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Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry

Lorenzo Sillari

TL;DR

The paper develops a suite of symplectic-inspired cut-and-paste techniques—generalized Luttinger surgery, generalized Gluck twist, and branched coverings—to construct stable generalized complex structures on high-dimensional manifolds. By working along $\mathcal J$-symplectic submanifolds and using careful $B$-field and local model arguments, the authors extend torus-surgery methods to $2m$-manifolds with $2m\ge6$, producing GC structures whose type-change loci can have multiple, non-homotopy-equivalent components. They also show how gluing diffeomorphism freedom and branched coverings increase the diversity of resulting manifolds, including explicit diffeomorphism types in dimension six and stable GC structures on elliptic surfaces and $T^2\times\Sigma_g$. Together these constructions enlarge the catalog of manifolds supporting stable GC structures and offer new tools for controlling fundamental groups and type-change loci. The results highlight a broad, flexible framework for importing symplectic techniques into generalized complex geometry in higher dimensions.

Abstract

Exploiting the affinity between stable generalized complex structures and symplectic structures, we explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting. We introduce generalized Luttinger surgery and generalized Gluck twist along $\mathcal{J}$-symplectic submanifolds. We also export branched coverings to the generalized complex setting. As an application, stable generalized complex structures are produced on a variety of high-dimensional manifolds. Remarkably, some of them have non-homotopy-equivalent path-connected components of their type change locus.

Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometry

TL;DR

The paper develops a suite of symplectic-inspired cut-and-paste techniques—generalized Luttinger surgery, generalized Gluck twist, and branched coverings—to construct stable generalized complex structures on high-dimensional manifolds. By working along -symplectic submanifolds and using careful -field and local model arguments, the authors extend torus-surgery methods to -manifolds with , producing GC structures whose type-change loci can have multiple, non-homotopy-equivalent components. They also show how gluing diffeomorphism freedom and branched coverings increase the diversity of resulting manifolds, including explicit diffeomorphism types in dimension six and stable GC structures on elliptic surfaces and . Together these constructions enlarge the catalog of manifolds supporting stable GC structures and offer new tools for controlling fundamental groups and type-change loci. The results highlight a broad, flexible framework for importing symplectic techniques into generalized complex geometry in higher dimensions.

Abstract

Exploiting the affinity between stable generalized complex structures and symplectic structures, we explain how certain constructions coming from symplectic geometry can be performed in the generalized complex setting. We introduce generalized Luttinger surgery and generalized Gluck twist along -symplectic submanifolds. We also export branched coverings to the generalized complex setting. As an application, stable generalized complex structures are produced on a variety of high-dimensional manifolds. Remarkably, some of them have non-homotopy-equivalent path-connected components of their type change locus.
Paper Structure (12 sections, 15 theorems, 47 equations)

This paper contains 12 sections, 15 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Luttinger surgery on -manifolds
  4. The underlying smooth manifold.
  5. The geometric structure.
  6. Examples and applications.
  7. Changing the gluing diffeomorphism.
  8. Generalized Gluck twist
  9. Branched coverings of generalized complex manifolds
  10. Branched coverings.
  11. The geometric structure.
  12. Examples and applications.

Key Result

Theorem A

Let $(M^{2m}, \mathcal{J}, H)$ be a stable generalized complex structure and let $S$ be its type change locus. Let $(\Sigma, \omega_\Sigma)$ be a symplectic $(2m-4)$-manifold. Assume that there exists an embedded $T^2 \times \Sigma \hookrightarrow M$ such that Then performing generalized Luttinger surgery along $T^2 \times \Sigma$ yields a stable generalized complex structure $(\Tilde{M}, \Tilde{

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5: CG18, Definition 2.10
  • Definition 2.6: cf. GH16 and Mun18, Definition 2.3.6
  • ...and 37 more