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 2021/10/12
Upcoming Talks:
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More Confirmed Speakers:
Organizers: Yujiro Kawamata
(Tokyo), Xiaotao Sun (Tianjin), JongHae
Keum (KIAS, Seoul), Jungkai Chen (NTU, Taipei), Conan
Nai Chung Leung (CUHK, Hong Kong), Phung Ho Hai
(VAST, Hanoi), De Qi Zhang (NUS, Singapore), Yusuke Nakamura (Tokyo), Baohua Fu (CAS, Beijing), Kiryong
Chung (Kyungpook National Univ., Daegu), Hsueh-Yung
Lin (IPMU, Tokyo)
Archive:
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