This seminar is a platform to present talks by visitors or Tsinghua Faculty on algebraic geometry and related topics. It is run by Dun Liang and Quan Xu and me. For suggestions, please contact one of us; for practical matters see Dun or Quan.

Time and Place: Thursdays 15:00-16:00, conference room 3, Floor 2, Jin Chun Yuan West Building.

As of April 2017 talks take place in the Lecture Hall, Floor 3.

Oct. 20: Robert Perlis (Louisiana State U., US): *The Yelton-Gaines Conjecture.* Abstract: It is well-known that two permutations of

*n*objects are conjugate if and only if they have the same cycle

structure. A much harder question is: When are two ordered pairs of permutations simultaneously conjugate? This will

be an expository talk with examples.

Oct. 27: Quan Xu (YMSC): *A functorial Riemann-Roch theorem in positive characteristic.** * Abstract: In this talk, I will recall Riemann-Roch theorems in different kinds of backgroud. Then, by the proof of the

Adams-Riemann-Roch theorem by Pink and Rossler, I will give an analogue of Deligne's functorial Riemann-Roch

theorem in positive characteristic.

Nov. 3: Zongbin Chen (YMSC): *Introduction to the affine grassmannian and the affine Springer fibers.** *Abstract: It dates back at least to Weil that the affine grassmannian uniformizes the moduli stack of vector bundles on

an algebraic curve. At first sight, this is a simple observation, but it turns out to be very fruitful: It has been the main

ingredient in the work of Beauville-Laszlo on the conformal block, it is also a starting point to a localisation approach of

Frenkel-Gaitsgory on the geometric Langlands program, etc. In the same spirit, the affine Springer fiber is the

uniformisation of the compactified jacobians. They have been the geometrisation of the orbital integrals, and appeared in

the proof of the fundamental lemma of Langlands-Shelstad. In this talk, I will try to give an elementary introduction to

these two objects. The talk should be accessible to a large audience.

Nov. 10: Yinbang Lin (YMSC): *Moduli spaces of stable pairs.*

A stable pair on a smooth projective variety is a coherent sheaf with a morphism from a fixed coherent sheaf. I will talk

about the existence of the moduli space under a stability condition. The construction is via geometric invariant theory.

Then I will describe the related deformation and obstruction theory. In some examples, there are virtual fundamental

classes. I will also discuss the integration of some tautological classes.

Nov. 17: Yao Yuan (YMSC): *Determinant line bundles and Strange duality on curves.*

*This is an introductory talk on the strange duality conjecture. We start from the definition and some basic properties*

of determinant line bundles, leading to the set-up of the strange duality conjecture. We state the conjecture for

curves and review its proof by Marian-Oprea.

Nov. 24: Yao Yuan (YMSC): *Strange duality on rational surfaces.* Although the strange duality conjecture has been proved for curves, very few cases of it are known for surfaces. We

will talk about some cases known for rational surfaces, where one of the two moduli spaces involved parametrizes

1-dimensional semistable sheaves, and the other parametrizes rank 2 semistable sheaves.

Dec. 1: Sz-Sheng Wang (YMSC):

*Complete intersection threefolds in Projective bundles (work in progress).*

We study complete intersection threefolds with a small contraction in projective bundles over a smooth fourfold. It is

a straightforward generalization of the construction of complete intersection Calabi–Yau (CICY) threefolds in a product

of projective spaces (Candelas et al. 1988). As an application, we supply a proof of the result by P.S. Green and

T. Hubsch (1988) that all CICY threefolds in product of projective spaces are connected through projective conifold

transitions.

Dec. 8: Han Wu (ETH Zürich, Switzerland):

*Subconvexity and Equidistribution of Heegner points.*

We will discuss the subconvexity problems for

**GL**

_{2}ⅹ

**GL**

_{2},

**GL**

_{2}ⅹ

**GL**

_{1}and their relation to the problem of

equidistribution of Heegner points. Emphasis will be put on the translation of the equidistribution problem into a

subconvexity problem, and the idea of amplification used in solving the subconvexity problems. Other technical

points will be avoided.

Dec. 15: Qizheng Yin (BICMR Peking University): *The Chow ring of hyper-Kähler varieties.*

We survey recent progress in the study of algebraic cycles on hyper-Kähler varieties, centering around the

Beauville-Voisin conjecture. We also discuss the use of moduli spaces of curves in the subject.

Dec. 22: Ziyang Gao (Princeton U., US & CNRS France): *Family version of the Mordell-Lang conjecture.*

In this talk I will explain how a conjecture of Pink is the appropriate family version of the Mordell-Lang conjecture.

Then I will explain the situation for this conjecture. In particular it is proven for curves in any family of abelian varieties.

Dec. 29: Xiaolei Zhao (Northwestern U., US): *Stability conditions and Bogomolov inequality in dimension three.* Bridgeland stability conditions are a central tool in algebraic geometry in recent years. Despite of its successful

applications to surfaces, the existence in dimension three is still an open question. In this talk, I will introduce the

construction of stability conditions on threefolds by Bogomolov inequality, and explain the recent progress on Fano

threefolds and certain toric threefolds. This is based on my joint work with Bernardara, Macri and Schmidt.

