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 (cokerB)∨ and its cokernel with (ker B)∨. 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 E1. Every manifold in this class fails the ∂∂̅ 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 h1(Θ)=19. The primary obstruction map is nonzero. The additional Cech class integrates in a holomorphic family.
Related
- Dolbeault cohomology of Endo-Pajitnov manifolds
- Laplace and Dolbeault operators on Sasakian manifolds
- A Dolbeault complex for the contact quantised flag manifolds of type C2 and D4
- Cohn-Vossen theory for locally conformally flat manifolds
- The recent counterexamples to the Eisenbud-Schreyer-Weyman Ulrich problem (2003) and to Hochster's small Cohen-Macaulay conjecture (1973): a survey, with a generalization to arbitrary dimension
BibTeX
@misc{katzarkov2026dolbeaulthochster,
title = {{Dolbeault-Hochster Theory of Polytopal LVM Manifolds}},
author = {Ludmil Katzarkov and Kyoung-Seog Lee and Ernesto Lupercio and Laurent Meersseman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37936},
eprint = {2609.37936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37936},
}
RIS
TY - UNPB
TI - Dolbeault-Hochster Theory of Polytopal LVM Manifolds
AU - Ludmil Katzarkov
AU - Kyoung-Seog Lee
AU - Ernesto Lupercio
AU - Laurent Meersseman
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37936
UR - https://arxiv.org/abs/2609.37936
AB - 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 (cokerB)^∨ and its cokernel with (ker B)^∨. 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 ∂∂̅ 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.
ER -