*Algebraic
Geometry in East Asia*

*Online Seminar*

This
is a joint effort of many algebraic geometers in East Asia. We aim to create a
platform for algebraic geometers and students for further interaction and
cooperation.

The
seminar will take place on Friday at GMT 1:00 or GMT 7:00, unless otherwise
specified, in order to accommodate most participants in East Asia.

To
receive the announcement and the Zoom password, please send an empty mail with
the title "Subscription" to the following address:

**ageaseminar
A@T gmail.com**

This is a mirror
site of

https://sites.google.com/ncts.ntu.edu.tw/agea-seminar

Last updated 2022/2/7

Upcoming Talks:

**2022/02/11 GMT 7:00-8:00**

**Speaker:** Hsin-Ku
Chen (NTU)

**Title:** Classification
of three-dimensional terminal divisorial contractions to curves

**Abstract: **We classify all
divisorial contractions to curves between terminal threefolds by describing
them as weighted blow-ups. This is a joint work with Jungkai Alfred Chen and
Jheng-Jie Chen.

**2022/02/11 GMT 8:15-9:15**

**Speaker:** Iacopo Brivio (NCTS)

**Title:** Invariance
of plurigenera in positive and mixed characteristic

**Abstract:** A famous
theorem of Siu states that the m-plurigenus P_m(X) of a complex projective
manifold is invariant under deformations for all m\geq 0. It is well-known that
in positive or mixed characteristic this can fail for m=1. In this talk I will
construct families of smooth surfaces over a DVR X/R such that
P_m(X_k)>>P_m(X_K) for all m>0 divisible enough. If time permits, I
will also explain how the same ideas can be used to prove (asymptotic) deformation
invariance of plurigenera for certain families of threefold pairs in positive
and mixed characteristic.

**Other upcoming talks:**

**Date Time Speaker **** **

2022/02/11 GMT 7:00-8:00 Hsin-Ku
Chen (NTU)

2022/02/11 GMT 8:15-9:15 Iacopo
Brivio (NCTS)

2022/02/25 GMT 7:00-8:00 Quoc
Ho (Institute of Sci. & Tech., Austria)

2022/02/25 GMT 8:15-9:15 Jinhyun
Park (KAIST)

**Organizers:** Yujiro Kawamata (Tokyo), Xiaotao
Sun (Tianjin), JongHae Keum (KIAS, Seoul), Jungkai Chen (NTU, Taipei), Conan
Nai Chung Leung (CUHK, Hong Kong), Phung Ho Hai (VAST, Hanoi), De Qi Zhang
(NUS, Singapore), Yusuke Nakamura (Tokyo), Baohua Fu (CAS, Beijing), Kiryong
Chung (Kyungpook National Univ., Daegu), Hsueh-Yung Lin (IPMU, Tokyo)

**Archive:**

**2022/01/14 GMT 7:00-8:00**

**Speaker: **Kien Huu
Nguyen (KU Leuven, Belgium)** (video)**

**Title: **Exponential
sums modulo p^m for Deligne polynomials

**Abstract: **

**2022/01/14 GMT 8:15-9:15**

**Speaker: **Xuan Viet Nhan Nguyen (BCAM, Spain)
**(video)**

**Title: **Moderately
discontinuous homology and Lipschitz normal embeddings

**Abstract: **In this talk, we will
present a simple example showing that for homomorphisms between MD-homologies
induced by the identity map, being isomorphic is not enough to ensure that the
given germ is Lipschitz normally embedded. This is a negative answer to the
question asked by Bobadilla et al. in their paper about Moderately
Discontinuous Homology.

**2021/12/31 GMT 7:00-8:00**

**Speaker:
**Lei Wu (KU Leuven, Belgium) **(video)**

**Title: **D-modles,
motivic integral and hypersurface singularities

**Abstract: **This talk is an
invitation to the study of monodromy conjecture for hypersurfaces in complex
affine spaces. I will recall two different ways to understand singularities of
hypersurfaces in complex affine spaces. The first one is to use D-modules to
define the b-function (also known as the Bernstein-Sato polynomial) of a polynomial
(defining the hypersurface). The other one uses motivic integrals and
resolution of singularities to obtain the motivic/topological zeta function of
the hypersurface. The monodromy conjecture predicts that these two ways of
understanding hypersurface singularities are related. Then I will discuss some
known cases of the conjecture for hyperplane arrangements.

**2021/12/31 GMT 8:15-9:15**

**Speaker: **Wenhao Ou (AMSS, CAS) **(video)**

**Title: **Projective varieties with
strictly nef anticanonical divisor

**Abstract: **A conjecture of
Campana-Peternell presumes that, if the anticanonical divisor of a projective
variety X has strictly positive intersection with all curves, then the manifold
is Fano. We show that if X is klt, then it is rationally connected. This
provides an evidence to the conjecture. Furthermore, if the dimension is at
most three, then we prove that X is Fano. This is joint with Jie Liu, Juanyong
Wang, Xiaokui Yang and Guolei Zhong.

**2021/12/17 GMT 7:00-8:00**

**Speaker: **Shuai Guo
(Peking University) **(video)**

**Title: **Structure of higher genus
Gromov-Witten invariants of the quintic threefolds

**Abstract: **The computation of the Gromov-Witten (GW) invariants of the
compact Calabi Yau 3-folds is a central and yet difficult problem in geometry
and physics. In a seminal work in 1993, Bershadsky, Cecotti, Ooguri and Vafa
(BCOV) introduced the higher genus B-model in physics. During the subsequent
years, a series of conjectural formulae was proposed by physicists based on the
BCOV B-model, which effectively calculates the higher genus GW potential from
lower genus GW potentials and a finite ambiguity. In this talk, we will
introduce some recent mathematical progresses in this direction. This talk is
based on the joint works with Chang-Li-Li and the joint works with Janda-Ruan.

**2021/11/19 GMT 7:00-8:00**

**Speaker: **Keiji
Oguiso (University of Tokyo) **(video)**

**Title: **Smooth
complex projective rational varieties with infinitely many real forms

**Abstract:**

This is a
joint work with Professors Tien-Cuong Dinh and Xun Yu.

