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 2022/9/27
Upcoming Talks:
2022/09/30
GMT 1:00-
Speaker:
Yuchen Liu (Northwestern University)
Title:
Wall crossing for K-moduli spaces
Abstract:
Recent developments in K-stability provide a nice moduli space, called a
K-moduli space, for log Fano pairs. When the coefficient of the divisor varies,
these K-moduli spaces demonstrate wall crossing phenomena. In this talk, I will
discuss the general principle of K-moduli wall crossings, and show in examples
that it provides a bridge connecting various moduli spaces of different
origins, such as GIT, KSBA, and Hodge theory. Based on joint works with Kenny
Ascher and Kristin DeVleming.
2022/09/30
GMT 2:15-
Speaker:
Ming Hao Quek (Brown University)
Title:
Around the motivic monodromy conjecture for
non-degenerate hypersurfaces
Abstract:
I will discuss my ongoing effort to comprehend, from a geometric viewpoint, the
motivic monodromy conjecture for a
"generic" complex multivariate polynomial f, namely any polynomial f
that is non-degenerate with respect to its Newton polyhedron. This conjecture,
due to Igusa and Denef--Loeser, states that for every pole s of the motivic zeta
function associated to f, exp(2πis) is a "monodromy
eigenvalue" associated to f. On the other hand, the non-degeneracy
condition on f ensures that the singularity theory of f is governed, up to a
certain extent, by faces of the Newton polyhedron of f.
The extent to which the former is governed by the latter is one key aspect of
the conjecture, and will be the main focus of my talk.
Other upcoming talks:
Date Time Speaker
2022/09/30 UTC
1:00- Yuchen Liu (Northwestern University)
2022/09/30 UTC
2:15- Ming
Hao Quek (Brown University)
2022/10/14 UTC
7:00- Yen-An Chen (National Center for Theoretical Science)
2022/10/14 UTC
8:15- Dang
Quoc Huy (Vietnam Institute for Advanced Study in
Mathematics)
2022/10/28 UTC
0:45- Bao
Viet Le Hung (Northwestern University)
2022/10/28 UTC
2:00- Hiromu Tanaka (University of Tokyo)
2022/11/11 UTC
7:00- Yao
Yuan (Capital Normal University)
2022/11/11 UTC
8:15- Rong
Du (East China Normal University)
2022/11/25 UTC
7:00- Ziming Ma (Southern University of Science and Technology)
2022/11/25 UTC
8:15- Zheng
Hua (University of Hong Kong)
2022/12/09 UTC
7:00- Jaehyun Kim (Ewha Womans University)
2022/12/09 UTC
8:15- Olivier
Benoist (CNRS, École Normale Supérieure)
2022/12/23 UTC
7:00- Tasuki Kinjo (Kavli IPMU)
2022/12/23 UTC
8:15- Hyeonjun Park (Korea Institute For
Advanced Study)
Organizers: Yujiro Kawamata
(Tokyo), Xiaotao Sun (Tianjin), JongHae
Keum (KIAS, Seoul), Jungkai
Chen (NTU, Taipei), Meng Chen
(Fudan,
Shanghai), 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), Yoshinori Gongyo (Tokyo), LE Quy Thuong (Vietnam National
University, Hanoi), Wei-Ping Li (HKUST, Hong Kong), Joonyeong
Won (KIAS, Seoul)
Archive:
2022/07/01
GMT 7:00-8:00
Speaker:
Dô Viêt Cuong
(University of Science, Vietnam National University) (video)
Title:
On the moduli spaces of parabolic Higgs bundles on a curve.
Abstract:
Let $C$ be a projective curve. The moduli space of Higgs bundles on $C$,
introduced by Hitchin, is an interesting object of
study in geometry. If $C$ is defined over the complex numbers, the moduli space
of Higgs bundles is diffeomorphic to the space of representations of the
fundamental group of the curve. If $C$ is defined over finite fields, the adelic description of the stack of Higgs bundles on $C$ is
closely related to spaces occurring in the study of the trace formula. It is a
start point to Ngo's proof for the fundamental lemma for Lie algebras.
A natural generalization of the Higgs bundles is the parabolic Higgs bundles
(that we shall equip each bundle of a parabolic structure, i.e
the choice of flags in the fibers over certain marked points, and some
compatible conditions). Simpson proved that there is analogous relation between
the space of representations of the fundamental group of a punctured curve (the
marked points are the points that are took out from the curve) with the moduli
space of parabolic Higgs bundles.
