I noticed that the holomorphic Euler characteristic $\chi(C,\mathcal{O}_C)=1-g$ of a smooth complex curve $C$ of genus $g$ is just a half of the topological Euler characteristic $\chi_{top}(C)=2-2g$. Do they coincide by accident or is there any good explanation of this?
Holomorphic Euler characteristics and topological Euler characteristics of curves.
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algebraic-geometry
algebraic-curves
complex-geometry