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A homotopy formula for $a_q$ domains in complex manifolds

Xianghong Gong, Ziming Shi

TL;DR

This work proves a global homotopy formula for relatively compact $a_q$ domains in complex manifolds by adapting Polyakov's local-homotopy approach to a global setting. The authors construct finite-dimensional projections and a decomposition of the $(0,q)$-form space, enabling the definition of global homotopy operators $P_q$ and $P_{q+1}$ that improve regularity by exactly $1/2$ derivative under suitable boundary smoothness. By analyzing convex/concave boundary regions and combining these constructions, they obtain a final pair of operators $(P_q,P_{q+1})$ on $a_q$ domains yielding $f=\bar{\partial}P_q f+P_{q+1}\bar{\partial}f+H_q f$ with $H_q$ finite-dimensional. The results advance the global Newlander–Nirenberg program by providing a robust, derivative-gaining homotopy framework that depends only on the boundary regularity and not on the particular form, with potential applications to global CR embedding and deformations of complex structures.

Abstract

We construct a global homotopy formula for $a_q$ domains in a complex manifold. The homotopy operators in the formula will gain $1/2$ derivative in Hölder-Zygmund spaces $Λ^{r}$ when the boundaries of the domains are in $Λ^{r+3}$ with $r>0$.

A homotopy formula for $a_q$ domains in complex manifolds

TL;DR

This work proves a global homotopy formula for relatively compact domains in complex manifolds by adapting Polyakov's local-homotopy approach to a global setting. The authors construct finite-dimensional projections and a decomposition of the -form space, enabling the definition of global homotopy operators and that improve regularity by exactly derivative under suitable boundary smoothness. By analyzing convex/concave boundary regions and combining these constructions, they obtain a final pair of operators on domains yielding with finite-dimensional. The results advance the global Newlander–Nirenberg program by providing a robust, derivative-gaining homotopy framework that depends only on the boundary regularity and not on the particular form, with potential applications to global CR embedding and deformations of complex structures.

Abstract

We construct a global homotopy formula for domains in a complex manifold. The homotopy operators in the formula will gain derivative in Hölder-Zygmund spaces when the boundaries of the domains are in with .

Paper Structure

This paper contains 14 sections, 24 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximate homotopy formula with compact term
  4. Existence of projection operators and their norms
  5. A decomposition for the total space $\Lambda_{(0,q)}^s$
  6. Inverting $A^{(1)}$ on $\mathcal{R} ^ s+\mathcal{S}$
  7. Homotopy operators on the subspace $\mathcal{R}\oplus \mathcal{S}$
  8. The projection $\Pi_{d\mathcal{P}}$
  9. Analogue of Polyakov's homotopy operators on $\tilde{\Lambda}^+_q$
  10. An improved Polyakov homotopy formula
  11. Improved $P_q$
  12. Improved $P_q$ and $P_{q+1}$ on convex side of the boundary
  13. Improved $P_q$ and $P_{q+1}$ on concave side of the boundary
  14. Improved $P_q$ and $P_{q+1}$ for $a_q$ domains

Key Result

Theorem 1.1

Let $M$ be a relatively compact $a_q$ domain in a complex manifold $\mathcal{M}$ of dimension $n$. Assume that $1\leq q\leq n-2$, $0<r<\infty$, and $b M\in\Lambda^{r_0+3}$ with $r\leq r_0\leq\infty$. Then there exists a decomposition for all $f\in \Lambda^r_{q}(\overline M, V)$ with $\bar{\partial} f\in \Lambda^r$. Furthermore, we have the following

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.4
  • ...and 42 more