2022/02/17 by Dmitry Sustretov, Sustretov, Dmitry
Mathematics · #14D06 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2202.08888
openalex publication_date 2022/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f: X → S be a unipotent degeneration of projective complex manifolds over a disc such that the reduction of the central fibre Y=f-1(0) is simple normal crossings, and let X_∞ be the canonical nearby fibre. Building on the work of Kontsevich, Tschinkel, Mikhalkin and Zharkov, I introduce a sheaf of graded algebras Λ^\bullet on the dual intersection complex of Y, denoted ΔX. I show that there exists a map Hq(ΔX, Λp) → grW2p Hp+q(X_∞, ℚ), where W is the monodromy weight filtration, which is injective whenever there exists a class ω∈ H2(Y) which is combinatorial and Lefschetz, a certain technical condition. When f is a Type III Kulikov degeneration of K3 surfaces, the sheaf Λ1 recovers the affine structure with singularities of Engel and Friedman on ΔX. In this case, I show that existence of such class follows from the existence of a positive d''-closed (1,1)-superform or supercurrent in the sense of Lagerberg on ΔX. The latter is established in the case of simple affine structure singularities in \citehessian, in fact, the cohomology of sheaves Λp coincides with the full nearby fibre cohomolgy then.