Higher-Degree Holomorphic Contact Structures
Hisashi Kasuya, Dan Popovici, Luis Ugarte
TL;DR
The paper develops a new dual framework of higher-degree holomorphic geometric structures by introducing holomorphic $p$-contact and holomorphic $s$-symplectic manifolds, showing these generalize classical holomorphic contact and symplectic structures and yield Calabi–Yau manifolds via $u_{\Gamma}=\boldsymbol\Gamma\wedge\partial\boldsymbol\Gamma$. It develops foundational cohomology and deformation tools, including a generalized Tian–Todorov theory for vector-valued forms and a detailed study of the sheaves ${\cal F}_{\Gamma}$ and ${\cal G}_{\Gamma}$, which control horizontal and vertical directions. The authors establish two central themes: structure theorems connecting holomorphic $p$-contact manifolds to holomorphic $s$-symplectic bases (and conversely under suitable fibrations) and unobstructedness results for essential horizontal deformations, extending Bogomolov–Tian–Todorov to this non-Kähler setting. These results open avenues toward a non-Kähler mirror-symmetry picture where deformations of $\Gamma$ and $\partial\Gamma$ interact with the foliation geometry and base $s$-symplectic structures.
Abstract
We introduce the classes of holomorphic $p$-contact manifolds and holomorphic $s$-symplectic manifolds that generalise the classical holomorphic contact and holomorphic symplectic structures. After observing their basic properties and exhibiting a wide range of examples, we give two types of general conceptual results involving the former class of manifolds: structure theorems and unobstructedness theorems. The latter type generalises to our context the classical Bogomolov-Tian-Todorov theorem for a type of small deformations of complex structures that generalise the small essential deformations previously introduced for the Iwasawa manifold and for Calabi-Yau page-$1$-$\partial\bar\partial$-manifolds.