AI 中文总结
本文通过有限曲率复形计算极小稳定多面体LVM流形的Dolbeault上同调，揭示其与可见单纯球面子复形及甲板格的关系，并证明该类流形均不满足$\partial\bar\partial$引理，同时构造了曲率秩下降的全纯族。
AI 中文摘要
我们利用有限曲率复形计算了极小稳定多面体LVM流形的Dolbeault上同调。设$B$为不可或缺的权块。一个典范正合序列将曲率映射的核等同于$(\operatorname{coker}B)^\vee$，其余核等同于$(\ker B)^\vee$。在满秩情形下，Dolbeault群是可见单纯球面的诱导子复形的约化上同调群与甲板格的复对偶上的外代数的张量和的直和。一个证明使用曲率模型和Stanley--Reisner环的Tor；第二个证明通过完备Laurent展开和全纯下降给出加性比较。满曲率秩等价于在$E_1$处的Frolicher退化。该类中的每个流形都不满足$\partial\bar\partial$引理。对于多边形，面数和曲率秩决定了整个Hodge菱形。我们构造了一个在圆盘上的真全纯族，其固定可见正方形在原点处曲率秩下降。Dolbeault上同调的总维数在那里增加了十六。在满秩时，全纯下降也计算切丛上同调，包括正Cech度中的共振贡献。我们确定了归一化权族的Kodaira--Spencer映射，并给出了具有光滑基底的半普适性的上同调判据。对于正方形上的一个共振例子，全局向量场具有九个元素的多项式基，且$h^1(\Theta)=19$。主要障碍映射非零。额外的Cech类在全纯族中积分。
英文摘要
We compute the Dolbeault cohomology of minimally stable polytopal LVM manifolds using a finite curvature complex. Let $B$ be the indispensable weight block. A canonical exact sequence identifies the kernel of the curvature map with $(\operatorname{coker}B)^\vee$ and its cokernel with $(\ker B)^\vee$. At full rank, the Dolbeault groups are direct sums of reduced cohomology groups of induced subcomplexes of the visible simplicial sphere, tensored with the exterior algebra on the complex dual of the deck lattice. One proof uses the curvature model and Stanley--Reisner Tor; a second gives an additive comparison through completed Laurent expansions and holomorphic descent. Full curvature rank is equivalent to Frolicher degeneration at $E_1$. Every manifold in this class fails the $\partial\bar\partial$ lemma. For polygons, the number of facets and the curvature rank determine the entire Hodge diamond. We construct a proper holomorphic family over a disk with fixed visible square whose curvature rank drops at the origin. The total dimension of Dolbeault cohomology increases there by sixteen. At full rank, holomorphic descent also computes tangent-sheaf cohomology, including resonant contributions in positive Cech degree. We determine the Kodaira--Spencer map of the normalized weight family and give a cohomological criterion for semiuniversality with smooth base. For a resonant example over the square, the global vector fields have a polynomial basis of nine elements and $h^1(Θ)=19$. The primary obstruction map is nonzero. The additional Cech class integrates in a holomorphic family.
