黎曼—羅赫定理的一個代數方法之證明

A Proof of Riemann-Roch Theorem by Algebraic Methods

黃皓偉(Hao-Wei Huang)

指導教授 ： 蔡宜洵

國立臺灣大學/理學院/數學研究所/碩士(2018年)

https://doi.org/10.6342/NTU201801660

摘要

無資料

關鍵字

同調代數 ； 指標定理

並列摘要

We start from some basic notions, like sheaves and cohomology, and try to introduce and prove Riemann-Roch theorem in the 2-dimension case. The definition of cohomology of a sheaf is more difficult to compute in some situation. However, the Čech cohomology of a sheaf over a paracompact space is isomorphic to the usual definition of cohomology,and Čech cohomology gives us a more concrete way to think what the cohomology of a sheaf is. In chapter 3 we introduce the concept of twisted complexes. We will use it to compute Ext and the class in Čech cohomology which is in the statement of Riemann-Roch theorem, and identify this class with characteristic class Td in cochain level by direct computation.

並列關鍵字

sheaf ； homological algebra ； spectral sequence ； Riemann-Roch theorem

參考文獻

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2. Toledo, D. , Tong, Y. L. , A parametrix for overline partial and Riemann-Roch in Cech theory. Topology, v. 15, 1976.

3. Toledo,D. , Tong, Y. L. , Duality and intersection theory in complex manifolds. I. Math. Ann. 237, 1978.

4. O'Brian, N. , Toledo, D. , Tong, Y. L. , The trace map and characteristic classes for coherent sheaves. Amer. J. Math. 103, 1981.

5. O'Brian, N. , Toledo, D. , Tong, Y. L. , Hirzebruch-Riemann-Roch for coherent sheaves. Ibid. , 103, 1981.