Despite their good applications, the cohomology of
the moduli space of (parabolic) Higgs bundles has not yet been determined. In
this talk, I shall explain an algorithm to calculate the (virtual) motive (i.e in a suitable Grothendieck
group) of the moduli spaces of (parabolic) Higgs bundles. In the case when the
moduli space is quasi-projective, the virtual motive allows us to read off the
dimensions of its cohomology spaces.
2022/07/01
GMT 8:15-9:15
Speaker:
Nguyên Tât Thang (Vietnam
Academy of Science and Technology) (video)
Title:
Contact loci and Motivic nearby cycles of nondegenerate singularities
Abstract:
In this talk, we study polynomials with complex coefficients which are
nondegenerate in two senses, one of Kouchnirenko and
the other with respect to its Newton polyhedron, through data on contact loci
and motivic nearby cycles. We introduce an explicit description of these
quantities in terms of the face functions. As a consequence, in the
nondegeneracy in the sense of Kouchnirenko, we give
calculations on cohomology groups of the contact
loci. This is a joint work with Le Quy Thuong.
2022/06/17
GMT 7:00-8:00
Speaker:
Qifeng Li (Shandong University) (video)
Title:
Deformation rigidity of wonderful group compactifications
Abstract:
For a complex connected semisimple linear algebraic
group G of adjoint type, De Concini and Procesi constructed its wonderful compactification, which
is a smooth Fano equivariant embedding of G enjoying many interesting
properties. In this talk, we will discuss on the properties of wonderful group
compactifications, especially the deformation rigidity of them. This is a joint
work with Baohua Fu.
2022/06/17
GMT 8:15-9:15
Speaker:
Xin Lu (East China Normal University) (video)
Title:
Sharp bound on the abelian automorphism groups of surfaces of general type
Abstract:
We prove that the order of any abelian (resp. cyclic) automorphism group of a
smooth complex projective of general type is bounded from above by
$12.5c_1^2+100$ (resp. $12.5c_1^2+90$) provided that its geometric genus $p_g>6$. The upper bounds can be both reached by
infinitely many examples whose geometric genera can be arbitrarily large. This
is a joint work with Sheng-Li Tan.
2022/06/03
GMT 7:00-8:00
Speaker:
Euisung Park (Korea University) (video)
Title:
On the rank of quadratic equations of projective varieties
Abstract:
For many classical varieties such as Segre-Veronese embeddings, rational normal
scrolls and curves of high degree, the defining homogeneous ideal can be
generated by quadratic polynomials of rank 3 and 4. In this talk, I will speak
about the question whether those ideals can be generated by quadratic
polynomials of rank 3. We prove that the ideal of the Veronese variety has this
property and explain the geometric structure of the rank 3 locus as a
projective algebraic set.
2022/06/03
GMT 8:15-9:15
Speaker:
Shunsuke Takagi (The University of Tokyo) (video)
Title:
Deformations of klt and slc
singularities
Abstract:
Esnault-Viehweg (resp. S. Ishii) proved that
two-dimensional klt (resp. lc)
singularities are stable under small deformations. Unfortunately, an analogous
statement fails in higher dimensions, because the generic fiber is not
necessarily Q-Gorenstein if the special fiber is klt. In this talk, I present a generalization of the
results of Esnault-Viehweg and Ishii under the
assumption that the generic fiber is Q-Gorenstein
(but the total space is not necessarily Q-Gorenstein).
This talk is based on joint work with Kenta Sato.
2022/05/20
GMT 7:00-8:00
Speaker:
Tong Zhang (East China Normal University) (video)
Title:
Noether-Severi inequality and equality for irregular threefolds of general type
Abstract:
In this talk, I will introduce the optimal Noether-Severi
inequality for all smooth and irregular threefolds of
general type. It answers in dimension three an open question of Z. Jiang. I
will also present a complete description of canonical models of smooth and
irregular threefolds of general type attaining the Noether-Severi equality. This is a joint work with Yong Hu.
