Consider the complex algebraic moduli space $X:=\mathcal A_g^n$ of ppav's of dimension $g$ with some high enough level $n$ structure (so that it represents the corresponding functor).

Can one compute the dimension of the infinitesimal space of deformations to $X$ (as a function of $g$ and the level)?

If $V$ is a smooth projective variety, then I'm basically asking about $\dim H^1(V,T_V)$. I presume that, as $X$ is not projective, the dimension of its infinitesimal deformation space is given by $\dim H^1(\overline X,T_{\overline X}(log D))$, where $\overline X$ is a good compactification of $X$ and $D$ is its boundary divisor. [Edit: This is wrong as is explained below by Jason Starr. The space of infinitesimal deformations of $X$ as a $\mathbb C$-scheme is $\mathrm{Ext}^1(\Omega_{X/\mathbb C}, \mathcal O_X)$.

Computing the space $\mathrm{Ext}^1(\Omega_{X/\mathbb C}, \mathcal O_X)$ explicitly will require writing down $\Omega_{X/\mathbb C}$. Can this be done by pulling-back the relative cotangent bundle from the universal family along the zero section?