The real
form problem asks how many different ways one can describe a given complex
variety by polynomial equations with real coefficients up to isomorphisms over
the real number field. For instance, the complex projective line has exactly
two real forms up to isomorphisms. This problem is in the limelight again after
a breakthrough work due to Lesieutre in 2018.

In this
talk, among other relevant things, we would like to show that in each dimension
greater than or equal to two, there is a smooth complex projective rational
variety with infinitely many real forms. This answers a question of Kharlamov
in 1999.

**2021/11/19 GMT 8:15-9:15**

**Speaker: **Yuki
Hirano (Kyoto University) **(video)**

**Title: **Equivariant
tilting modules, Pfaffian varieties and noncommutative matrix factorizations

**Abstract:**

It is known
that a tilting bundle T on a smooth variety X induces a derived equivalence of coherent
sheaves on X and finitely generated modules over the endomorphism algebra
End(T). We prove that, in a suitable setting, a tilting bundle also induces an
equivalence of derived matrix factorization categories. As an application, we
show that the derived category of a noncommutative resolution of a linear
section of a Pfaffian variety is equivalent to the derived matrix factorization
category of a noncommutative gauged Landau-Ginzburg model.

**2021/10/08 GMT 7:00-8:00**

**Speaker: **Christian Schnell (Stony Brook University) **(video)**

**Title: **Finiteness for self-dual classes in
variations of Hodge structure

**Abstract:**

I will talk about a new finiteness theorem
for variations of Hodge structure. It is a generalization of the
Cattani-Deligne-Kaplan theorem from Hodge classes to so-called self-dual (and
anti-self-dual) classes. For example, among integral cohomology classes of
degree 4, those of type (4,0) + (2,2) + (0,4) are self-dual, and those of type
(3,1) + (1,3) are anti-self-dual. The result is suggested by considerations in
theoretical physics, and the proof uses o-minimality and the definability of
period mappings. This is joint work with Benjamin Bakker, Thomas Grimm, and
Jacob Tsimerman.

**2021/10/08 GMT 8:15-9:15**

**Speaker: **Nguyen-Bac Dang (Université Paris-Saclay) **(video)**

**Title: **Spectral interpretations of dynamical
degrees

**Abstract:**

This talk is based on a joint work with
Charles Favre. I will explain how one can control the degree of the iterates of
rational maps in arbitrary dimension by applying method from functional
analysis. Namely, we endow some particular norms on the space of b-divisors and
on the spaces of b-classes and study the eigenvalues of the pullback operator
induced by a rational map.

**2021/9/24 GMT 7:00-8:00**

**Speaker: **Kenta Hashizume (University of Tokyo) **(video)**

**Title:** Adjunction and inversion of adjunction

**Abstract:**

Finding a relation between
singularities of a variety and singularities of subvarietes is a natural
problem. An answer to the problem, called adjunction and inversion of
adjunction for log canonical pairs, plays a critical role in the recent
developments of the birational geometry. In this talk, I will introduce a
generalization of the result, that is, adjunction and inversion of adjunction
for normal pairs. This is a joint work with Osamu Fujino.

**2021/9/24
GMT 8:15-9:15**

**Speaker: **Takuzo Okada (Saga University) **(video)**

**Title:** Birational geometry of sextic double solids
with cA points

**Abstract:**

A sextic double solid is a Fano
3-fold which is a double cover of the projective 3-space branched along a
sextic surface. Iskovskikh proved that a smooth sextic double solid is
birationally superrigid, that is, it does not admit a non-biregular birational
map to a Mori fiber space. Later on Cheltsov and Park showed that the same
conclusion holds for sextic double solids with ordinary double points. In this
talk I will explain birational (non-)superrigidity of sextic double solids with
cA points. This talk is based on a joint work with Krylov and Paemurru.

**2021/09/10 GMT
1:00-2:00**

**Speaker: **Kyoung-Seog
Lee (Miami University)** (****video****)**

**Title:**
Derived categories and motives of moduli spaces of vector bundles on curves

**Abstract: **Derived
categories and motives are important invariants of algebraic varieties invented
by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured
that they will be closely related and now there are several evidences
supporting his conjecture. On the other hand, moduli spaces of vector bundles
on curves provide attractive and important examples of algebraic varieties and
there have been intensive works studying them. In this talk, I will discuss
derived categories and motives of moduli spaces of vector bundles on curves.
This talk is based on several joint works with I. Biswas, T. Gomez, H.-B. Moon
and M. S. Narasimhan

**2021/09/10 GMT
2:15-3:15**

**Speaker:** Insong
Choe (Konkuk University)** (****video****)**

**Title: **Symplectic
and orthogonal Hecke curves

**Abstract:** A
Hecke curve is a rational curve on the moduli space $SU_C(r,d)$ of vector
bundles over an algebraic curve, constructed by using the Hecke transformation.
The Hecke curves played an important role in Jun-Muk Hwang's works on the
geometry of $SU_C(r,d)$. Later, Xiaotao Sun proved that they have the minimal
degree among the rational curves passing through a general point. We construct
rational curves on the moduli spaces of symplectic and orthogonal bundles by
using symplecitic/orthogonal version of Hecke transformation. It turns out that
the symplectic Hecke curves are special kind of Hecke curves, while the
orthogonal Hecke curves have degree $2d$, where $d$ is the degree of Hecke
curves. Also we show that those curves have the minimal degree among the
rational curves passing through a general point. This is a joint work with
Kiryong Chung and Sanghyeon Lee.

**2021/09/03 GMT
1:00-2:00**

**Speaker:** Jingjun
Han (Johns Hopkins University)** (****video****)**

**Title:**
Shokurov's conjecture on conic bundles with canonical singularities

**Abstract: **A
conic bundle is a contraction $X\to Z$ between normal varieties of relative
dimension $1$ such that the anit-canonical divisor is relatively ample. In this
talk, I will prove a conjecture of Shokurov which predicts that, if $X\to Z$ is
a conic bundle such that $X$ has canonical singularities, then base variety $Z$
is always $\frac{1}{2}$-lc, and the multiplicities of the fibers over
codimension $1$ points are bounded from above by $2$. Both values $\frac{1}{2}$
and $2$ are sharp. This is a joint work with Chen Jiang and Yujie Luo.