Jan. 12: Rob de Jeu (VU Amsterdam, Neth.): * Generalisations of the Hodge conjecture to higher algebraic K-theory*.

The Hodge conjecture predicts the images of the Chow groups (with rational coefficients) of a smooth, projective

variety over the complex numbers in terms of the Hodge structure of its cohomology groups. The direct sum of these

Chow groups form the K_0 (tensored with the rationals) of the variety, and as such the Hodge conjecture is also a

conjecture about K_0. Beilinson formulated a generalisation of this conjecture to the higher algebraic K-groups of

smooth, quasi-projective varieties over the complex numbers, but this turned out to be too optimistic. We discuss when

this conjecture could still hold, which includes a relative version, and provide some evidence in those cases. This is

joint work with James Lewis and/or Deepam Patel.

March 2: Thomas Koberda (U. of Virginia, US): *Quotients of surface groups via TQFT. *I will show how to construct linear quotients of surface groups which are infinite, and where each simple loop has

finite order. As an application, I will construct finite covers of surfaces where the pullbacks of simple loops fail to

generate the integral homology. We thus answer questions due to Looijenga. This represents joint work with

R. Santharoubane.

March 9: Dasheng Wei (AMSS CAS Beijing):

*Strong approximation for certain norm varieties.*

In this talk, we will introduce some norm varieties which satisfy weak approximation or strong approximation.

These norm varieties are a generalization of universal torsors of Chatelet surfaces. Some applications were also given for

rational points and integral points by the descent theory.

*March 16: Chao Zhang (YMSC):*

*Truncated displays with additional structure and reductions of Shimura varieties.*

The theory of displays, invented by D. Mumford and later developped by P. Norman, T. Zink, and E .Lau, is a Dieudonné

theory dealing with

*p*-divisible groups. Truncated displays are introduced by Lau recently to study truncated

*p*-divisible

groups, i.e. BT-ms. In this talk, I will explain what are truncated displays, and how to put additional structure on them.

Then I will show how to use them to define and study level m tratifications on reductions of Shimura varieties.

March 23: Sheng-Li Tan (Eastern China Normal University, Shanghai): *Moduli Spaces of Curves and Holomorphic Foliations on an Algebraic Surface. *In the 19th century, Darboux, Painlev\’e and Poincar\’e studied differential equations of the first order by using

complex algebraic geometry. The set of holomorphic solutions of the differential equation is called a holomorphic

foliation. In fact, holomorphic foliations can be viewed as a generalization of pencils of algebraic curves, i.e.,

fibrations on an algebraic surface. Poincaré proposed the following research program:

— Study the (topological) properties of families of algebraic curves on a complex algebraic surface, and check if they

are the properties of differential equations.

— Find numerical invariants of complex differential equations.

— Classify complex differential equations according to their invariants.

— Characterize those complex differential equations which are algebraically integrable

— Apply to some problems on real differential equations.

In this lecture, I will talk about some birational invariants of foliations coming from the moduli spaces of curves.

March 30: Eduard Looijenga (YMSC):

*Some Algebraic Geometry related to the mapping class group.*

Since this talk is aimed at algebraic geometers, we begin with reviewing some basic constructions of

Teichmueller theory. We then explain how Hodge theory helps us to understand certain representations of

the mapping class group. But this talk is more about open problems than about definitive results.

April 6: no seminar. NB: as of now the talks take place in the Lecture Hall, Floor 3.

April 13: Jin Cao (YMSC):

*Constructing Elliptic Motives by Cycle Algebras*(in Lecture Hall, Floor 3)

*.*

In this talk, we recall the theory of cdgas over a reductive group. As one application, we show that how to use this theory to reconstruct motives of an elliptic curve without CM in the sense of Voevodsky.

April 20: Minxian Zhu (YMSC):

*Periods and hypergeometric equations.*

We will investigate the periods of Calabi-Yau hypersurfaces in toric varieties. It turns out that they satisfy a system of

differential equations, known as the Gelfand-Kapranov-Zelevinsky (GKZ) hypergeometric equations. We will review

some basic properties of these equations and discuss a conjecture by Hosono-Lian-Yau (HLY) describing the precise

relation between periods and solutions to the GKZ system. This talk is intended as a survey of HLY's work in the

midnineties.

April 27: Jilong Tong (Capital Normal University & University of Bordeaux 1): *A glimpse of anabelian geometry.* After a quick review of the definition of algebraic fundamental groups, we shall survey certain aspects of the

anabelian geometry introduced by Grothendieck, including some important progress made by Tamagawa and

Mochizuki. If time allows, we will explain a new (but still vague) idea in my joint project with Mao SHENG aiming at

a better understanding of anabelian geometry in positive characteristic.

June 1: Weizhe Zheng (Morningside Center of Mathematics, CAS):

*Nearby cycles over general bases.*

Nearby cycles and vanishing cycles in étale cohomology were introduced by Grothendieck in the cohomological study

of singular families. Deligne extended these notions to bases of higher dimension. I will review this theory and discuss

some recent results.

* *

* *