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On manifold-like polyfolds as differential geometrical objects with applications in complex geometry

Per Åhag, Rafał Czyż, Håkan Samuelsson Kalm, Aron Persson

TL;DR

The paper argues for treating $M$-polyfolds as the primary differential-geometric objects, enabling smooth local changes of dimension and extending differential geometry to scale-smooth settings. It builds a tensor framework, proves the existence of strong Riemannian metrics on reflexive Hilbert $M$-polyfolds, and shows that even-dimensional class $ ext{Γ}$ $M$-polyfolds admit an integrable almost complex structure with a compatible symplectic form, thereby enabling complex-geometric structures on these spaces. It then defines complex $M$-polyfolds and develops a theory of sc-holomorphic maps, including explicit examples like $X=Ccup P$, and discusses integrability criteria and the limits of extending the Newlander–Nirenberg theorem to the infinite-dimensional M-polyfold setting. Overall, the work lays groundwork for complex-analytic geometry in the polyfold context, inspired by Lempert, and points toward broader applications in complex geometry and related fields.

Abstract

We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To establish their utility, we introduce tensors and prove the existence of Riemannian metrics, symplectic structures, and almost complex structures within the M-polyfold framework. Drawing inspiration from a series of highly acclaimed articles by László Lempert, we lay the foundation for advancing geometry and function theory in complex M-polyfolds.

On manifold-like polyfolds as differential geometrical objects with applications in complex geometry

TL;DR

The paper argues for treating -polyfolds as the primary differential-geometric objects, enabling smooth local changes of dimension and extending differential geometry to scale-smooth settings. It builds a tensor framework, proves the existence of strong Riemannian metrics on reflexive Hilbert -polyfolds, and shows that even-dimensional class -polyfolds admit an integrable almost complex structure with a compatible symplectic form, thereby enabling complex-geometric structures on these spaces. It then defines complex -polyfolds and develops a theory of sc-holomorphic maps, including explicit examples like , and discusses integrability criteria and the limits of extending the Newlander–Nirenberg theorem to the infinite-dimensional M-polyfold setting. Overall, the work lays groundwork for complex-analytic geometry in the polyfold context, inspired by Lempert, and points toward broader applications in complex geometry and related fields.

Abstract

We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To establish their utility, we introduce tensors and prove the existence of Riemannian metrics, symplectic structures, and almost complex structures within the M-polyfold framework. Drawing inspiration from a series of highly acclaimed articles by László Lempert, we lay the foundation for advancing geometry and function theory in complex M-polyfolds.
Paper Structure (6 sections, 8 theorems, 119 equations, 3 figures)

This paper contains 6 sections, 8 theorems, 119 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A primer on M-polyfolds
  3. Scale topology and scale Banach spaces
  4. M-polyfolds
  5. Riemannian Metrics, Symplectic and Almost Complex Structures
  6. Complex M-polyfolds

Key Result

Theorem 2.10

Let $(U,C,E)$, $(V,D,F)$, and $(W,Q,G)$ be open tuples. Furthermore, suppose that $f:U\rightarrow V$ and $g:V\rightarrow W$ are sc$^1$-maps. Then, the function $g\circ f$ is an sc$^1$-map and

Figures (3)

  • Figure 1: An M-polyfold in $\mathbb{R}^3$.
  • Figure 2: A projected image of the M-polyfold $X=C\cup P$ in $\mathbb{R}^3$.
  • Figure 3: An illustration of $\Gamma_0^1$.

Theorems & Definitions (58)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Theorem 2.10: hofer2017polyfold
  • ...and 48 more