Intuition: Topological notion of genus and Genus as the dimension of the complex vector space of holomorphic differentials

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In my self-study of Lec.7 of Belyi Maps and Dessins d'Enfants, I came across the following statement

Definition 3: Genus can be defined as the dimension of the complex vector space of holomorphic differentials.

Followed by the remark saying that

Remark 4: One can show that this agrees with the topological notion of genus.

Can someone point me to some work or book that helps me to understand this...? Or just answering what this is also super-appreciated!!

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