  Let $S$ be a connected closed oriented surface of genus $g$. Given a
triangulation (resp. quadrangulation) of $S$, define the index of each of its
vertices to be the number of edges originating from this vertex minus $6$
(resp. minus $4$). Call the set of integers recording the non-zero indices the
profile of the triangulation (resp. quadrangulation). If $\kappa$ is a profile
for triangulations (resp. quadrangulations) of $S$, for any $m\in
\mathbb{Z}_{>0}$, denote by $\mathscr{T}(\kappa,m)$ (resp.
$\mathscr{Q}(\kappa,m)$) the set of (equivalence classes of) triangulations
(resp. quadrangulations) with profile $\kappa$ which contain at most $m$
triangles (resp. squares). In this paper, we will show that if $\kappa$ is a
profile for triangulations (resp. for quadrangulations) of $S$ such that none
of the indices in $\kappa$ is divisible by $6$ (resp. by $4$), then
$\mathscr{T}(\kappa,m)\sim c_3(\kappa)m^{2g+|\kappa|-2}$ (resp.
$\mathscr{Q}(\kappa,m) \sim c_4(\kappa)m^{2g+|\kappa|-2}$), where $c_3(\kappa)
\in \mathbb{Q}\cdot(\sqrt{3}\pi)^{2g+|\kappa|-2}$ and $c_4(\kappa)\in
\mathbb{Q}\cdot\pi^{2g+|\kappa|-2}$. The key ingredient of the proof is a
result of J. Koll\'ar on the link between the curvature of the Hogde metric on
vector subbundles of a variation of Hodge structure over algebraic varieties,
and Chern classes of their extensions. By the same method, we also obtain the
rationality (up to some power of $\pi$) of the Masur-Veech volume of arithmetic
affine submanifolds of translation surfaces that are transverse to the kernel
foliation.
