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課程名稱 |
複幾何
Complex Manifolds |
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開課學期 |
102-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
齊震宇 |
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課號 |
MATH5336 |
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課程識別碼 |
221 U5890 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一7,8(14:20~16:20)星期四3,4(10:20~12:10) |
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上課地點 |
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備註 |
上課教室:數學科學中心101教室(原新數學館101教室)
總人數上限：20人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1021CG |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
This is an introduction to complex geometry, which studies the geometry of complex manifolds and related topics. Those who are interested in attending my introductory course in algebraic geometry next semester are strongly suggested to take this one, for the current course may provide some intuitive background for the abstract theory of algebraic schemes and coherent sheaves. |
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課程目標 |
1. Complex analysis of several variables: Holomorphic functions, Hortogs' theorem on separate holomorphic functions, and Weierstrass' preparation theorem and division theorem.
2. Complex manifolds, complex/hermitian multilinear algebra, vector bundles, and hermitian metrics
3. Sheaf theory: presheaves, sheaves, sheafification, Cech cohomology of sheaves, and basic exact sequences.
4. Connections on vectorbundles and their curvature tensors.
5. Harmonic forms, Hodge's decomposition, and Serre's duality theorem for holomorphic vector bundles.
6. The hard Lefschetz theorem.
7. Divisors, line bundles, and the first Chern class.
8. Positivity and vanishing theorems.
9. The Blow-up construction and Kodaira's embedding theorem.
If time permits we will try to cover some of the following topics.
10. The theory of Stein spaces.
11. The theory of analytic coherent sheaves.
12. Hormander's L^2 estimate.
13. Nadel's vanishing theorem.
14. Siu's invariance of plurigenera.
15. Kodaira-Spencer-Kuranishi's theory of deformation of complex structures.
16. Variation of Hodge structures.
17. Yau's solution to the Calabi conjecture.
18. The Donaldson-Uhlenbeck-Yau theorem |
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課程要求 |
1. Commutative algebra, such as commutative rings, ideals, modules and their tensor products over rings, noetherian rings, PID and UFD, Hilbert's basis theorem, the Hilbert Nullstellensatz: See [AM], [J], and [ZS].
2. General topology: Ch. I of [B].
3. The language of category and algebraic topology, such as simplicial/singular homology and cohomology groups: See Ch. 1 to Ch. 6 of [S] or Ch. II to Ch. VI [B]. Some of the topics can be found in my lectures on Youtube. (To get them, simply search "代數拓樸" or my name.) Categorical thinking may appear quite a few times and the participants are strongly suggested to get familiar to that as much and early as possible.
4. Algebraic topology on smooth manifolds: Ch. I of [BT] or Ch. II and V of [B].
5. Linear analysis on smooth manifolds, such as elliptic regularity: Ch. 3 of [N].
6. Complex analysis: Ch. 1 to Ch. 7 of [A].
7. Real analysis/measure theory: Ch. 1 to Ch. 8 of [R1].
8. Functional analysis: Ch. 1, 2, and 6 of [R2]. |
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預期每週課後學習時數 |
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Office Hours |
另約時間 |
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指定閱讀 |
Arapura, Algebraic Geometry over the Complex Numbers.
Demailly, Complex Analytic and Differential Geometry. This is an online book available on the homepage of the author. This is a useful reference when one does research. On the other hand, I do not suggest you to take it as a textbook to follow because it contains too many topics and lots of technical treatments.
Griffiths and Harris, Principles of Algebraic Geometry. This book is the most frequently used textbook of the subject but it contains many typo and inaccurate points. Be very careful when you read.
Huybrechts, Complex Geometry.
Kodaira, Complex Manifolds and Deformation of Complex Structures, Ch. 1, 2, and 3.
Kodaira and Morrow, Complex Manifolds, Ch. 1, 2, and 3. |
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參考書目 |
Reference for the prerequisite requirement:
[A] Ahlfors, Complex Analysis.
[AM] Atiyah and MacDonaldson, Introduction to Commutative Algebra.
[B] Bredon, Topology and Geometry.
[BT] Bott and Tu, Differential Forms in Algebraic Topology.
[N] Narasimhan, analysis on real and complex manifolds.
[R1] Rudin, Real and Complex Analysis.
[R2] Rudin, Functional Analysis.
[S] Spanier, Algebraic Topology.
[ZS] Zariski and Samuel, Commutative Algebra, Vol I and II. |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
期中考 1 |
25% |
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2. |
期中考 2 |
30% |
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3. |
期末考或報告 |
30% |
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4. |
習題 |
15% |
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