2022/05/20
GMT 8:15-9:15
Speaker:
Hélène Esnault (Freie
Universität Berlin) (video)
Title:
Recent developments on rigid local systems
Abstract:
We shall review some of the general problems which are unsolved on rigid local
systems and arithmetic $\ell$-adic local systems. We‘ll report briefly on a proof (2018 with Michael Groechenig) of Simpson's integrality conjecture for {\it
cohomologically rigid local systems}. While all rigid local systems in
dimension $1$ are cohomologically rigid (1996, Nick Katz), we did not know
until last week of a single example in higher dimension which is rigid but not
cohomologically rigid. We’ll present one series of examples (2022, joint with
Johan de Jong and Michael Groechenig).
2022/05/06
GMT 7:00-8:00
Speaker:
Adeel Khan (Academia Sinica) (video)
Title:
Microlocalization and Donaldson-Thomas theory
Abstract:
I will discuss a certain categorification of Kontsevich's
virtual fundamental class, which I call derived microlocalization.
Based on joint work with Tasuki Kinjo, I will explain
how this formalism can be used to prove a conjecture of D. Joyce about
categorified Donaldson-Thomas theory of Calabi-Yau threefolds. This has several consequences, including the
existence of cohomological Hall algebras à la Kontsevich-Soibelman
for Calabi-Yau threefolds.
2022/05/06
GMT 8:15-9:15
Speaker:
Hiroki Matsui (Tokushima University) (video)
Title:
Spectra of derived categories of Noetherian schemes
Abstract:
The spectrum of a tensor triangulated category (i.e., a triangulated category
with a tensor structure) has been introduced and studied by Balmer in 2005.
Balmer applied it to the perfect derived category with the derived tensor
products for a Noetherian scheme and proved that the tensor triangulated
category structure of the perfect derived category completely determines the
original scheme.
In this talk, I will introduce the notion of the spectrum of a triangulated category
without tensor structure and develop a ``tensor-free” analog of Balmer’s
theory.
Also, I will apply this to derived categories of Noetherian
schemes.
2022/04/22
GMT 7:00-8:00
Speaker:
Jun-Muk Hwang (Institute for Basic Science) (video)
Title:
Partial compactification of metabelian Lie groups with prescribed varieties of
minimal rational tangents
Abstract:
We study minimal rational curves on a complex manifold that are tangent to a
distribution. In this setting, the variety of minimal rational tangents (VMRT)
has to be isotropic with respect to the Levi tensor of the distribution. Our
main result is a converse of this: any smooth projective variety isotropic with
respect to a vector-valued anti-symmetric form can be realized as VMRT of
minimal rational curves tangent to a distribution on a complex manifold. The
complex manifold is constructed as a partial equivariant compactification of a
metabelian group, which is a result of independent interest.
2022/04/22
GMT 8:15-9:15
Speaker:
Qizheng Yin (Peking University) (video)
Title:
Perverse-Hodge symmetry for Lagrangian fibrations
Abstract:
For a Lagrangian fibration
from a projective irreducible symplectic variety, the
perverse numbers of the fibration are equal to the
Hodge numbers of the source variety. In my talk I will first explain how this
fact is related to hyper-Kähler geometry. Then I will
focus on the symplectic side of the story, especially
on how to enhance/categorify the perverse-Hodge symmetry. Joint work with Junliang Shen.
2022/04/08
GMT 1:00-2:00
Speaker:
Christopher Hacon (The University of Utah) (video)
Title:
Boundedness of polarized Calabi-Yau fibrations and generalized pairs
Abstract:
In this talk we will discuss recent results and work in progress related to the
boundedness of polarized Calabi-Yau fibrations and to the failure of the boundedness of moduli
spaces of generalized pairs.
2022/04/08
GMT 2:15-3:15
Speaker:
Ngô Bao Châu (Vietnam
Institute for Advanced Study in Mathematics) (video)
Title:
On the functional equation of automorphic L-functions
Abstract:
Automorphic L-functions introduced by Langlands in
the late 60' are expected to satisfy a functional equation similar to the
functional equation of Riemann's zeta function. The functional equation would
follow from the Langlands' functoriality conjecture,
which is one of the far-reaching goals of the Langlands
program, and in a sense is equivalent to it. Around 2000, Braverman and Kazhdan formulated a new approach to the functional
equation not following the route of functoriality but attempting to generalize
the Fourier analysis on adeles used by Tate to prove
the functional equation of the Riemann zeta function. I will report some recent
progress in this approach.