**2021/09/03 GMT
2:15-3:15**

**Speaker:** Jia
Jia (National University of Singapore)** (****video****)**

**Title: **Surjective
Endomorphisms of Affine and Projective Surfaces.

**Abstract:** In
this talk, we will give structure theorems of finite surjective endomorphisms of
smooth affine surfaces and normal projective surfaces. Combining with some
local dynamics and known results, we will talk about their applications to
Zariski Dense Orbit and Kawaguchi-Silverman Conjectures. These are joint work
with Takahiro Shibata, Junyi Xie and De-Qi Zhang.

**2021/08/13 ****GMT 7:00-8:00**

**Speaker:** Jihao Liu** **(University of Utah)** (****video****)**

**Title:** Minimal model program for
generalized lc pairs

**Abstract: **The theory of generalized pairs
was introduced by C. Birkar and D.-Q. Zhang in order to tackle the effective
Iitaka fibration conjecture, and has proven to be a powerful tool in birational
geometry. It has recently become apparent that the minimal model program for generalized
pairs is closely related to the minimal model program for usual pairs and
varieties. A folklore conjecture proposed by J. Han and Z. Li and recently
re-emphasized by Birkar asks whether we can always run the minimal model
program for generalized pairs with at worst generalized lc singularities. In
this talk, we will confirm this conjecture by proving the cone theorem,
contraction theorem, and the existence of flips for generalized lc pairs. As an
immediate consequence, we will complete the minimal model program for
generalized lc pairs in dimension <=3 and the pseudo-effective case in
dimension 4. This is joint work with C. D. Hacon.

**2021/08/13 ****GMT 8:15-9:15**

**Speaker:** Thomas Krämer (Humboldt-Universität
zu Berlin)** (****video****)**

**Title: **Big Tannaka groups on abelian
varieties

**Abstract:** Lawrence and Sawin have shown
that up to translation, any abelian variety over a number field contains only
finitely many smooth ample hypersurfaces with given fundamental class and good
reduction outside a given finite set of primes. A key ingredient in their proof
is that certain Tannaka groups attached to smooth hypersurfaces are big. In the
talk I will give a general introduction to Tannaka groups of perverse sheaves
on abelian varieties and explain how to determine them for subvarieties of
higher codimension (this is work in progress with Ariyan Javanpeykar, Christian
Lehn and Marco Maculan).

**2021/07/30
****GMT
1:00-2:00**

**Speaker:** Hsian-Hua Tseng** **(Ohio State
University)** (****video****)**

**Title:** Relative Gromov-Witten theory
without log geometry

**Abstract: **

We
describe a new Gromov-Witten theory of a space relative to a simple normal-crossing
divisor constructed using multi-root stacks.

**2021/07/30 ****GMT 2:15-3:15**

**Speaker:** Shusuke Otabe (Tokyo Denki
University)

**Title:** Universal triviality of the Chow group of
zero-cycles and unramified logarithmic Hodge-Witt cohomology

**Abstract:**

Auel-Bigazzi-Böhning-Graf
von Bothmer proved that if a proper smooth variety over a field has universally
trivial Chow group of zero-cycles, then its cohomological Brauer group is
trivial as well. Binda-Rülling-Saito recently prove that the same conclusion is
true for all reciprocity sheaves. For example, unramified logarithmic
Hodge-Witt cohomology has the structure of reciprocity sheaf. In this talk, I
will discuss another proof of the triviality of the unramified cohomology,
where the key ingredient is a certain kind of moving lemma. This is a joint
work with Wataru Kai and Takao Yamazaki.

**2021/07/16 ****GMT 7:00-8:00**

**Speaker:** Qingyuan Jiang (University of Edinburgh)** ****(****video****)**

**Title:** On the derived categories of Quot schemes
of locally free quotients

**Abstract: **

Quot schemes of locally
free quotients of a given coherent sheaf, introduced by Grothendieck, are
generalizations of projectivizations and Grassmannian bundles, and are closely
related to degeneracy loci of maps between vector bundles. In this talk, we
will discuss the structure of the derived categories of these Quot schemes in
the case when the coherent sheaf has homological dimension $\le 1$. This
framework not only allows us to relax the regularity conditions on various
known formulae -- such as the ones for blowups, Cayley's trick, standard flips,
projectivizations, and Grassmannain flips, but it also leads us to many new
phenomena such as virtual flips, and blowup formulae for blowups along determinantal
subschemes of codimension $\le 4$. We will illustrate the idea of proof in
concrete cases, and if time allowed, we will also discuss the applications to
the case of moduli of linear series on curves, and Brill-Noether theory for
moduli of stable objects in K3 categories.

**2021/07/16 ****GMT 8:15-9:15**

**Speaker:** Le Quy Thuong (Vietnam National
University)** (****video****)**

**Title:** The ACVF theory and motivic Milnor fibers

**Abstract:**

** **In this talk, I review recent studies
on the theory of algebraically closed value fields of equal characteristic zero
(ACVF theory) developed by Hrushovski-Kazhdan and Hrushovski-Loeser. More
precisely, I consider a concrete Grothendieck ring of definable subsets in the
VF-sort and prove the structure theorem of this ring which can be presented via
materials from extended residue field sort and value group sort. One can
construct a ring homomorphism HL from this ring to the Grothendieck ring of
algebraic varieties, from which the motivic Milnor fiber can be described in
terms of a certain definable subset in VF-sort. As applications, I sketch
proofs of the integral identity conjecture and the motivic Thom-Sebastiani
theorem using HL, as well as mention the recent work of Fichou-Yin in the same
topic.

