4th Meeting, March 5, 2016, AMSS, CAS
Kang Zuo (Mainz)
Periodic Higgs bundle in postive and mixed characteristic
I shall first explain briefly the notion "Higgs-de Rham flow" on a smooth quasiprojective scheme X/W(k) and the induced p-adic correspondence between the category of crystalline representations of the etale fundamental group of the generic fibre of X and the category of periodic Higgs bundles on X. As an application we construct an absolute irreducible rank-2 crystalline representation of the etale fundamental group of a so-called canonical lifted hyperbolic curve via the uniformization Higgs bundle, which should be regarded as a p-adic analogue of Hitchin-Simpson's uniformization theorem on complex hyperbolic curves via uniformization Higgs bundles. This talk is based on my joint papers with Lan, Sheng and Yang.
Hui-Wen Lin (NTU)
Quantum cohomology under smooth blow-ups along complete intersection centers
I would like to consider the Dubrovin (flat) connection on TH(X) and analyze its behavior under various maps including complete intersection imbedding and projective bundle maps. The essential mathematical tools are the corresponding Quantum Lefschetz Hyperplane Theorem and the Quantum Leray-Hirsch Theorem. By combining these two theorems, I can discuss an application on smooth blow-ups along complete intersection centers and succeed in determining a blow-up formula of quantum cohomology.
Chin-Lung Wang (NTU)
Simple flips and quantum cohomology
I will present a recent joint work with Yuan-Pin Lee and Hui-Wen Lin on on quantum cohomology rings under simple ordinary flips. There is a natural decomposition of quantum rings which refines the decomposition of motives. In contrast to the case of flops, where analytic continuation exists, the new phenomenon appeared here is the irregularity of the Dubrovin connections along the kernel factor under flips.
5th Meeting, April 9, 2016, BICMR, PKU
Shunsuke Takagi (Tokyo)
A Gorenstein criterion for strongly F-regular and log terminal singularities
Let (X, x) be a normal F-pure (resp. log canonical) singularity. We give a criterion for (X, x) to be quasi-Gorenstein in terms of an F-pure (resp. log canonical) threshold under the hypothesis that its anti-canonical cover is finitely generated.
Zhiyuan Li (Bonn)
Cones and tautological classes on moduli of K3 surfaces
In this lecture, I will review the cycle theory on moduli problems. In particular, I will talk about the recent questions for cycles on moduli space of K3 surfaces. This includes the Noether-Lefschetz (NL) conjecture, the effective cone question and the tautological conjecture.
Don Zagier (MPIM Bonn)
Teichmueller curves and twisted modular forms
6th Meeting, June 11, 2016, AMSS, CAS
Meng Chen (Fudan)
On minimal 3-folds of general type with maximal pluricanonical section index
Let X be a minimal 3-fold of general type. The pluricanonical section index $\delta(X)$ is defined to be the minimal integer m so that $P_m(X) \geq 2$. According to J. A. Chen and M. Chen, one has either $1 \leq \delta(X) \leq 15$ or $\delta(X) = 18$. In this talk, we explain further study on the case $\delta(X) = 18$. A direct corollary is that the 57th canonical map of every minimal 3-fold of general type is stably birational, which improves known results.
Lei Fu (Nankai)
Rigidity of $\ell$-adic sheaves
Let X be a smooth connected algebraic curve over an algebraically closed field k, let S be a finite closed subset in X, and let F be a lisse $\ell$-adic sheaf on X - S. We study the deformation of F. By studying the generic fiber of the universal deformation space, we prove a conjecture of Katz which says that if F is irreducible and physically rigid, then F has rigidity index 2.
Yuchen Liu (Princeton)
The volume of Kähler-Einstein Q-Fano varieties
A complex projective variety is Q-Fano if it has klt singularities and the anti-canonical divisor is Q-Cartier and ample. Starting from dimension 2, the anti-canonical volume of a Q-Fano variety can be arbitrarily large, e.g. weighted projective spaces. Recently, Fujita showed that an n-dimensional Kähler-Einstein Q-Fano variety has volume at most (n + 1)n. In this talk, I will discuss a refinement of Fujita’s volume upper bounds involving invariants of the local singularities. If time permits, I will also talk about an equivalent relation between K-semistability and de Fernex-Ein-Mustata type inequalities. Part of this work is joint with Chi Li.
7th Meeting, October 29, 2016, AMSS, CAS
Michel Brion (CNRS Grenoble)
Commutative algebraic groups up to isogeny
The commutative algebraic groups over a field k form an abelian category, which has been studied by Serre and Oort when k is algebraically closed. In particular, they showed that the homological dimension of this category is 1 if k has characteristic 0, and 2 in positive characteristics. Over an arbitrary field k, the homological dimension can be arbitrarily large, as showed by Milne. In this talk, we will consider the isogeny category of commutative algebraic groups, obtained by inverting all morphisms with finite kernel and cokernel. We will see that this category is equivalent to that of modules of finite length over an (explicit but huge) ring, and has homological dimension 1.
Frédéric Campana (Nancy)
Rational curves and negativity of the cotangent bundle
We will survey results relating the existence of covering/connecting families of rational curves on a complex projective manifold to the negativity of its canonical/cotangent bundle. In particular, foliations which are positive in a suitable birational sense have algebraic leaves, with connecting families of rational curves. This extension of former results of Miyaoka and Bogomolov-McQuillan has applications to families of canonically polarised manifolds.
Jun-Muk Hwang (KIAS)
Contact manifold, Legendrian variety and cubic hypersurface
We will discuss three objects: Fano contact manifold, projective Legendrian variety and nondegenerate cubic hypersurface with smooth singular locus. They are related by certain constructions of microlocal nature. We give an overview of these relations and their role in an approach to the LeBrun-Salamon problem on Fano contact manifolds.
8th Meeting, December 17, 2016, BICMR, PKU
Xiaolei Zhao (Northeastern)
Birational geometry of moduli spaces of sheaves and stability conditions
The birational geometry of moduli of sheaves is greatly studied in recent years, after the pioneer work on K3 surfaces by Bayer and Macrì. The key point is to do wall-crossing in the space of Bridgeland stability conditions. In this talk, I will recall the definition and construction of stability conditions, and explain how to carry out this idea of Bayer and Macrì for the projective plane, based on joint work with Chunyi Li. If time permits, I will also discuss some recent progress in higher dimensions.
Chunyi Li (Edinburgh)
Stability conditions on threefolds
The notion of stability condition on a triangulated category has been introduced by Bridgeland around fifteen years ago. The existence of stability conditions on threefolds has become a core problem of this field. The existence of such conditions is equivalent to some generalized version of Bogomolov inequalities, and will imply new bounds on Chern classes of stable sheaves. I will discuss the progress of this problem in recent years and some related open problems.
Junliang Shen (ETH Zürich)
Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories
By physical considerations, Huang, Katz, and Klemm conjectured that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by Jacobi forms. This gives strong structural results for curve counting invariants of elliptic CY 3-folds. I will explain a mathematical approach to proving (part of) the HKK conjecture. Our method is to construct an involution in the derived category and use wall-crossing techniques. As applications, we present new calculations of Gromov-Witten invariants in any genus for several compact CY 3-folds. Finally, we will discuss the connection to the Igusa cusp form conjecture of Oberdieck and Pandharipande, which concerns the enumerative geometry of K3 surfaces. Joint work with Georg Oberdieck.
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