I fear I'm missing something important here, so forgive me if my question is stupid.

Consider $\mathcal{M}_g$ the moduli space of Riemann surfaces of genus $g>2$ and $\mathcal{H}_g$ the moduli space of abelian differentials on Riemann surfaces of genus $g$. By that I mean $\mathcal{H}_g:=\{(C,\omega)\}/_\simeq$ with $C$ a Riemann surface and $\omega$ an abelian differential on $C$. The moduli space $\mathcal{H}_g$ can be considered as an holomorphic fibration on $\mathcal{M}_g$ of rank $g$. From this fact we get that, since $\mathcal{M}_g$ has complex dimension $3g-3$, that $\mathcal{H}_g$ has complex dimension $4g-3$.

An abelian differential on $C$ of genus $g$ has degree $2g-2$. For every partition $\underline{k}$ of $2g-2$ we can consider the stratum $\mathcal{H}_g(\underline{k})\subset\mathcal{H}_g$ of abelian differentials with degrees for the zeroes prescribed by $\underline{k}$. It is known that each stratum $\mathcal{H}_g(\underline{k})$ has complex dimension $2g+k-1$.

**From what I wrote follows that the dimensions of the strata should add up to the dimension of $\mathcal{H}_g$.**

But this doesn't follow from my computations..

Consider the easiest case of $g=2$, then $dim_{\mathbb{C}}\mathcal{H}_2=5$ and $dim_{\mathbb{C}}\mathcal{H}_2(2,0)=4$, $dim_{\mathbb{C}}\mathcal{H}_2(1,1)=5$, but $9\neq 5$.

**What am I getting wrong?**