2022/12/20 by Dmitry Sustretov, Sustretov, Dmitry
Mathematics · #14D06 #53A15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2212.10640
openalex publication_date 2022/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Δ be a 2-sphere endowed with an affine structure away from a finite set of points P ⊂ Δ, and assume that the monodromy of the associated connection ∇ on Δ∖ P around any point from P is unipotent. I show that there exists a pseudo-metric tensor with distribution coefficients on Δ that is non-degenerate on Δ∖ P and that locally is of the form ∇ d f for some convex function f. In particular, if X_∞ is the canonical nearby fibre of a Type III degeneration of K3 surfaces in Kulikov form, ΔX ≅ S2 is the dual intersection complex of the central fibre and ΔX has simple affine structure singularities, existence of such ``Hessian metric'' on ΔX implies that the map H1(ΔX, Λ1) → gr2W H2(X_∞), constructed previously in \citesus22, where W is the monodromy weight filtration on H2(X_∞) and Λ1 is the push-forward of the sheaf of parallel 1-forms along the open embedding Δ∖ P \hookrightarrow Δ, is an isomorphism.