課程資訊
課程名稱
反常層導論
Introduction to perverse sheaves 
開課學期
107-1 
授課對象
理學院  數學研究所  
授課教師
余正道 
課號
MATH5097 
課程識別碼
221 U8420 
班次
 
學分
1.0 
全/半年
半年 
必/選修
選修 
上課時間
第13,14,15,16 週
星期一8,9(15:30~17:20)星期四8,9(15:30~17:20) 
上課地點
 
備註
密集課程。合授教師史丹佛大學蔡政江老師。上課教室:新數102。
總人數上限:10人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1071perv 
課程簡介影片
 
核心能力關聯
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課程大綱
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課程概述

** The short course will be leaded by Professor Cheng-Chiang Tsai (Stanford University).

The goal of this short course is to give a quick introduction to perverse sheaves, intersection cohomology, and their application to algebraic geometry. On one hand, perverse sheaves provide a replacement of the singular cohomology for non-smooth algebraic varieties and morphisms, that in some occasions works as if they were smooth. On the other hand, perverse sheaves were first discovered for linear differential equations with regular singularities, a setting which is connected to ours via the Riemann-Hilbert correspondence.

We will mostly work with algebraic varieties over the complex numbers and the analytic topology on them. We begin with a review on Grothendieck’s six functors. After that we work on the triangulated category of complexes of constructible sheaves, t-structures, and eventually the abelian subcategory of perverse sheaves and their construction from intersection cohomology sheaves.

The most important result about perverse sheaves is the decomposition theorem. In spirit this is the result that allows us to sometimes work with proper morphisms of varieties as if they are also smooth morphisms, or rather to measure the failure of smoothness for the sake of cohomology. The original proof of the decomposition theorem by Beilinson, Bernstein, Deligne and Gabber uses the theory of weight in Deligne’s second proof of the Weil conjectures.

For the sake of time we will necessarily skip the detail of some proofs, which likely include the proof that the six functors preserve constructible sheaves and the proof of the decomposition theorem.

As an application, we will work out the Betti numbers of the Hilbert scheme (a compactified moduli space) of points on a smooth surface. If time permits, we will discuss other applications, possibly on topics in geometric representation theory and toric varieties. 

課程目標
Introduction to perverse sheaves, with an eye toward applications in algebraic geometry. 
課程要求
The audience will be expected to be familiar with:
• Sheaves of abelian groups on a topological space and their cohomology theory.
• Basic properties of algebraic varieties and schemes at the level of Hartshorne, “​Algebraic Geometry,​” Chapter I, and Chapter II.1-II.4.

** If you are interested in enrolling the course, please talk to Jeng-Daw Yu beforehand. 
預期每週課後學習時數
 
Office Hours
另約時間 
參考書目
• M. Kashiwara, P. Schapira, ​Sheaves on Manifold​s.
• R. Kiehl, R. Weissauer, ​Weil Conjectures, Perverse Sheaves and l-adic Fourier Transform​.
• A. Beilinson, J. Bernstein, P. Deligne, O. Gabber, ​Faisceaux pervers​. 
指定閱讀
 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
Problem sessions 
100% 
There will be an extra weekly problem session. 
 
課程進度
週次
日期
單元主題
Week 1
12/03, 12/06  Introduction, derived categories 
Week 2
12/10, 12/13  extraordinary pullback, Verdier dual 
Week 3
12/17, 12/20  Perverse t-structure, perverse extension 
Week 4
12/24, 12/27  [Notice the time change of last lecture: 12/27 (Thursday), 8:30 - 10 am]
Intersection complex as a perverse sheaf, applications