Hessian metrics with distribution coefficients on a 2-sphere

Abstract: 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.