论文标题
附近光纤共同体的组合部分
Combinatorial part of the cohomology of the nearby fibre
论文作者
论文摘要
令$ f：x \ to s $是光盘上投影复杂的歧管的一项单位变性，使中央光纤$ y = f^{ - 1}（0）$的减少是简单的普通横梁，让$ x_ \ infty $为典型的附近光纤。在Kontsevich，Tschinkel，Mikhalkin和Zharkov的作品的基础上，我在$ Y $的双交叉点数上引入了分级代数$λ^\ bullet $，表示为$Δ_x$。 I show that there exists a map $H^q(Δ_X, Λ^p) \to \mathrm{gr}^W_{2p} H^{p+q}(X_\infty, \mathbb{Q})$, where $W$ is the monodromy weight filtration, which is injective whenever there exists a class $ω\in H^2(Y)$ which is组合和Lefschetz，一定的技术状况。当$ f $是$ k3 $表面的III型kulikov变性时，捆绑$λ^1 $以$Δ_x$以恩格尔（Engel）和弗里德曼（Friedman）的奇异性恢复了仿射结构。在这种情况下，我表明，从$Δ_x$上的Lagerberg意义上讲，从正面的$ d''$ - 封闭$（1,1）$ - 超级或超流。后者是在\ cite {hessian}中简单的仿射结构奇异性的情况下确定的，实际上，或骨的同时$λ^p $与附近的完整的纤维同时同时同时发生。
Let $f: X \to S$ 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 $X_\infty$ 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 $Δ_X$. I show that there exists a map $H^q(Δ_X, Λ^p) \to \mathrm{gr}^W_{2p} H^{p+q}(X_\infty, \mathbb{Q})$, where $W$ is the monodromy weight filtration, which is injective whenever there exists a class $ω\in H^2(Y)$ which is combinatorial and Lefschetz, a certain technical condition. When $f$ is a Type III Kulikov degeneration of $K3$ surfaces, the sheaf $Λ^1$ recovers the affine structure with singularities of Engel and Friedman on $Δ_X$. 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 $Δ_X$. The latter is established in the case of simple affine structure singularities in \cite{hessian}, in fact, the cohomology of sheaves $Λ^p$ coincides with the full nearby fibre cohomolgy then.
