## Workshop on Algebraic and Arithmetic Geometry

**Speaker(s): ** Beijing International Center for Mathematical Research

**Time: **July 31 - August 3, 2012

**Venue: ** Room 82J04, Bld #82 Jiayibing, JingChunYuan, BICMR

### Group Photo

### Organizers

**Huayi Chen** - BICMR/Paris 7

**Ruochuan Liu** - BICMR

**Chenyang Xu** - BICMR

**Zhiwei Yun** - Stanford

### Speakers

**Dawei Chen** - Boston College

**Tsao-Hsien Chen** - IAS

**Yifei Chen** - Chinese Academy

**Baohua Fu** - Chinese Academy

**Xuhua He** - HKUST

**Jiuzu Hong** - Yale

**Wen-Wei Li** - Chinese Academy

**Peng Shan** - CNRS/MIT

**Fucheng Tan** - Michigan State

**Shun Tang** - Max Planck

**Ye Tian** - Chinese Academy

**Shanwen Wang** - Padova

**Xinyi Yuan** - Berkeley

**Wei Zhang** - Columbia

**Xinwen Zhu** - Northwestern

### Schedule

Tuesday, July 31, 2012

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9:00 - 9:30 Coffee and Welcome

9:30 - 10:30 **Baohua Fu**, On automorphism groups of smooth projective varieties

10:30 - 11:00 Coffee

11:00 - 12:00 **Dawei Chen**, Flat surfaces, moduli of differentials and Teichmueller dynamics

12:00 - 14:00 Lunch

14:00 - 15:00 **Shun Tang**, On the determinant bundles of abelian schemes

15:00 - 15:30 Coffee

15:30 - 16:30 **Yifei Chen**, The moduli part of the Kawamata-Kodaira canonical bundle formula

Wednesday, August 1, 2012

-----------------------------

9:00 - 9:30 Coffee

9:30 - 10:30 **Ye Tian**, Congruent numbers and Heegner Points

10:30 - 11:00 Coffee

11:00 - 12:00 **Shanwen Wang**, Familly of Kato's Euler systems

12:00 - 14:00 Lunch

14:00 - 15:00 **Wei Zhang**, An arithmetic intersection problem and orbital integrals

15:00 - 15:30 Coffee

15:30 - 16:30 **Fucheng Tan**, Overconvergent families of Siegel-Hilbert modular forms

18:30 Banquet - The lakeview hotel

Thursday, August 2, 2012

-----------------------------

9:00 - 9:30 Coffee

9:30 - 10:30 **Peng Shan**, Affine Lie algebras and Rational Cherednik Algebras

10:30 - 11:00 Coffee

11:00 - 12:00 **Jiuzu Hong**, The notion of categorical Heisenberg representation at positive characteristic and root of unity

12:00 - 14:00 Lunch

14:00 - 15:00 **Xinyi Yuan**, Introduction to the non-archimedean Calabi--Yau theorem

15:00 - 15:30 Coffee

15:30 - 16:30 **Wen-Wei Li**, On the weighted fundamental lemma

Friday, August 3, 2012

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8:30 - 9:30 **Xuhua He**, Cohen-Macaulayness of special fibers of local models of Shimura variety

9:30 - 10:00 Coffee

10:00 - 11:00 **Tsao-Hsien Chen**, Localization theorem for affine Lie algebras

11:00 - 11:30 Coffee

11:30 - 12:30 **Xinwen Zhu**, TBA

### Abstracts

**Dawei Chen**, Boston College

**Flat surfaces, moduli of differentials and Teichmueller dynamics**

An abelian differential defines a flat structure on a Riemann surface such that it can be realized as a plane polygon. Varying the shape of the polygon induces a dynamical action on their moduli space, called the Teichmueller dynamics. In this talk I will illustrate a beautiful interplay between these objects. As an application, we prove a conjecture posed by Kontsevich and Zorich a decade ago about the numerical property of Teichmueller curves in low genus.

**Tsao-Hsien Chen**, IAS

**Localization theorem for affine Lie algebras**

I will first recall a theorem of Frenkel and Gaitsgory about localization theorem for affine Lie algebra modules at critical level using Affine Grassmannian. Then I will explain a generalization of their result which allows more general central characters. It is a joint work with Giorgia-Fortuna and Sasha-Tsymbaliuk.

**Yifei Chen**, Chinese Academy

**The moduli part of the Kawamata-Kodaira canonical bundle formula**

We discuss Kawamata-Kodaira canonical bundle formula, which is a higher-dimension analogue of Kodaira bundle formula for surfaces. We also introduce a conjecture related by Kawamata and Shokurov. As an application, we shall prove a conjecture due to Fujino and Gongyo, that is, if f: X -> Y is a smooth surjective morphism between smooth projective varieties over C, then -K_X is semi-ample implies that -K_Y is semi-ample. This is a joint work with Caucher Birkar.

