| Abstracts |
Mini-courses
- Yujiro Kawamata (University of Tokyo): On non-commutative deformations
(1) I will explain deformations of sheaves over non-commutative (NC) base. The point is that there are more NC deformations than usual (C) deformations, and NC deformations
yield more invariants. The moduli space of sheaves become larger in the sense that it has additional formal structure.
(2) I will explain deformations of a usual variety, based on commutative algebra, to NC varieties. Though the localizations of NC algebras are usually not possible, the
deformations are constructed by gluing NC associative algebras. I will also consider NC deformations of McKay correspondence.
- Hsueh-Yung Lin (National Taiwan University): Motivic invariants of birational maps
The motivic invariant \(c(f)\) of a birational map \(f\colon X \dashrightarrow Y\) is an additive invariant which measures the difference between the birational types of
the exceptional divisors of \(f\) and those of its inverse \(f^{−1}\). After explaining the construction and the motivic nature of the invariant \(c(f)\), we will focus on the case where f is a birational
automorphism of a variety \(X\). On one hand, we will show that the motivic invariants vanish on the birational automorphism group \(\textrm{Bir}(X)\) when \(X\) is a surface over a perfect
field or a complex threefold. On the other hand, we will construct some Cremona transformations \(f\) such that \(c(f)\) is nonzero, and derive new consequences on the Cremona
groups. If time permits, we will talk about some possible refinements of the motivic invariants. This is joint work with E. Shinder and partly with S. Zimmermann.
- Junyi Xie (Peking University): DAO for curves
With Zhuchao Ji, we prove the Dynamical Andre-Oort (DAO) conjecture proposed by Baker and DeMarco for families of rational maps parameterized by an algebraic curve.
In fact, we prove a stronger result, which is a Bogomolov type generalization of DAO for curves.
Talks
- Keping Huang (Harbin Institute Technology): A new Diophantine approximation inequality on surfaces and its applications
We prove a Diophantine approximation inequality for closed subschemes on surfaces which can be viewed as a joint generalization of recent inequalities
of Ru-Vojta and HeierLevin in this context. As applications, we study various Diophantine problems on affine surfaces given as the complement of three
numerically parallel ample projective curves: inequalities involving greatest common divisors, degeneracy of integral points, and related Diophantine
equations including families of S-unit equations. We state analogous results in the complex analytic setting, where our main result is an inequality of Second
Main Theorem type for surfaces, with applications to the study and value distribution theory of holomorphic curves in surfaces. This is a joint work
with Aaron Levin and Zheng Xiao.
- Renjie Lyu (Xiamen University): Generic triviality of automorphisms of algebraic varieties: complete intersections
The automorphism group is an important invariant for algebraic varieties. In this talk, we consider automorphism groups for a family of smooth projective
varieties. I will show that, under certain assumptions on the associated moduli stacks and monodromy groups, the automorphism group of a general member
in the family is minimal. In particular, we can prove that, in most cases, a general smooth complete intersection has no non-trivial automorphisms. This
is a joint work with Dingxin Zhang.
- Hao Sun (South China University of Technology): A nonabelian Hodge correspondence for principal bundles in positive characteristic
Ogus-Vologodsky established the nonabelian Hodge correspondence in positive characteristic in the case of \(G = GL_n\). It is hard to generalize their approach
to principal bundles due to an absent analogue of Azumaya algebras. Based on the approach called exponential twisting introduced by Lan-Sheng-Zuo, we established
a correspondence between nilpotent \(G\)-Higgs bundles and nilpotent flat \(G\)-bundles of exponent smaller than \(p\). When \(G = GL_n\), it goes back to Ogus-Vologodsky result.
As an application, when \(G\) is classical group, mod p reductions of rigid flat \(G\)-bundles has nilpotent p-curvature. The first part is joint work with Mao Sheng
and Jianping Wang, and the second part is an ongoing project joint work with Pengfei Huang and Yichen Qin.
- Bin Wang (The Chinese University of Hong Kong): Picard groups of spectral varieties, instanton and monopole branches
In this talk, we will discuss moduli spaces of Higgs pairs on surfaces. We will focus on the associated Hitchin maps, especially on generic fibers via a
Noether-Lefschetz theorem for spectral varieties. We then apply them to study instanton and monopole branches. This is a joint work with Xiaoyu Su.
- Long Wang (Fudan University): The movable cone of Schoen’s Calabi-Yau threefold
The cone conjecture of Morrison and Kawamata concerns the structure of nef and movable cones of Calabi-Yau manifolds. In this talk, after giving an overview of
the conjecture, I will report the joint work in progress with Cécile Gachet, Hsueh-Yung Lin and Isabel Stenger about the movable cone conjecture for
Schoen’s Calabi-Yau threefold.
- Jianshi Yan (Northeastern University): On the pluricanonical map and the canonical volume of projective 4-folds of general type
Understanding the behavior of pluricanonical maps and the lowest bound of canonical volumes of projective varieties has been a major question in birational
geometry. For curves, surfaces and 3-folds, there are classical results. In this talk, I will introduce some results on the birationality of pluricanonical
maps and the lowest bound of canonical volumes of projective 4-folds of general type with geometric genus \(p_g \geq 2\) or with plurigenus \(P_{m_0}(V )\geq 2\)
for some positive integer \(m_0\).
- Hang Zhao (Yunnan University): Boundedness of numericallytrivial automorphism group of irregularthreefolds of general type
An automorphism of an algebraic variety is called numerically trivial, if it acts identically on the cohomology ring \(H^0(X,\mathbb{Q})\). The groups
\(Aut_{\mathbb{Q}}(X)\) of numerically trivial automorphism of irregular surfaces of general type are uniformly bounded, and for \(X\) has maximal Albanese
dimension, \(Aut_{\mathbb{Q}}(X)\) is either tirival or \(\mathbb{Z}_2\). In dimension three, there are examples of irregular threefold of general type with
terminal singularities such that \(Aut_{\mathbb{Q}}(X)\) can be arbitrarily large. Therefore, it is natural to conjecture that for a given natural number \(n\geq 3\),
there exists a constant \(M\) which depending on \(n\), such that \(|Aut_{\mathbb{Q}}(X)| \leq M\) for all smooth varieties of general type \(X\).
In this talk, I will discuss some results related to this problem for smooth irregular threefolds of general type.
- Tong Zhang (East China Normal University): On the minimal slope problem for threefolds fibred by surfaces of general type
In the 1980s, Cornalba-Harris and Xiao proved the slope inequality which gives the minimal slope for surfaces fibred by curves of genus \(g\geq 2\).
In this talk, I would like to address a similar problem in dimension three. That is, to study the minimal slope for threefolds fibred by surfaces of
general type with \(p_g > 0\). I will also introduce some recent results on it. This is a joint work in progress with Y. Hu.
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