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An observation on the existence of stable generalized complex structures on ruled surfaces

Rafael Torres

TL;DR

The paper proves that on sphere bundles over closed surfaces of genus $g\ge 2$, every stable generalized complex structure is of constant type, so no type-change locus can occur and the structure is induced by a Kähler form. The argument combines Li–Li’s minimal-genus results for surfaces in irrational ruled four-manifolds, Cavalcanti–Gualtieri’s GC surgery that links type-change loci to symplectic models, Oszváth–Szabó’s Thom conjecture, and Liu’s classification of irrational ruled four-manifolds to contradict the existence of a nonempty type-change locus. Specifically, any hypothetical non-constant-type GC structure would force a genus-minimizing torus in a negative-Euler-characteristic, irrational-ruled model, which is impossible. Consequently, these negative-Euler-characteristic sphere bundles admit only constant-type GC structures, clarifying the landscape of generalized complex geometries on ruled surfaces and connecting GC theory with classical 4-manifold topology.

Abstract

We point out that any stable generalized complex structure on a sphere bundle over a closed surface of genus at least two must be of constant type.

An observation on the existence of stable generalized complex structures on ruled surfaces

TL;DR

The paper proves that on sphere bundles over closed surfaces of genus , every stable generalized complex structure is of constant type, so no type-change locus can occur and the structure is induced by a Kähler form. The argument combines Li–Li’s minimal-genus results for surfaces in irrational ruled four-manifolds, Cavalcanti–Gualtieri’s GC surgery that links type-change loci to symplectic models, Oszváth–Szabó’s Thom conjecture, and Liu’s classification of irrational ruled four-manifolds to contradict the existence of a nonempty type-change locus. Specifically, any hypothetical non-constant-type GC structure would force a genus-minimizing torus in a negative-Euler-characteristic, irrational-ruled model, which is impossible. Consequently, these negative-Euler-characteristic sphere bundles admit only constant-type GC structures, clarifying the landscape of generalized complex geometries on ruled surfaces and connecting GC theory with classical 4-manifold topology.

Abstract

We point out that any stable generalized complex structure on a sphere bundle over a closed surface of genus at least two must be of constant type.
Paper Structure (6 sections, 6 theorems, 4 equations)

This paper contains 6 sections, 6 theorems, 4 equations.

Table of Contents

  1. Introduction.
  2. Acknowledgements:
  3. Background results.
  4. A relation between the symplectic Kodaira dimension and a stable generalized complex surgery
  5. The minimal genus of symplectic surfaces in irrational ruled symplectic four-manifolds
  6. Proof of Theorem \ref{['Theorem A']}

Key Result

Theorem 1

The type change locus of a stable generalized complex structure on a sphere bundle over a closed surface of genus strictly greater than one is empty. In particular, any stable generalized complex structure on such a four-manifold arises from a Kähler structure.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Remark 7