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NCTS-Taiwan Math School-Derived and Homological Algebraic GeometryWinter School, 2018
 
9:00 - 16:30, January 29 - February 1, 2018
Room 201, Astronomy-Mathematics Building, NTU

Organizers:
Jiun-Cheng Chen (National Tsing Hua University)
Jungkai Chen (National Taiwan University & NCTS)
Wu-yen Chuang (National Taiwan University)
Shin-Yao Jow (National Tsing Hua University)
Ching-Jui Lai (National Cheng Kung University)


Course Description: 
The purpose of this winter school is to introduce some cohomological aspects of algebraic geometry.  With some explicit examples, we hope to provide a hand-on approach for participants to get acquaintance with some exciting results in algebraic geometry involving derived category and homological algebra.  For example, cohomologies of vector bundles on projective spaces and elliptic curves are relatively easy to compute. Moreover, vector bundle on elliptic curves provide an illuminating example of the notion of stability and Fourier-Mukai transform. These starting examples will provide some insight into some recent development of algebraic geometry. 
 
Topics: 
A.Vector bundles on projective spaces  
    Lecturer: Sheng-Fu Chiu邱聖夫 (AS)
    Ref: C. Okonek, H. Spindler, M. Schneider, Vector Bundles on Complex Projective Spaces, 
            Progress in Mathematics, Birkauser. 
1. Basic notions of vector bundles. (Ref) Section 1.1~1.2 of Chapter One.
2. The splitting of vector bundles. (Ref) Section 2.1~2.3 of Chapter One. 
 
B. Vector bundles on elliptic curves 
    Lecturer: Jungkai Chen陳榮凱 (NTU/NCTS), Hong-Yu Yeh葉弘裕 (AS)
    Ref: M. Atiyah, Vector Bundles over an Elliptic Curve, Proceedings of the London 
            Mathematical Society, Issue 1, 1957, 414–452
1. introduction and general properties of vector bundles on elliptic curves and abelian varieties [Jungkai Chen]
2. additive structure of vector bundles on elliptic curves [Yeh]
3. multiplicative structure of vector bundles on elliptic curves. [Yeh]
4. applications to moduli spaces and Fourier-Mukai transform [Jungkai Chen]
 
C. Reconstruction theorem of canonical polarized manifolds 
     Lecturer: Jiun Cheng Chen陳俊成 (NTHU), Ching-jui Lai賴青瑞(NCKU)
     Ref: A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of 
             autoequivalences, Comp. Math. 125 (2001), 327-344
             D. Huybretch, Fourier-Mukai Transforms in Algebraic Geometry
1. Ampleness, Serre's Theorem, Construction of derived functions for Db(X)
2. Canonical line bundle and its relation to birational classification, a crush course.
3. Filtration by cohomology and point objects.
4. Spectral sequence and line bundle objects.
5. Proof of main theorem.
6. Second proof: approach via Fourier-Mukai transforms
 
D. Discussion sessions with exercises 
Leader: Ching-jui Lai 賴青瑞 (NCKU), Hong-Yu Yeh 葉弘裕 (AS)
 

Time

1/29

1/30

1/31

2/1

9:00~10:30

A1

A2

B3

B4

10:50~12:20

B1

B2

C3

C5

Lunch

2:00~3:30

C1

C2

C4

C6

3:45~4:30

Discussion

Discussion

Discussion

Discussion

 
Reference:
[1] C. Okonek, H. Spindler, M. Schneider, Vector Bundles on Complex Projective Spaces, Progress in Mathematics, Birkauser. 
[2] M. Atiyah, Vector Bundles over an Elliptic Curve, Proceedings of the London Mathematical Society, Issue 1, 1957, 414–452,
[3] A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Comp. Math. 125 (2001), 327-344
 





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