: "\(10/8\)-type inequalities and TQFT" **

Manolescu and Lin gave \(10/8\)-type inequalities for spin \(4\)-manifolds
with boundary.
In this talk I formulate the inequalities in terms of a \(3+1\) TQFT.
I announced the formulation two years ago, but there was a gap which I
did not realize at that moment. Due to works of Khandhawit and
Sasahira, the gap is now filled in. I would also like to explain how
to extend the TQFT for a family of \(3+1\) manifolds. Joint work with T.J. Li.

**Title: "Cartan-Fubini type extension theorems" **

Cartan-Fubini type extension theorems give various settings where local structure-preserving maps
can be extended to global holomorphic maps. They can be viewed as holomorphic generalizations of
Liouville's theorem in conformal geometry. We will give an introductory survey o f recent progress on this topic.

**Title: "Constructing mirrors via shift operators" **

In this talk, I will explain a mirror construction for
big equivariant quantum cohomology of toric varieties via
shift operators of equivariant parameters. Shift operators
in equivariant quantum cohomology have been introduced
in the work of Braverman, Maulik, Okounkov and Pandharipande
and can be viewed as an equivariant lift of the Seidel representation.
These operators naturally define a mirror Landau-Ginzburg potential
together with a primitive form and yield an almost tautological proof
of toric mirror symmetry. They are also closely related to the
Gamma structure in quantum cohomology.

**Title: "Moduli space of smoothable Kähler-Einstein Q-Fano varieties" **

The study of moduli spaces of polarized projective varieties
is an important subject in complex algebraic geometry. The classical
examples are the moduli spaces of Riemann surfaces, from which we know
that situations differ depending on the genus or the positivity
property of the canonical class. I will discuss the higher dimensional
generalizations and focus on the construction of moduli spaces of
Q-smoothable Kähler-Einstein Q-Fano varieties. I will then discuss the
projectivity of moduli spaces, and prove in particular the
quasi-pojectivity of moduli spaces of Kähler-Einstein Fano manifolds.
Note that here "Kähler-Einstein" can be replaced by "K-polystable" by
the recent resolution of Yau-Tian-Donaldson conjecture. This talk is
based on the joint work with Xiaowei Wang and Chenyang Xu.

**Title: "Morse index and Geometry" **

The relation between the Morse index and topology has been very fruitful over the last 100 years.
I will survey some of these developments concerning the theory of geodesics
and minimal surfaces and also talk about some new results.
This is joint work with Fernando Marques.

**Title: "Bulk deformations, tropicalizations and non-displaceable Lagrangian toric fibers" **

In this talk, I will first explain how we can deform Lagrangian
Floer homology by cycles from the given ambient symplectic manifold, and apply this to
Lagrangian torus fibers of toric manifolds. Then we will characterize what FOOO
call `bulk-balanced Lagrangian torus fibers' as the intersections of certain
collection of tropical curves selected purely in terms of the associated moment polytope.
This talk is based on a joint work with Fukaya-Ohta and Ono (for the first part), and also on the work of
my two graduate students, Yoosik Kim and Jaeho Lee (for the latter part).

**Title: "Localization method and isoperimetric inequalities" **

This talk will be a survey on recent extensions of the “localization method”
in convex geometry to curved spaces.
The localization is a classical method reducing (isoperimetric and functional)
inequalities on \(n\)-dimensional spaces to those on \(1\)-dimensional spaces.
Klartag (arXiv:1408.6322) generalized this method to Riemannian manifolds
and gave a beautiful alternative proof of Lévy-Gromov’s isoperimetric inequality
without relying on the regularity theorem in geometric measure theory.
In a different way directly along optimal transport theory,
Cavalletti and Mondino (arXiv:1502.06465) further generalized
the localization method to non-branching metric measure spaces satisfying
the curvature-dimension condition. This class contains limits of Riemannian manifolds,
Alexandrov spaces and reversible Finsler manifolds, for all of those the
Lévy-Gromov type isoperimetric inequality was previously unknown.
We also discuss non-reversible Finsler manifolds studied by the speaker (arXiv:1506.05876).

**Title: "Potential of Takhtajan-Zograf Kähler form" **

In 1991, Takhtajan and Zograf calculated the first Chern form of the determinant line bundle on
Teichmüller space for Riemann surface with punctures. They discovered that the first Chern form is given by
the Weil-Petersson Kähler form and additional new Kähler form. This new form originates from punctures of
Riemann surface and is now called Takhtajan-Zograf Kähler form.
In this talk, I will explain a construction of potential of the Takhtajan-Zograf Kähler form, which naturally combines
with a potential of Weil-Petersson Kähler form given by the classical Liouville action functional.
This is a joint work with Takhtajan and Teo.