2022/03/25
GMT 7:00-8:00
Speaker:
Yang Zhou (Fudan University) (video)
Title:
Wall-crossing for K-theoretic quasimap invariants
Abstract:
For a large class of GIT quotients, the moduli of epsilon-stable quasimaps is a proper Delinge-Mumford
stack with a perfect obstruction theory. Thus K-theoretic epsilon-stable quasimap invariants are defined.
As epsilon tends to infinity, it recovers the K-theoretic invariants; and as
epsilon decreases, fewer and fewer rational tails are allowed in the domain
curves. There is a wall and chamber structure on the space of stability
conditions.
In this talk, we will decribe a master space
construction involoving the moduli spaces on the two
sides of a wall, leading to the proof of a wall-crossing formula.
A key ingredient is keeping track of the S_n-equivariant
structure on the K-theoretic invariants.
2022/03/25
GMT 8:15-9:15
Speaker:
Yong Hu (Shanghai Jiao Tong University) (video)
Title:
Algebraic threefolds of general type with small
volume
Abstract:
It is known that the optimal Noether inequality
$\vol(X) \ge \frac{4}{3}p_g(X)
- \frac{10}{3}$ holds for every $3$-fold $X$ of
general type with $p_g(X) \ge
11$. In this talk, we give a complete classification of $3$-folds $X$ of
general type with $p_g(X) \ge
11$ satisfying the above equality by giving the explicit structure of a
relative canonical model of $X$. This model coincides with the canonical model
of $X$ when $p_g(X) \ge
23$. I would also introduce the second and third optimal Noether
inequalities for $3$-folds $X$ of general type with $p_g(X)
\ge 11$. This is a joint work with Tong Zhang.
2022/02/25
GMT 7:00-8:00
Speaker:
Quoc Ho (Hong Kong Univ. Science and Technology) (video)
Title:
Revisiting mixed geometry
Abstract:
I will present joint work with Penghui Li on our
theory of graded sheaves on Artin stacks. Our sheaf
theory comes with a six-functor formalism, a perverse
t-structure in the sense of Beilinson--Bernstein--Deligne--Gabber, and a weight (or co-t-)structure
in the sense of Bondarko and Pauksztello,
all compatible, in a precise sense, with the six-functor
formalism, perverse t-structures, and Frobenius
weights on ell-adic sheaves. The theory of graded
sheaves has a natural interpretation in terms of mixed geometry à la Beilinson--Ginzburg--Soergel and
provides a uniform construction thereof. In particular, it provides a general
construction of graded lifts of many categories arising in geometric
representation theory and categorified knot invariants. Historically,
constructions of graded lifts were done on a case-by-case basis and were
technically subtle, due to Frobenius' non-semisimplicity. Our construction sidesteps this issue by
semi-simplifying the Frobenius action itself. As an
application, I will conclude the talk by showing that the category of
constructible B-equivariant graded sheaves on the flag variety G/B is a
geometrization of the DG-category of bounded chain complexes of Soergel bimodules.
2022/02/25
GMT 8:15-9:15
Speaker:
Jinhyun Park (KAIST) (video)
Title:
On motivic cohomology of singular algebraic schemes
Abstract:
Motivic cohomology is a hypothetical cohomology theory for algebraic schemes, including
algebraic varieties, over a given field, that can be seen as the counterpart in
algebraic geometry to the singular cohomology theory
in topology. It‘s construction was completed for
smooth varieties, but for singular ones the situation was not clear.
In this talk, I will sketch some recent attempts of mine to
provide an algebraic-cycle-based functorial model for the motivic cohomology of singular algebraic schemes, via formal
schemes and some ideas from derived algebraic geometry. As this is very
complicated, as an illustration I will give an example on the concrete case of
the fat points, where the situation is simpler, but not still trivial.
2022/02/11
GMT 7:00-8:00
Speaker:
Hsin-Ku Chen (NTU) (video)
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 8:00-9:15
Speaker:
Iacopo Brivio (NCTS) (video)
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.
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 8:00-9:15
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.
2021/5/28
GMT 7:00-8:00
Speaker:
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.
2021/5/28
GMT 8:15-9:15
Speaker:
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.
2021/5/14
GMT 6:00-7:00
Speaker:
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.
