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 $\nabla df$ 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 $H^1({\Delta }_X,{\Lambda }^1)\to {\mathrm{gr}}_W^2{H}^2(X_{\infty })$, 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.

15 pages, comments welcome!