**Title: "K-stability and parabolic stability" **

Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces.
We show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature
Kähler metrics on the iterated blowup are equivalent, for certain polarizations close to the boundary of
the Kähler cone.
This proves a version of the Donaldson-Tian-Yau conjecture for this particular class of complex ruled surfaces.

**Title: "Tian's properness conjectures" **

In the 90's, Tian introduced a notion of properness in the space of Kähler metrics
in terms of Aubin's nonlinear Dirichlet energy and Mabuchi's K-energy
and put forward several conjectures on the relation between properness
and Kähler-Einstein metrics.
In joint work with T. Darvas we disprove one of these conjectures, and prove the remaining ones.
Our results extend to Kähler-Einstein edge metrics and Kähler-Ricci solitons.
Moreover, we formulate a corresponding conjecture for constant scalar curvature metrics
and reduce it to purely PDE problem of regularity of minimizers of the K-energy
in a certain Finsler metric completion of the space of Kähler metrics introduced by Darvas.

**Title: "Residue mirror symmetry for Grassmannians" **

Toric Residue Mirror Symmetry of Batyrev and Materov
relates the generating functions of intersection numbers
on certain compactifications of the spaces of maps
into toric varieties with residues of some rational functions.
Based on recent progress on sphere partition functions
of A-twisted gauged linear sigma models,
we formulate a generalization of this conjecture
to non-abelian GIT quotients of vector spaces,
which is closely related to Vafa-Intriligator formula,
Verlinde algebra, Bethe/gauge correspondence, and so on.
This is a joint work with Yutaka Yoshida.

**Title: "Critical metrics on the blow-up of \(CP^2\)" **

A construction of critical metrics on the connected sum of
Einstein manifolds was given in joint work of the author with Matt
Gursky. In this talk, I will give a more precise description of the
moduli space of metrics obtained in the simplest case, that of the one
point blow-up of \(CP^2\). This is done by finding the next coefficient in
the expansion of the Kuranishi map for this gluing problem. The
computation is quite non-trivial, since (i) this coefficient is
fundamentally a global geometric invariant and is not determined alone
by local geometric invariants, and (ii) this coefficient depends on the
nonlinear structure of the critical equation. There is an interesting
connection between these critical metrics, and a family of constant
scalar curvature Kähler metrics with edge-cone singularities along two
divisors.

**Title: "A higher order mass and hyperbolic Alexandrov-Fenchel inequalities" **

With a higher order scalar curvature, the Gauss-Bonnet-Chern Curvature,
we introduced a higher order mass for asymptotically flat and
asymptotically hyperbolic manifolds. This mass is closely related to
many geometric inequalities, one of them is the Alexandrov-Fenchel
inequality in the hyperbolic space, which we want also to discuss
in this talk. The talk is based on the joint work with Yuxin Ge (Paris),
Jie Wu (Hangzhou) and also Chao Xia (Xiamen).

**Title: "Analytic torsion for K3 surfaces with involution" **

In 2004, I introduced a holomorphic torsion invariant of K3 surfaces with involution
and proved its automorphy viewed as a function on the moduli space.
Very recently, its explicit formula is completely determined. It is expressed as
the product of an explicit Borcherds lift and a classical Siegel modular form.
In my talk, I will report this progress. This is a joint work with Shouhei Ma.

**Title: "Asymptotic behavior of positively curved steady Ricci Solitons" **

We will discuss the behavior of \(\kappa\)-noncollapsed and positively curved steady Ricci solitons.
As an application, we prove that any \(2\) dimensional \(\kappa\)-noncollapsed steady
Kähler-Ricci soliton with non-negative sectional curvature must be a flat metric on \(\mathbb C^2\).
The same result holds for higher dimensions if in addition the scalar curvature decays uniformly.
This is a joint work with Yuxing Deng.

Access

The symposium will be held at Lecture Hall, Graduate School of Mathematical Sciences Building in
University of Tokyo (Komaba Campus) ,
Campus Map (Bldg no.53 in PDF ).

For some participants (especially, for foreign participants), accommodation will be booked at Tokyu Stay Shibuya (東急ステイ渋谷)
@GoogleMap.
Here are route maps:
Narita Airport ↔ Shibuya (渋谷) Station , and
Shibuya (渋谷) Station → the hotel (10 min by walk).
Search Site Train Route Finder is available.

**From hotel to university (less than 1 km):**
The gate closest to the Math Sci Build is the east gate (Suiji gate, no.6 in PDF ).
See the map below.

The last half of the symposium will be held at Sunvalley Resort Nasu (ホテルサンバレー那須) .

Akito Futaki (Tokyo)

Nobuhiro Honda (Tokyo Tech)

Toshiki Mabuchi (Osaka)

Hajime Ono (Saitama)

Yuji Sano (Kumamoto)

Jeff Viaclovsky (Madison)