2021/5/14
GMT 7:15-8:15
Speaker:
YongJoo Shin (Chungnam
National University) (video)
Title:
Complex minimal surfaces of general type with pg= 0
and K2 = 7 via bidouble covers
Abstract:
Let S be a minimal surface of general type with pg(S)
= 0 and K2S = 7 over the field of complex numbers. Inoue firstly constructed
such surfaces S described as Galois Z2×Z2-covers over the four-noda cubic surface. Chen later found different surfaces S
constructed as Galois Z2×Z2-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
Z2×Z2-covers over rational surfaces with Picard number three, with eight nodes
and with two elliptic fibrations. This is a joint
work with Yifan Chen.
2021/4/30
GMT 7:00-8:00
Speaker:
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.
2021/4/30
GMT 8:15-9:15
Speaker: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.
2021/4/16
GMT 1:00-2:00
Speaker:
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).
2021/4/16
GMT 2:15-3:15
Speaker:
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.
2021/4/2
GMT 7:00-8:00
Speaker:
Weizhe Zheng (Morningside Center of Mathematics) (video)
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.
2021/4/2
GMT 8:15-9:15
Speaker:
Kestutis Cesnavicius (Universite Paris Sud) (video)
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.
2021/3/19
GMT 1:00-2:00
Speaker:
Zhiyuan Li (Shanghai Center for Mathematical
Sciences) (video)
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.
2021/3/19
GMT 2:15-3:15
Speaker:
Michael Kemeny (University of Wisconsin-Madison) (video)
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.
2021/3/5
GMT 7:00-8:00
Speaker:
Junyi Xie (CNRS Rennes)
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.
2021/3/5
GMT 8:15-9:15
Speaker:
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.
2021/2/19
GMT 7:00-8:00
Speaker:
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.
2021/2/19
GMT 8:15-9:15
Speaker:
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.
2021/2/5
GMT 1:00-2:00
Speaker:
Lei Zhang (USTC) (video)
Title:
Counterexample to Fujita conjecture in positive characteristic
Abstract:
Fujita conjecture was proposed over complex numbers, which predicts that for a
smooth projective variety X and an ample line bundle L on X, K_X + (dim X+1)L is base point free and K_X + nL
is very ample if n > dim X+1. Joint with Yi Gu, Yongming Zhang, we find
counterexamples to this elegant conjecture in positive characteristic. These
examples stem from Raynaud’s surfaces. I will first report some related results
on this topic and explain the construction and the proof.
2021/1/22
GMT 7:00-8:00
Speaker:
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.
2021/1/22
GMT 8:15-9:15
Speaker:
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.
2021/1/8
GMT 1:00-2:00
Speaker:
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.
2021/1/8
GMT 2:15-3:15
Speaker:
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.
2020/12/18
GMT 7:00-8:00
Speaker:
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.
2020/12/18
GMT 8:15-9:15
Speaker:
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.
2020/12/4
GMT 1:00-2:00
Speaker:
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.
2020/12/4
GMT 2:15-3:15
Speaker:
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.
2020/11/20
GMT 7:00-8:00
Speaker:
Jinhyung Park (Sogang
University) (video)
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.
2020/11/20
GMT 8:15-9:15
Speaker:
Evgeny Shinder (University of Sheffield) (video)
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.
2020/11/06
GMT 1:00-2:00
Speaker:
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.
2020/11/06
GMT 2:15-3:15
Speaker:
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.
2020/10/23
GMT 7:00-8:00
Speaker:
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.
2020/10/23
GMT 8:15-9:15
Speaker:
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.
2020/10/09
GMT 7:00-8:00
Speaker:
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.
2020/10/09
GMT 8:15-9:15
Speaker:
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.).
2020/09/25
GMT 7:00-8:00
Speaker:
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).
2020/09/25
GMT 8:30-9:30
Speaker:
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.
2020/09/11
GMT 2:15-3:15
Speaker:
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.
2020/08/28
GMT 7:00-8:00
Speaker:
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.
2020/08/28
GMT 8:15-9:15
Speaker:
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).
2020/08/14
GMT 7:00-8:00
Speaker:
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.
2020/08/14
GMT 8:15-9:15
Speaker:
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.
2020/08/07
GMT 8:00-9:00
Speaker:
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