Let $\mathcal{M}$ be a projective smooth moduli space over $\mathbb{C}$ (the specific example I have in mind is the moduli of curves $\mathcal{M}_g)$. Consider a point $[X]\in \mathcal{M}(\mathbb{C})$. Then via deformation theory in algebraic geometry, we say that the tangent space $T_{[X]}\mathcal{M}$ consists of lifts (or rather isomorphism classes of lifts) of $X$ to $\mathbb{C}[\varepsilon]/(\varepsilon^2)$. This is naturally a $\mathbb{C}$-vector space. My question is in two parts:
Does this construction match with the tangent space of the analytification $\mathcal{M}^{\mathrm{an}}$ at $X$? (This question doesn't need compact, I think)
Since I assume that $\mathcal{M}$ is projective, then $\mathcal{M}^{\mathrm{an}}$ is compact. If we choose a basis of the projective space, we get an induced metric on $\mathcal{M}^{\mathrm{an}}$ (thanks to the comments for pointing me to this gap!). By Hopf-Rinow, the exponential map $T_{[X]}\mathcal{M}^{\mathrm{an}}\rightarrow \mathcal{M}^{\mathrm{an}}$ is well-defined (if we assume that $[X]$ does not lie in the boundary of $\mathcal{M}^{\mathrm{an}}$. If part 1) is true, the tangent space corresponds to infinitesimal lifts of $X$. Can we explicitly say what the image of such a lift is under the exponential map?
If this isn't possible in general, is there perhaps a way to do it for the moduli of curves, moduli of abelian varieties or the Hilbert scheme?
