The arithmetic genus of a projective scheme $X$ is the alternating sum $\chi(\mathcal{O}_X) = \sum_{i=0}^{\dim X} (-1)^i \dim H^i(X, \mathcal{O}_X)$. Note that some authors call this quantity the holomorphic Euler characteristic. Additionally, some authors refer to the quantity $p_a(X) = (-1)^{\dim X} (\chi(\mathcal{O}_X) - 1)$ as the arithmetic genus.
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- Last edited by Sam Schiavone on 2023-07-12 20:40:28
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- 2023-07-12 20:40:28 by Sam Schiavone
- 2023-07-12 15:40:39 by Sam Schiavone
- 2023-06-02 18:51:50 by Avi Kulkarni
- 2023-06-02 18:47:46 by Avi Kulkarni
- 2023-06-02 13:50:29 by Juanita Duque-Rosero
- 2023-06-02 13:50:09 by Juanita Duque-Rosero