10th Meeting, September 29, 2018, Tsinghua

Ahmed Abbes (CNRS & IHÉS)

The p-adic Simpson correspondence

The p-adic Simpson correspondence, initiated by Gerd Faltings in 2005, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra - namely Higgs bundles. My lecture will be an introduction to this topic. I will focus on the approach that I developed with Michel Gros relying on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J.-M. Fontaine.

Daxin Xu (Caltech)

Generalized Kloosterman sheaves and their p-adic variants

I will first review the relationship between the classical Bessel equation and the classical Kloosterman sum. Such a relation can be regarded as an instance of the geometric Langlands correspondence for GL2. I will then explain the recent generalizations of this story for arbitrary reductive groups, based on the works by Frenkel-Gross, Heinloth-Ngo-Yun, and X. Zhu. In the end, I will discuss some joint work in progress with X. Zhu, where we study the p-adic aspect of this theory.

11th Meeting, October 17, 2018, CAS

Constantin Shramov (Steklov Institute)

Finite groups of birational selfmaps of surfaces

I will speak about finite groups acting by birational automorphisms of surfaces. In particular, we will consider surfaces over function fields. One of our important observations here is that for a smooth geometrically rational surface S, either it is birational to a product of a projective line and a conic (so that S is rational provided that it has a point), or finite subgroups of its birational automorphism group are bounded. I will also discuss some particular types of surfaces with interesting automorphism groups, including Severi-Brauer surfaces.

Jian Xiao (YMSC, Tsinghua)

Local positivity for curves

One of the most important invariants measuring the local positivity of a nef line bundle is the Seshadri constant. We first give a brief introduction to this invariant. Then using the duality of positive cones, we show that applying the polar transform to local positivity invariants for divisors gives interesting and new local positivity invariants for curves. These new invariants, studied also independently by M. Fulger, have nice properties similar to those for divisors. In particular, this enables us to obtain a Seshadri type ampleness criterion for movable curves, and give a characterization of the divisorial components of the non-ample locus of a big class. (Joint work with N. McCleerey.)

12th Meeting, November 3, 2018, PKU

Matteo Ruggiero (Paris Diderot)

Algebraically stable models for holomorphic maps on normal surface singularities

Let f be a holomorphic selfmap of a normal surface singularity (X, x_0), and let π: X_π -> (X, x_0) be a proper birational map. We say that the lift f_π of f on X_π is algebraically stable if for every compact curve E in X_π, its image through high iterates of f_π does not belong to the indeterminacy set of f_π. In joint work with William Gignac, we show that for any proper birational map π', there exists a higher model π for which f_π is algebraically stable, but for one class of exceptions, where such models do not exist. The proof relies on the study of the action f_* induced by f on a suitable space of valuations V, following techniques previously introduced by Charles Favre and Mattias Jonsson. In our setting, we construct a distance on V for which f_* is non-expanding. This allows to deduce fixed point theorems for f_*.

Chen Jiang (Kavli IPMU)

Anti-canonical geometry of Fano 3-folds

We are interested in the explicit geometry of Fano 3-folds given by the pluri-anti-canonical systems. For a Q-factorial terminal Fano 3-fold of Picard number 1, we show that the 39-th pluri-anti-canonical map is birational onto its image. For a canonical weak Fano 3-fold, we show that the 97-th pluri-anti-canonical map is birational, and the 52-nd pluri-anti-canonical map is birational if changed by a birational model. I will explain some ideas of the proof and further applications. This talk is based on joint work with Meng Chen.

Junwu Tu (ShanghaiTech)

Categorical Saito theory

In this talk, I shall present a categorical construction of Saito's theory of primitive forms. The key observation is to use Kontsevich/Tsygan formality to study the category of matrix factorizations. Indeed, as an application of this new technology, we prove a comparison result that the categorical construction is equivalent to Saito's original one, through a kind of B-model open-closed map. Joint work with Andrei Căldăraru.

13th Meeting, December 12, 2018, CAS

Qifeng Li (KIAS)

Fano deformation rigidity of rational homogeneous spaces of submaximal Picard numbers

A Fano manifold is said to be rigid under Fano deformation if any deformation of it, which is also a Fano manifold, must be biholomorphic to itself. The deformation rigidity property of rational homogeneous spaces are get attentions since the series works of J.-M. Hwang and N. Mok on Picard number one cases (whose Kahler deformations are automatically Fano manifolds). Fano deformation rigidity of rational homogeneous spaces becomes more interesting since counterexamples of small Picard numbers are found. Recently A. Weber and J. A. Wisniewski proved that rational homogeneous spaces with maximal Picard numbers (i.e. complete flag manifolds) are rigid under Fano deformation. In this talk, I will present a program to verify the Fano deformation rigidity of a homogeneous space via that property of homogeneous submanifolds. Then the Fano deformation rigidity of rational homogeneous spaces of submaximal Picard numbers will be checked.

KyeongDong Park (IBS-CGP)

Equivariant Ulrich bundles and moduli spaces of Ulrich bundles

Ulrich bundles are vector bundles which enjoy many special features, and existence and properties of Ulrich bundles on a given algebraic variety tell us properties of the variety. Equivariant Ulrich bundles on rational homogeneous varieties of classical type were studied by many people. We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the (complex) Cayley plane E_6/P_1 and the E_7-adjoint variety E_7/P_1. And I will describe the moduli space of stable Ulrich bundles on the smooth Fano 3-fold of Picard number 1, degree 5 and index 2 using the moduli space of stable quiver representations. This talk is based on joint works with Kyoung-Seog Lee.

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