**2021/07/02 ****GMT 7:00-8:00**

**Speaker:** Han-Bom Moon (Fordham University, New York)** (****video****)**

**Title:** Point configurations, phylogenetic trees, and dissimilarity
vectors

**Abstract:**

In 2004 Pachter and Speyer introduced
the dissimilarity maps for phylogenetic trees and asked two important questions
about their relationship with tropical Grassmannian. Multiple authors answered
affirmatively the first of these questions, showing that dissimilarity vectors
lie on the tropical Grassmannian, but the second question, whether the set of
dissimilarity vectors forms a tropical subvariety, remained opened. In this
talk, we present a weighted variant of the dissimilarity map and show that
weighted dissimilarity vectors form a tropical subvariety of the tropical
Grassmannian in exactly the way that Pachter-Speyer envisioned. This tropical
variety has a geometric interpretation in terms of point configurations on rational
normal curves. This is joint work with Alessio Caminata, Noah Giansiracusa, and
Luca Schaffler.

**2021/07/02 ****GMT 8:15-9:15**

**Speaker:** Yifei Chen (Chinese Academy of Sciences)** (video)**

**Title:** Jordan property of automorphism groups of surfaces of
positive characteristic

**Abstract:**

A classical theorem of C. Jordan
asserts the general linear group G over a field of characteristic zero is
Jordan. That is, any finite subgroup of G contains a normal abelian subgroup of
index at most J, where J is an integer only depends on the group G. J.-P. Serre
proved that the same property holds for the Cremona group of rank 2. In this
talk, we will discuss Jordan property for automorphism groups of surfaces of positive
characteristic. This is a joint work with C. Shramov.

**2021/06/18 ****GMT 7:00-8:00**

**Speaker:** Mingshuo Zhou (Tianjin University) **(****video****)**

**Title:** Moduli space of parabolic bundles over a
curve

**Abstract:**

In this talk, we will review a
program (by Narasinhan-Ramadas and Sun) on the proof of Verlinde formula by
using degeneration of moduli space of parabolic bundles over a curve. We will
also show how the degeneration argument can be used to prove F-splitting of
moduli space of parabolic bundles (for generic choice of parabolic points) over
a generic curve in positive charactersitic. This is a joint work with Professor
Xiaotao Sun.

**2021/06/18 ****GMT 8:15-9:15**

**Speaker:** Zhi Jiang (Shanghai Center for Mathematical
Sciences) **(****video****)**

**Title:** On syzygies of homogeneous varieties

**Abstract:**

We discuss some recent progress on
syzygies of ample line bundles on homogeneous varieties, including abelian
varieties and rational homogeneous varieties.

**Zhiyu Tian** (BICMR-Beijing
University) **(****video****)**

**Title:** Some conjectures about Kato homology of rationally connected
varieties and KLT singularities

**Abstract:**

A natural question about zero cycles on a variety defied over an
arithmetically interesting field is the injectivity/surjectivity of the cycle
class map. This leads to the study of a Gersten type complex defined by
Bloch-Ogus and Kato. I will present some conjectures about this complex for
rationally connected varieties and Kawamata log terminal (KLT) singularities. I
will also present some evidence for the conjectures, and explain how they fit
into a variety of conjectures about the stability phenomenon observed in
topology and number theory.

**Joao Pedro dos Santos** (Universite de Paris) **(****video****)**** **

**Title:** Group schemes from ODEs defined over a discrete valuation
ring.

**Abstract:**

Differential Galois theory has the objective to study linear
ODEs (or connections) with the help of algebraic groups. Roughly and
explicitly, to a matrix $A\in \mathrm{Mat}_n( \mathbb C(x) )$ and a
differential system $y'=Ay$, we associate a subgroup of $GL_n(\mathbb C)$, the
differential Galois group, whose function is to measure the complexity of the
solutions. There are three paths to this theory: Picard-Vessiot extensions,
monodromy representations and Tannakian categories.

If instead of working with complex coefficients we deal with a
discrete valuation ring $R$, the construction of the differential Galois groups
are less obvious and the theory of groups gives place to that of group schemes.
This puts forward the Tannakian approach and relevant concepts from algebraic
geometry like formal group schemes and blowups. In this talk, I shall explain
how to associate to these differential equations certain flat $R$-group
schemes, what properties these may have--what to expect from a group having a
generically faithful representation which becomes trivial under
specialisation?--and how to compute with the help of the analytic method of
monodromy. The talk is a horizontal report on several works done in
collaboration with P.H.Hai and his students N.D.Duong and P.T.Tam over the past
years.

**Yuji Odaka** (Kyoto University)** ****(****video****)**

**Title:** On (various) geometric compactifications of moduli of K3
surfaces

**Abstract:**

What we mean by “geometric compactifications” in the title is it
still parametrizes “geometric objects” at the boundary. In algebraic geometry,
it is natural to expect degenerate varieties as such objects. For the moduli of
polarized K3 surfaces (or K-trivial varieties in general) case, it is
natural to expect slc and K-trivial degenerations, but there are many such
compactifications for a fixed moduli component, showing flexibility / ambiguity
/ difficulty of the problem. This talk is planned to mainly focus the
following. In K3 surfaces (and hyperKahler varieties), there is a canonical
geometric compactification whose boundary and parametrized objects are Not
varieties but tropical geometric or with more PL flavor. This is ongoing joint
work with Y.Oshima (cf., arXiv:1810.07685, 2010.00416).

In general, there is a canonical PARTIAL compactification
(quasi-projective variety) of moduli of polarized K-trivial varieties
(essentially due to Birkar and Zhang), as a completion with respect to the
Weil-Petersson metric. This is characterized by K-stability.

**YongJoo Shin** (Chungnam National
University) **(****video****)**

**Title:** Complex minimal surfaces of general type with p_{g}= 0 and K^{2 }= 7 via bidouble covers

**Abstract:**

Let S be a minimal surface of general type with p_{g}(S) = 0 and K^{2}_{S} = 7 over the field of complex
numbers. Inoue firstly constructed such surfaces S described as Galois Z_{2}×Z_{2}-covers over the four-noda cubic surface. Chen later found different
surfaces S constructed as Galois Z_{2}×Z_{2}-covers over six nodal del Pezzo surfaces of degree one. In this talk
we construct a two-dimensional family of surfaces S different from ones by
Inoue and Chen. The construction uses Galois Z_{2}×Z_{2}-covers over rational surfaces
with Picard number three, with eight nodes and with two elliptic fibrations.
This is a joint work with Yifan Chen.

