Dolbeault-Hochster Theory of Polytopal LVM Manifolds

Abstract

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.