**Baohua Fu**, Chinese Academy

**On automorphism groups of smooth projective varieties**

We shall show that the continuous part and the discret part of the automorphism group of a smooth projective variety X cannot be both infinite unless X is an abelian variety or a fibration. This is a joint work with De-Qi Zhang.

**Xuhua He**, HKUST

**Cohen-Macaulayness of special fibers of local models of Shimura variety**

We establish a connection between affine flag variety and De Concini-Procesi compactification. As a consequence, we prove that in the unramified case, local models of Shimura variety with parahoric level structure are normal and Cohen Macaulay.

**Jiuzu Hong**, Yale

**The notion of categorical Heisenberg representation at positive characteristic and root of unity**

By work of Khovanov, Licata & Savage there is a notion of a strong categorical Heisenberg action on an abelian category. Their theory applies to categories defined over the field of characteristic zero and with generic parameter. We show how to extend their theory to positive characteristic and roots of unity by introducing constructions using (co)-equalizers instead of idempotents. In particular, we deduce categorical Heisenberg relations from a given Heisenberg action in general case. This is a joint work with Oded Yacobi.

**Wen-Wei Li**, Chinese Academy

**On the weighted fundamental lemma**

The weighted fundamental lemma is introduced by Arthur in order to stabilize the full trace formula. Its variant for Lie algebras has been proved by Chaudouard and Laumon in positive characteristics, by extending Ngô's approach. I will try to present some aspects of this important generalization of the well-known fundamental lemma.

**Peng Shan**, CNRS/MIT

**Affine Lie algebras and Rational Cherednik Algebras**

I will explain some recent progress in the proof of a conjecture of Varagnolo-Vasserot on the equivalence between the category O of cyclotomic rational Cherednik algebras and a parabolic category O of affine Lie algebras. I will also discuss how to deduce from this a conjecture of Rouquier on the characters of simple modules for cyclotomic rational Cherednik algebras and a conjecture of Chuang-Miyachi on the Koszulity of the corresponding category O. This is a joint work with M. Varagnolo and E. Vasserot.

**Fucheng Tan**, Michigan State

**Overconvergent families of Siegel-Hilbert modular forms**

We construct one dimensional families of overconvergent Siegel-Hilbert modular forms, i.e. forms on GSp(2g) over arbitrary totally real fields . In particular, for any classical Siegel-Hilbert modular eigenform one can find a rigid analytic disc centered at this point, on which an infinite family of classical points with varying weights accumulate at the center. Time permitting, we will remark on certain approach to the construction of the whole eigenvariety. This is the joint work with Chung Pang Mok.

**Shun Tang**, Max Planck

**On the determinant bundles of abelian schemes**

Let $pi: A o S$ be a projective abelian scheme over a locally noetherian scheme $S$ with unit section $e: S o A$ and let $L$ be a symmetric, rigidified, relatively ample line bundle on $A$. Denote by $omega_A$ the determinant of the sheaf of differentials of $pi$ and by $d$ the rank of the locally free sheaf $pi_*L$. It is well known that the line bundle $Delta(L):={
m det}(pi_*L)^{otimes2}otimes(e^*omega_A)^{otimes d}$ has a torsion class in the Picard group ${
m Pic}(S)$. In this talk, we will discuss its arithmetic analogue in the context of Arakelov geometry.

**Ye Tian**, Chinese Academy

**Congruent numbers and Heegner Points**

For any given integer k>=1, we construct infinitely many squarefree congruent numbers with exact k odd prime divisors.

**Shanwen Wang**, Padova

**Family of Kato's Euler Systems**

We construct a familly of Kato's Euler systems (resp. a familly of Kato's dual exponential maps) over the weight space which interpolate Kato's original one by a specialisation map.

**Xinyi Yuan**, Berkeley

**Introduction to the non-archimedean Calabi--Yau theorem**

In this talk, we will introduce the recent results on the non-archimedean Calabi--Yau theorem by Yuan--Zhang, Yifeng Liu, and Boucksom--Favre--Jonsson.

**Wei Zhang**, Columbia

**An arithmetic intersection problem and orbital integrals**

We describe an arithmetic intersection problem on some Rapoport-Zink space (a formal moduli of some p-divisible groups). The ``arithmetic fundamental lemma" predicts that the intersection numbers should be equal to some relative orbital integrals. This is a local version of a global problem concerning height pairing and L-value in analogue to the Birch and Swinnerton-Dyer conjecture.

**Xinwen Zhu**, Northwestern

TBA