Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Obstructions of deforming complex structures and cohomology contractions

Xueyuan Wan, Wei Xia

TL;DR

The paper analyzes obstructions to deformations of compact complex manifolds through an analytic lens, refining the Kodaira principle by linking obstructions to the Frölicher spectral sequence via a finite-step deformation framework for $(p,q)$-forms. It shows that, under partial vanishing of the differentials $d_r^{p,q}$, obstruction classes land in the kernel of the cohomology contraction map, yielding a degree-wise Kodaira principle and new unobstructedness criteria for Calabi–Yau manifolds and certain non-Kähler settings. The results extend classical unobstructedness theorems and provide tools to study deformation spaces with potential singularities, including detailed analysis for complex parallelisable manifolds. Concrete examples such as the Iwasawa and Nakamura manifolds illustrate the applicability and limits of the criteria.

Abstract

The Kodaira principle asserts that suitable cohomological contraction maps annihilate obstructions to deforming complex structures. In this paper, we revisit these phenomena from a purely analytic point of view, developing a refined power series method for the deformation of $(p,q)$-forms and complex structures. Working with the Frölicher spectral sequence, we show that under natural partial vanishing conditions on its differentials, all obstruction classes lie in the kernel of the corresponding contraction maps. This yields a refined Kodaira principle that recovers and strictly extends the known results. As a main application, we obtain new unobstructedness criteria for compact complex manifolds with trivial canonical bundle.

Obstructions of deforming complex structures and cohomology contractions

TL;DR

The paper analyzes obstructions to deformations of compact complex manifolds through an analytic lens, refining the Kodaira principle by linking obstructions to the Frölicher spectral sequence via a finite-step deformation framework for -forms. It shows that, under partial vanishing of the differentials , obstruction classes land in the kernel of the cohomology contraction map, yielding a degree-wise Kodaira principle and new unobstructedness criteria for Calabi–Yau manifolds and certain non-Kähler settings. The results extend classical unobstructedness theorems and provide tools to study deformation spaces with potential singularities, including detailed analysis for complex parallelisable manifolds. Concrete examples such as the Iwasawa and Nakamura manifolds illustrate the applicability and limits of the criteria.

Abstract

The Kodaira principle asserts that suitable cohomological contraction maps annihilate obstructions to deforming complex structures. In this paper, we revisit these phenomena from a purely analytic point of view, developing a refined power series method for the deformation of -forms and complex structures. Working with the Frölicher spectral sequence, we show that under natural partial vanishing conditions on its differentials, all obstruction classes lie in the kernel of the corresponding contraction maps. This yields a refined Kodaira principle that recovers and strictly extends the known results. As a main application, we obtain new unobstructedness criteria for compact complex manifolds with trivial canonical bundle.

Paper Structure

This paper contains 9 sections, 24 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Frölicher spectral sequence
  3. Deformations of $(p,q)$-forms for finite steps
  4. Obstructions and the cohomology contractions
  5. Obstructions to deforming complex structures
  6. The cohomology contraction map
  7. The contraction map annihilates obstructions
  8. Kodaira principles
  9. Deformations of complex parallelisable manifolds

Key Result

Theorem 1.1

Let $X$ be a compact Kähler manifold and let $\sum_{1\le j\le N}\phi_j$ be an $N$-th order deformation of $X$. Then the obstruction of $\sum_{1\le j\le N}\phi_j$ lies in the kernel of the cohomological contraction map eqn1.1, i.e. In particular, if $\ker\mu=0$, then the deformations of $X$ are unobstructed.

Theorems & Definitions (32)

  • Theorem 1.1: Cle05
  • Theorem 1.2: Man, Kodaira principle
  • Theorem 1.3
  • Theorem 1.4: MR3978322
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: WX2023
  • Lemma 2.4: WX2023
  • ...and 22 more