First, I never learned complex manifold properly (I am more familiar with complex variety) hence the questions will be very elementary. I apologise for this.
Let $X$ be a complex smooth manifold of dimension $n$. Then the singular cohomology of $X$ with complex coefficient has top degree $2n$.
Also, we can consider holomorphic de Rham cohomology of $X$. In this case, top degree is $n$.
How can they coincide if de Rham cohomology only tells about the even degree?
So my question is that how are holomorphic de Rham cohomology and singular cohomology with complex coefficient related.
Also, where can I get more information about this?