**Yi Gu** (Suzhou University)** ****(****video****)**

**Title: **On the equivariant
automorphism group of surface fibrations

**Abstract:**

Let f:X→C be a relatively minimal surface fibration with smooth
generic fibre. We will discuss the finiteness of its equivariant automorphism
group, which is the group of pairs {ÎAut(X)´Aut(C)|} with natural group law. We
will give a complete classification of those surface fibrations with infinite
equivariant automorphism group in any characteristic. As an application, we
will show how this classification can be used to study the bounded subgroup
property and the Jordan property for automorphism group of algebraic surfaces.

**Takehiko Yasuda **(Osaka University) **(****video****)**

**Title:** On the isomorphism problem of projective schemes

**Abstract:**

I will talk about the isomorphism problem of projective schemes;
is it algorithmically decidable whether or not two given projective (or, more
generally, quasi-projective) schemes, say over an algebraic closure of Q, are
isomorphic? I will explain that it is indeed decidable for the following
classes of schemes: (1) one-dimensional projective schemes, (2) one-dimensional
reduced quasi-projective schemes, (3) smooth projective varieties with either
the canonical divisor or the anti-canonical divisor being big, and (4) K3
surfaces with finite automorphism group. Our main strategy is to compute Iso
schemes for finitely many Hilbert polynomials. I will also discuss related
decidability problems concerning positivity properties (such as ample, nef and
big) of line bundles.

**Kuan-Wen Lai **(University of
Massachusetts Amherst) **(****video****)**

**Title: **On the irrationality of
moduli spaces of K3 surfaces

**Abstract:**

As for moduli spaces of curves, the moduli space of polarized K3
surfaces of genus g is of general type and thus is irrational for g
sufficiently large. In this work, we estimate how the irrationality grows with
g in terms of the measure introduced by Moh and Heinzer. We proved that the
growth is bounded by a polynomial in g of degree 15 and, for three sets of
infinitely many genera, the bounds can be refined to polynomials of degree 10.
These results are built upon the modularity of the generating series of these
moduli spaces in certain ambient spaces, and also built upon the existence of
Hodge theoretically associated cubic fourfolds, Gushel–Mukai fourfolds, and
hyperkähler fourfolds. This is a collaboration with Daniele Agostini and
Ignacio Barros (arXiv:2011.11025).

**Yu-Shen Lin** (Boston University) **(****video****)**

**Title:** Special Lagrangian Fibrations in Log Calabi-Yau Surfaces
and Mirror Symmetry

**Abstract:**

Strominger-Yau-Zaslow conjecture predicts that the Calabi-Yau
manifolds admit special Lagrangian fibrations and the mirror can be constructed
via the dual torus fibration. The conjecture has been the guiding principle for
mirror symmetry while the original conjecture has little progress. In this
talk, I will prove that the SYZ fibration exists in certain log Calabi-Yau
surfaces and their mirrors indeed admit the dual torus fibration under suitable
mirror maps. The result is an interplay between geometric analysis and complex
algebraic geometry. The talk is based on joint works with T. Collins and A.
Jacob.

**Title:** Ultraproduct
cohomology and the decomposition theorem

**Abstract:**

Ultraproducts of étale cohomology provide a large family of Weil
cohomology theories for algebraic varieties. Their properties are closely
related to questions of l-independence and torsion-freeness of l-adic
cohomology. I will present recent progress in ultraproduct cohomology with
coefficients, such as the decomposition theorem. This talk is based on joint
work with Anna Cadoret.

**Title: **Grothendieck--Serre in the quasi--split unramified case

**Abstract:**

The Grothendieck--Serre conjecture predicts that every
generically trivial torsor under a reductive group scheme G over a regular
local ring R is trivial. We settle it in the case when G is quasi-split and R
is unramified. To overcome obstacles that have so far kept the mixed
characteristic case out of reach, we adapt Artin's construction of "good
neighborhoods" to the setting where the base is a discrete valuation ring,
build equivariant compactifications of tori over higher dimensional bases, and
study the geometry of the affine Grassmannian in bad characteristics.

**Title: **Twisted derived equivalence for abelian surfaces

**Abstract:**

Over complex numbers, the famous global Torelli theorem for K3
surfaces says that two integral Hodge isometric K3 surfaces are isomorphic.
Recently, Huybrechts has shown that two rational Hodge isometric K3 surfaces
are twisted derived equivalent. This is called the twisted derived Torelli
theorem for K3. Natural questions arise for abelian varieties. In this talk, I
will talk about the twisted derived equivalence for abelian surfaces, including
the twisted derived Torelli theorem for abelian surfaces (over all fields) and
its applications. This is a joint work with Haitao Zou.

**Title:** Universal Secant Bundles
and Syzygies

**Abstract:**

We describe a universal approach to the secant bundle
construction of syzygies provided by Ein and Lazarsfeld. As an application, we
obtain a quick proof of Green's Conjecture on the shape of the equations of
general canonical curves. Furthermore, we will explain how the same technique
resolves a conjecture of von Bothmer and Schreyer on Geometric Syzygies of
canonical curves.

**Title: **Some boundedness problems in Cremona group

**Abstract:**

This talk is based on a work with Cantat and Deserti. According
to the degree sequence, there are 4 types (elliptic, Jonquieres, Halphen and
Loxodromic) of elements f in Bir(P^2). For a fixed degree d>=1, we
study the set of these 4 types of elements of degree d. We show that for
Halphen twists and Loxodromic transformations, such sets are constructible.
This statement is not true for elliptic and Jonquieres elements.We also show
that for a Jonquieres or Halphen twist f of degree d, the degree of the unique
f-invariant pencil is bounded by a constant depending on d. This result may be
considered as a positive answer to the Poincare problem of bounding the degree
of first integrals,but for birational twists instead of algebraic foliations.
As a consequence of this, we show that for two Halphen twists f and g, if they
are conjugate in Bir(f), then they are conjugate by some element of degree
bounded by a constant depending on deg(f)+deg(g). This statement is not true
for Jonquieres twists.

