  Let $\Delta$ be a 2-sphere endowed with an affine structure away from a
finite set of points $P \subset \Delta$, and assume that the monodromy of the
associated connection $\nabla$ on $\Delta \setminus P$ around any point from
$P$ is unipotent. I show that there exists a pseudo-metric tensor with
distribution coefficients on $\Delta$ that is non-degenerate on $\Delta
\setminus P$ and that locally is of the form $\nabla d f$ for some convex
function $f$. In particular, if $X_\infty$ is the canonical nearby fibre of a
Type III degeneration of K3 surfaces in Kulikov form, $\Delta_X \cong S^2$ is
the dual intersection complex of the central fibre and $\Delta_X$ has simple
affine structure singularities, existence of such ``Hessian metric'' on
$\Delta_X$ implies that the map $H^1(\Delta_X, \Lambda^1) \to \mathrm{gr}^2_W
H^2(X_\infty)$, constructed previously in \cite{sus22}, where $W$ is the
monodromy weight filtration on $H^2(X_\infty)$ and $\Lambda^1$ is the
push-forward of the sheaf of parallel 1-forms along the open embedding $\Delta
\setminus P \hookrightarrow \Delta$, is an isomorphism.
