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 2022/8/31
Upcoming Talks:
2022/09/30 GMT 1:00-
Speaker: Yuchen Liu (Northwestern University)
Title:
Abstract:
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 GMT 7:00-8:00
2022/09/30 GMT 8:15-9:15
2022/10/14 GMT 7:00-8:00
2022/10/14 GMT 8:15-9:15
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 1:00-
2022/10/14
UTC 2:15- Dang
Quoc Huy (VIASM)
2022/10/28
UTC 2:00- Hiromu Tanaka (University of Tokyo)
2022/10/28
UTC 7:00- Yen-An Chen (NCTS)
2022/10/28
UTC 8:15- Jaehyun Kim (Ewha Womans Univ.)
2022/11/11
UTC 7:00- Yuan
Yao (Capital Normal University)
2022/11/11
UTC 8:15- Rong
Du (East China Normal Univ.)
2022/11/25
UTC 7:00- Ziming Ma (Southern Univ. Sci. Tech.)
2022/11/25
UTC 8:15- Zheng
Hua (University of Hong Kong)
2022/12/09
UTC 7:00-
2022/12/09
UTC 8:15- Olivier
Benoist (École Norm. Sup.)
2022/12/23
UTC 7:00- Tasuki Kinjo (Kavli IPMU)
2022/12/23
UTC 8:15- Hyeonjun Park (KIAS)
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:-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.
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 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.
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