**Guolei Zhong **(National University of Singapore)

**Title: **Fano threefolds and fourfolds admitting non-isomorphic
endomorphisms.

**Abstract: **

In this talk, we first show that a smooth Fano threefold X admits
a non-isomorphic surjective endomorphism if and only if X is either toric or a
product of a smooth rational curve and a del Pezzo surface. Second, we show
that a smooth Fano fourfold Y with a conic bundle structure is toric if and
only if Y admits an amplified endomorphism. The first part is a joint work with
Sheng Meng and De-Qi Zhang, and the second part is a joint work with Jia jia.

**Joonyeong Won**
(KIAS) **(****video****)**

**Title:** Sasaki-Einstein and Kähler-Einstein metric on
5-manifolds and weighted hypersurfaces

**Abstract:**

By developing the method introduced by Kobayashi in 1960's,
Boyer, Galicki and Kollár found many examples of simply connected Sasaki-
Einstein 5-manifolds. For such examples they verified existence of orbifold Kähler-Einstein
metrics on various log del Pezzo surfaces, in particular weighted log del Pezzo
hypersurfaces. We discuss about recent progresses of the existence
problem of Sasaki -Einstein and Kähler-Einstein metric on 5-manifold and
weighted del Pezzo hypersurfaces respectively.

**Soheyla Feyzbakhsh**
(Imperial College) **(****video****)**

**Title:** An application of a Bogomolov-Gieseker type inequality to
counting invariants

**Abstract:**

In this talk, I will work on a smooth projective threefold X
which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such
as the projective space P^3 or the quintic threefold. I will show certain
moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over
Hilbert schemes of ideal sheaves of curves and points in X. When X is
Calabi-Yau this gives a simple wall crossing formula expressing curve counts
(and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0
branes. This is joint work with Richard Thomas.

**Lei Zhang**
(USTC) **(****video****)**

**Title:** Counterexample to Fujita conjecture in positive
characteristic

**Sho Tanimoto** (Kumamoto University) **(****video****)**

**Title:** Classifying sections of del Pezzo fibrations

**Abstract:**

Mori invented a technique called as Bend and Break lemma which
claims that if we deform a curve with fixed points, then it breaks into the
union of several curves such that some of them are rational. This technique has
wide applications ranging from rationally connectedness of smooth Fano
varieties, Cone theorem for smooth projective varieties, to boundedness of
smooth Fano varieties. However, a priori there is no control on breaking curves
so in particular, an outcome of Bend and Break could be a singular point of the
moduli space of rational curves. With Brian Lehmann, we propose Movable Bend
and Break conjecture which claims that a free rational curve of enough high
degree can degenerate to the union of two free rational curves in the moduli
space of stable maps, and we confirm this conjecture for sections of del Pezzo
fibrations over an arbitrary smooth projective curve. In this talk I will
explain some of ideas of the proof of MBB for del Pezzo fibrations as well as
its applications to Batyrev’s conjecture and Geometric Mann’s conjecture. This
is joint work with Brian Lehmann.

**Fei Hu** (University
of Oslo) **(****video****)**

**Title:** Some comparison problems on correspondences

**Abstract:**

Although the transcendental part of Weil's cohomology theory
remains mysterious, one may try to understand it by investigating the pullback
actions of morphisms, or more generally, correspondences, on the cohomology
group and its algebraic part.

Inspired by a result of Esnault and Srinivas on automorphisms of
surfaces as well as recent advances in complex dynamics, Truong raised a
question on the comparison of two dynamical degrees, which are defined using
pullback actions of dynamical correspondences on cycle class groups and
cohomology groups, respectively. An affirmative answer to his question
would surprisingly imply Weil’s Riemann hypothesis.

In this talk, I propose more general comparison problems on the
norms and spectral radii of the pullback actions of certain correspondences
(which are more natural in some sense). I will talk about their
connections with Truong’s dynamical degree comparison and the standard
conjectures. Under certain technical assumption, some partial results
will be given. I will also discuss some applications to Abelian varieties
and surfaces. This talk is based on joint work with Tuyen Truong.

**Junliang Shen**
(MIT) **(****video****)**

**Title:** Intersection cohomology of the moduli of of 1-dimensional
sheaves and the moduli of Higgs bundles

**Abstract:**

In general, the topology of the moduli space of semistable
sheaves on an algebraic variety relies heavily on the choice of the Euler
characteristic of the sheaves. We show a striking phenomenon that, for the
moduli of 1-dimensional semistable sheaves on a toric del Pezzo surface (e.g.
P^2) or the moduli of semistable Higgs bundles with respect to a divisor of
degree > 2g-2 on a curve, the intersection cohomology (together with the
perverse and the Hodge filtrations) of the moduli space is independent of the
choice of the Euler characteristic. This confirms a conjecture of Bousseau for
P^2, and proves a conjecture of Toda in the case of certain local Calabi-Yau
3-folds. In the proof, a generalized version of Ngô's support theorem plays a
crucial role. Based on joint with Davesh Maulik.

**Yohsuke Matsuzawa**
(Brown University) **(****video****)**

**Title:** Vojta's conjecture and arithmetic dynamics

**Abstract:**

I will discuss applications of Vojta's conjecture to some
problems in arithmetic dynamics, concerning the growth of sizes of coordinates
of orbits, greatest common divisors among coordinates, and prime factors of
coordinates. These problems can be restated and generalized in terms of
(local/global) height functions, and I proved estimates on asymptotic behavior
of height functions along orbits assuming Vojta's conjecture. One of the key
inputs is an asymptotic estimate of log canonical thresholds of (X, f^{-n}(Y)),
where f : X->X is a self-morphism and Y is a closed subscheme of X.

As corollaries, I showed that Vojta's conjecture implies
Dynamical Lang-Siegel conjecture for projective spaces (the sizes of
coordinates grow in the same speed),and existence of primitive prime divisors
in higher dimensional setting.

