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Properties of Holomorphic $p$-Contact Manifolds

Hisashi Kasuya, Dan Popovici, Luis Ugarte

TL;DR

This work develops a cohesive framework for holomorphic $p$-contact manifolds beyond the Kähler setting by introducing a vector-valued Lie-derivative calculus and exploiting it to study hyperbolicity and deformations. It generalizes classical Tian–Todorov theory through a vector-valued Cartan-type calculus, yielding generalized deformation formulae and new unobstructedness results for $p$-contact deformations under page-$1$-$dard$ assumptions. The paper also analyzes the obstructions posed by positive currents, constructs and analyzes the horizontal/vertical sheaves ${ m F}_ Gamma$ and ${ m G}_ Gamma$ in explicit 3- and higher-dimensional examples, and investigates two notions of $p$-contact hyperbolicity, linking metric and map-based perspectives via Ahlfors currents. Collectively, these results advance non-Kähler mirror-symmetry programs and offer tools for probing deformations, hyperbolicity, and geometric structures on complex manifolds with holomorphic $p$-contact forms, including canonical examples like the Iwasawa manifold and related nilmanifolds.

Abstract

We continue the study of compact holomorphic $p$-contact manifolds $X$ that we introduced recently by expanding the discussion to include non-Kähler hyperbolicity issues and a differential calculus based on what we call the Lie derivative with respect to a $(0,\,q)$-form with values in the holomorphic tangent bundle of $X$. We also propose the notion of $p$-contact deformations for which we prove a Bogomolov-Tian-Todorov-type unobstructedness theorem to order two. This kind of small deformations of the complex structure is related to the essential horizontal deformations that we introduced in our previous work and forms part of a wider on-going project aimed at developing a non-Kähler mirror symmetry theory that was first tested on the Iwasawa manifold and subsequently on Calabi-Yau page-$1$-$\partial\bar\partial$-manifolds.

Properties of Holomorphic $p$-Contact Manifolds

TL;DR

This work develops a cohesive framework for holomorphic -contact manifolds beyond the Kähler setting by introducing a vector-valued Lie-derivative calculus and exploiting it to study hyperbolicity and deformations. It generalizes classical Tian–Todorov theory through a vector-valued Cartan-type calculus, yielding generalized deformation formulae and new unobstructedness results for -contact deformations under page-- assumptions. The paper also analyzes the obstructions posed by positive currents, constructs and analyzes the horizontal/vertical sheaves and in explicit 3- and higher-dimensional examples, and investigates two notions of -contact hyperbolicity, linking metric and map-based perspectives via Ahlfors currents. Collectively, these results advance non-Kähler mirror-symmetry programs and offer tools for probing deformations, hyperbolicity, and geometric structures on complex manifolds with holomorphic -contact forms, including canonical examples like the Iwasawa manifold and related nilmanifolds.

Abstract

We continue the study of compact holomorphic -contact manifolds that we introduced recently by expanding the discussion to include non-Kähler hyperbolicity issues and a differential calculus based on what we call the Lie derivative with respect to a -form with values in the holomorphic tangent bundle of . We also propose the notion of -contact deformations for which we prove a Bogomolov-Tian-Todorov-type unobstructedness theorem to order two. This kind of small deformations of the complex structure is related to the essential horizontal deformations that we introduced in our previous work and forms part of a wider on-going project aimed at developing a non-Kähler mirror symmetry theory that was first tested on the Iwasawa manifold and subsequently on Calabi-Yau page---manifolds.

Paper Structure

This paper contains 15 sections, 17 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incompatibility with certain currents
  4. Integrability
  5. Examples in dimension $3$
  6. Lie derivatives with respect to vector-valued forms
  7. Background
  8. Lie derivatives w.r.t. $T^{1,\,0}X$-valued $(0,\,1)$-forms
  9. Applications of the Lie derivative calculus w.r.t. $T^{1,\,0}X$-valued forms
  10. Generalised Tian-Todorov formulae
  11. The sheaves ${\cal F}_\Gamma$ and ${\cal G}_\Gamma$: examples and computations
  12. Contact hyperbolicity
  13. $p$-contact small deformations
  14. Meaning of unobstructedness for arbitrary small deformations
  15. Unobstructedness to order two

Key Result

Proposition 3.2

Let $X=G/\Gamma$ be the quotient of a nilpotent Lie group of real dimension $6$ endowed with a left invariant complex structure. Suppose that $X$ has a holomorphic contact structure. Then, $X$ is the Iwasawa manifold $(1)$ or the non complex parallelisable nilmanifold $(4)$.

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Proposition 3.2
  • Corollary 3.3
  • Proposition 3.4
  • Lemma 4.1
  • Definition 4.2
  • Proposition 4.3
  • Lemma 4.4
  • Proposition 4.5
  • ...and 20 more