I just begin to learn about Hodge Theory. The following statement is heard from somewhere, which I know it is true. But I don't understand the exact detail of how and why.
Let $M$ be a closed Riemann Surface. Then $$H^1(M,\mathbb{C})=\mathcal H^{0,1}(M)\oplus \mathcal H^{1,0}(M),$$ where $\mathcal H$ is the space of harmonic forms.
The question is why then $\dim \mathcal H^{1,0}(M,\mathbb{C})=\operatorname{genus}(M)$? How to understand it intuitively, and how can I prove it?