**Chen Jiang**
(Fudan University) **(****video****)**

**Title:** Positivity in hyperk\"{a}hler manifolds via
Rozansky—Witten theory

**Abstract:**

For a hyperk\"{a}hler manifold $X$ of dimension $2n$,
Huybrechts showed that there are constants $a_0, a_2, \dots, a_{2n}$ such that

$$\chi(L)
=\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q_X(c_1(L))^{i}$$

for
any line bundle $L$ on $X$, where $q_X$ is the Beauville--Bogomolov--Fujiki
quadratic form of $X$. Here the polynomial
$\sum_{i=0}^n\frac{a_{2i}}{(2i)!}q^{i}$ is called the Riemann--Roch polynomial
of $X$.

In
this talk, I will discuss recent progress on the positivity of coefficients of
the Riemann--Roch polynomial and also positivity of Todd classes. Such
positivity results follows from a Lefschetz-type decomposition of the root of
Todd genus via the Rozansky—Witten theory, following the ideas of Hitchin,
Sawon, and Nieper-Wißkirchen.

**Ya Deng**
(IHES) **(****video****)**

**Title:** Big Picard theorem for varieties admitting a variation of
Hodge structures

**Abstract:**

In 1972, A. Borel proved generalized big Picard theorem for any
hermitian locally symmetric variety $X$: any holomorphic map from the punctured
disk to $X$ extends to a holomorphic map of the disk into any projective
compactification of $X$. In particular any analytic map from a quasi-projective
variety to $X$ is algebraic. Period domains, introduced by Griffiths in 1969,
are classifying spaces for Hodge structures. They are transcendental
generalizations of hermitian locally symmetric varieties. In this talk, I will
present a generalized big Picard theorem for period domains, which extends the
recent work by Bakker-Brunebarbe-Tsimerman.

**Jakub Witaszek**
(University of Michigan) **(****video****)**

**Title:** On the four-dimensional Minimal Model Program for
singularities and families in positive characteristic

**Abstract:**

I will discuss new developments on the four-dimensional Minimal
Model Program in positive characteristic. This is based on a joint work with Christopher
Hacon.

**Yu-Wei Fan** (UC
Berkeley) **(****video****)**

**Title:** Stokes matrices, surfaces, and points on spheres

**Abstract:**

Moduli spaces of points on n-spheres carry natural actions of
braid groups. For n=0,1, and 3, we prove that these symmetries extend to actions
of mapping class groups of positive genus surfaces, through exceptional
isomorphisms with certain moduli of local systems. This relies on the existence
of group structure for spheres in these dimensions. We also apply the
exceptional isomorphisms to the study of Stokes matrices and exceptional
collections of triangulated categories. Joint work with Junho Peter Whang.

**Jinhyung Park** (Sogang University)

**Title:** A Castelnuovo-Mumford regularity bound for threefolds
with mild singularities

**Abstract:**

The Eisenbud-Goto regularity conjecture says that the
Castelnuovo-Mumford regularity of an embedded projective variety is bounded
above by degree - codimension +1, but McCullough-Peeva recently constructed
highly singular counterexamples to the conjecture. It is natural to make a
precise distinction between mildly singular varieties satisfying the regularity
conjecture and highly singular varieties not satisfying the regularity
conjecture. In this talk, we consider the threefold case. We prove that every
projective threefold with rational singularities has a nice regularity bound,
which is slightly weaker than the conjectured bound, and we show that every
normal projective threefold with Cohen-Macaulay Du Bois singularities in codimension
two satisfies the regularity conjecture. The codimension two case is
particularly interesting because one of the counterexamples to the regularity
conjecture appears in this case. This is joint work with Wenbo Niu.

**Evgeny Shinder**
(University of Sheffield)

**Title:** Semiorthogonal decompositions for singular varieties

**Abstract:**

I will explain a semiorthogonal decomposition for derived
categories of singular projective varieties into derived categories of finite-dimensional
algebras, due to Professor Kawamata, generalizing the concept of an exceptional
collection in the smooth case. I will present known constructions of these for
nodal curves (Burban), torsion-free toric surfaces (Karmazyn-Kuznetsov-Shinder)
and two nodal threefolds (Kawamata). Finally, I will explain obstructions
coming from the K_{-1} group, and how it translates to maximal nonfactoriality
in the nodal threefold case. This is joint work with M. Kalck and N. Pavic.

**Hong Duc Nguyen**
(Thang Long University) **(****video****)**

**Title:** Cohomology of contact loci

**Abstract:**

We construct a spectral sequence converging to the
cohomology with compact support of the $m$-th contact locus of a complex
polynomial. The first page is explicitly described in terms of a log
resolution and coincides with the first page of McLean's spectral sequence
converging to the Floer cohomology of the $m$-th iterate of the monodromy, when
the polynomial has an isolated singularity. Inspired by this connection we
conjecture that the Floer cohomology of the $m$-th iterate of the monodromy of
$f$ is isomorphic to the compactly supported cohomology of the $m$-th contact
locus of $f$, and that this isomorphism comes from an isomorphism of McLean
spectral sequence with ours.

**Qizheng Yin**
(Peking University) **(****video****)**

**Title:** The Chow ring of Hilb(K3) revisited

**Abstract:**

The Chow ring of hyper-Kähler varieties should enjoy similar
properties as the Chow ring of abelian varieties. In particular, a Beauville
type decomposition is believed (by Beauville himself) to exist for all
hyper-Kähler varieties. In this talk, we discuss a general approach towards the
Beauville type decomposition of the Chow ring. We carry it out explicitly for
the Hilbert scheme of points of K3 surfaces, and prove the multiplicativity of
the resulting decomposition. Joint work with Andrei Negut and Georg Oberdieck.

**Takahiro Shibata**
(National University of Singapore) **(****video****)**

**Title:** Invariant subvarieties with small dynamical degree

**Abstract:**

Given a self-morphism on an algebraic variety, we can consider
various dynamical problems on it. Motivated by an arithmetic-dynamical problem,
we consider invariant subvarieties whose first dynamical degree is less than
that of the ambient variety. We give an estimate of the number of them in
certain cases.

