Note: The lecture on April 5 will be adjusted to April 7. 

Venue: 

2024-04-02  A404, Science Building (Math Department), Tsinghua University

2024-04-07~2024-04-26 A304, Science Building (Math Department), Tsinghua University

Description:

The McKay Correspondence gathers together group theory,representations, algebraic geometry, and commutative algebra. The original McKay’s observation establishes a correspondence between the finite subgroups of SL(2, C) and the ADE type Dynkin diagrams. Later, Artin and Verdier, reformulate the McKay correspondence in a more geometrical setting. For rational double singularities, the McKay correspondence by Artin and Verdier gives a complete classification of the indecomposable reflexive modules. The classification of reflexive modules has been studied by several people in different cases: Esnault improved the results of Artin and Verdier for rational surface singularity, and classified rank one reflexive modules for quotient singularities. Wunram introduced and classified the family of special reflexive modules for rational surface singularities, nevertheless the complete classification of reflexive modules on rational singularities remains open. For a non rational singularity, Khan classified all the reflexive modules in the simply elliptic case using the Atiyah’s classification of vector bundles on elliptic curves. For a general normal Gorenstein singularity, Bobadilla and Romano classified the family of cohomological special reflexive modules. Recently, Nemethi and Romano classified all the rank one reflexive modules on rational and minimally elliptic singularities. For a general surface singularity, the classification problem for reflexive modules remains open.

In this course, we will do a review of the original McKay’s  observation and its different generalizations. Moreover, we will see how to use the theorem of Atiyah-Patodi-Singer to classify all the indecomposable reflexive module on quotient surface singularities.

Reference：

J. A. Arciniega-Nevárez, J. L. Cisneros-Molina, and A. Romano-Velázquez.Cheeger-Chern-Simons classes of representations of finite subgroups of SL(2, C)  

and the spectrum of rational double point singularities. arXiv:2302.02000, (2023).

M. Artin and J.L. Verdier. Reflexive modules over rational double points. Math. Ann.  270, 79-82, (1985).

H. Esnault. Reflexive modules on quotient surface singularities. J. Reine Angew. Math. 362 63-71, (1985).

J. Fernández de Bobadilla and A. Romano-Velázquez. Reflexive modules on normal Gorenstein Stein surfaces, their deformations and moduli. To appear in Memoirs of 

AMS.

C. Kahn. Reflexive modules on minimally elliptic singularities. Math. Ann. 285 (1), 141-160, (1989).

A. Némethi and A. Romano-Velázquez. The classification of reflexive modules of rank one over rational and minimally elliptic singularities. arXiv:2302.08768,

(2023).

J. Wunram. Reflexive modules on quotient surface singularities. Math. Ann. 279 (4), 583-598, (1988).

Bio:

Agustin Romano received his Ph.D. degree from the Universidad Nacional Autónoma de México in 2018. He completed postdoctoral stays at CIMAT (2018-2019), Tata  Institute of Fundamental Research (India, 2019-2021) and Alfréd Rényi Institute of Mathematics (Hungary, 2021-2022). His current research focuses on singularity theory: Classification of MCM modules on Gorenstein singularities, Auslander-Reiten theory, moduli problems and topological invariants of real and complex singularities.

Registration: https://www.wjx.top/vm/thx8iSF.aspx# 

Videos: http://archive.ymsc.tsinghua.edu.cn/pacm_course?html=Generalizations_of_the_McKay_correspondence.html