**Sheng Meng**
(KIAS) **(****video****)**

**Title:** Dynamical equivariant minimal model program

**Abstract:**

I will describe the minimal model program (MMP) in the study of complex
dynamics and how MMP can be applied to many conjectures with dynamical or
arithmetical flavours. Several open questions will also be proposed in this
talk.

**Frank Gounelas**
(Göttingen University) **(****video****) **

**Title:** Curves on K3 surfaces

**Abstract:**

I will survey the recent completion (joint with Chen-Liedtke) of
the remaining cases of the conjecture that a projective K3 surface contains
infinitely many rational curves. As a consequence of this along with the
Bogomolov-Miyaoka-Yau inequality and the deformation theory of stable maps, I
will explain (joint with Chen) how in characteristic zero one can deduce the
existence of infinitely many curves of any geometric genus moving with maximal
variation in moduli on a K3 surface. In particular this leads to an algebraic
proof of a theorem of Kobayashi on vanishing of global symmetric differentials
and applications to 0-cycles.

**Toshiyuki Katsura**
(University of Tokyo) **(****video****) **

**Title:** Counting Richelot isogenies of superspecial curves of
genus 2

**Abstract:**

Recently, supersingular elliptic curve isogeny cryptography has
been extended to the genus-2 case by using superspecial curves of genus 2 and
their Richelot isogeny graphs. In view of this situation, we examine the
structure of Richelot isogenies of superspecial curves of genus 2, and
give a characterization of decomposed Richelot isogenies. We also give a
concrete formula of the number of such decomposed Richelot isogenies up
to isomorphism between superspecial principally polarized abelian surfaces.
This is a joint work with Katsuyuki Takashima (Mitsubishi Electic Co.).

**Jie Liu**
(Chinese Academy of Sciences)** (****video****) **

**Title:** Strictly nef subsheaves in tangent bundle

**Abstract:**

Since the seminal works of Mori and Siu-Yau on the solutions to
Hartshorne conjecture and Frankel conjecture, it becomes apparent that the
positivity of the tangent bundle of a complex projective manifold carries
important geometric information. In this talk, we will discuss the structure of
projective manifolds whose tangent bundle contains a locally free strictly nef
subsheaf and present a new characterisation of projective spaces. This is a
joint work with Wenhao Ou (AMSS) and Xiaokui Yang (YMSC).

**Junyan Cao** (Université Côte d'Azur)
**(****video****) **

**Title:** On the Ohsawa-Takegoshi extension theorem

**Abstract:**

Since it was established, the Ohsawa-Takegoshi extension theorem
turned out to be a fundamental tool in complex geometry. We establish a new
extension result for twisted canonical forms defined on a hypersurface with
simple normal crossings of a projective manifold with a control on its L^{^2} norme. It is a joint work with
Mihai Păun.

**Jeongseok Oh** (KIAS) **(****video) **

**Title:** Counting sheaves on Calabi-Yau 4-folds

**Abstract: **

We define a localised Euler class for isotropic sections, and
isotropic cones, in SO(N) bundles. We use this to give an algebraic definition
of Borisov-Joyce sheaf counting invariants on Calabi-Yau 4-folds. When a torus
acts, we prove a localisation result. This talk is based on the joint work with
Richard. P. Thomas.

**Xun Yu**
(Tianjin University) **(****video****)**

**Title:** Automorphism groups of smooth hypersurfaces

**Abstract: **

I will discuss automorphism groups of smooth hypersurfaces in
the projective space and explain an approach to classify automorphism groups of
smooth quintic threefolds and smooth cubic threefolds. This talk is based on my
joint works with Professor Keiji Oguiso and Li Wei.

**Tuyen Trung Truong**
(University of Oslo) **(****video****)**

**Title:** Rationality of quotients of Abelian varieties and
computer algebra

**Abstract:**

This talk concerns the question of what variety of the form X/G,
where X is an Abelian variety and G a finite subgroup of Aut(X), is rational.
It is motivated by some interesting geometric and dynamical system questions.
Most of the work so far concerns the case where X is of the form E^m where E is
an elliptic curve, and G is a finite subgroup of Aut(E). I will review the
current known results and approaches, and explain why it could be necessary to
use computer algebra to resolve the question, and a brief discussion on works
in this direction (including my ongoing joint work with Keiji Oguiso).

**Yukinobu Toda**
(Kavli IPMU) **(****video****)**

**Title:** On d-critical birational geometry and categorical DT
theories

**Abstract:**

In this talk, I will explain an idea of d-critical birational
geometry, which deals with certain "virtual" birational maps among
schemes with d-critical structures. One of the motivations of this new
framework is to categorify wall-crossing formulas of Donaldson-Thomas
invariants. I will propose an analogue of D/K equivalence conjecture in
d-critical birational geometry, which should lead to a categorification of wall-crossing
formulas of DT invariants.

The main result in this talk is to realize the above story for
local surfaces. I will show the window theorem for categorical DT theories on
local surfaces, which is used to categorify wall-crossing invariance of genus
zero GV invariants, MNOP/PT correspondence, etc.

**Huai-Liang Chang**
(HKUST) **(****video****)**

**Title:** BCOV Feynman structure for Gromov Witten invariants

**Abstract:**

Gromov Witten invariants Fg encodes the numbers of genus g
curves in Calabi Yau threefolds. They play a role in enumerative geometry and
are not easy to be determined.

In 1993 Bershadsky, Cecotti, Ooguri, Vafa exhibited a hidden
"Feynman structure" governing all Fg’s at once. Their argument was
via path integral while its counterpart in mathematics had been missing for
decades.

In 2018, considering the moduli of a special kind of algebro
geometric objects, "Mixed Spin P fields", is developed and provides
the wanted "Feynman structure". In this talk we will see genuine
ideas behind these features.

**Caucher Birkar** (University of
Cambridge)

**Title:** Geometry and moduli of polarised varieties

**Abstract:**

In this talk I will discuss projective varieties polarised by
ample divisors (or more generally nef and big divisors) and outline some recent
results about the geometry and moduli spaces of such varieties.

Sponsors: National Center for Theoretical Sciences