[Paper Review] C^\infty-logarithmic transformations and generalized complex structures

Ryushi Goto, Kenta Hayano|arXiv (Cornell University)|May 17, 2013

Homotopy and Cohomology in Algebraic Topology12 references3 citations

TL;DR

This paper introduces a new construction of twisted generalized complex structures on 4-manifolds using $C^inity$-logarithmic transformations along symplectic 2-tori with trivial normal bundle. By successively applying multiplicity-0 logarithmic transformations, the authors show that manifolds such as $(2m-1)\mathbb{C}P^2\#l\overline{\mathbb{C}P}^2$, $(2m-1)S^2\times S^2$, and $S^1\times S^3$ admit generalized complex structures with an arbitrary number $n$ of connected components in the type-changing locus, even when no such structures existed before.

ABSTRACT

Applying logarithmic transformations along 2-tori, we construct a generalized complex structure J_n with n type changing luci for every $n\geq 0$ on genus 1-Lefschetz fibrations with a cusp neighborhood, which include elliptic surfaces with non-zero euler characteristic. Applying a technique of broken Lefschetz fibrations, we further obtain twisted generalized complex structures with arbitrary large numbers of connected components of type changing loci on the manifold which is obtained from a symplectic manifold by logarithmic transformations of multiplicity 0 on a symplectic 2-torus with trivial normal bundle. The connected sums $(2m+1)S^2 imes S^2$ for $m\geq 0$, $(2n-1)\C P^2# (10n-1)\ol{\C P^2}$ and $S^1 imes S^3$ admit twisted generalized complex structures J_n with n type changing luci for arbitrary large n.

Motivation & Objective

• To extend existing constructions of generalized complex structures to arbitrary logarithmic transformations, particularly multiplicity-0 and multiplicity-1 transformations.
• To demonstrate that logarithmic transformations preserve the diffeomorphism type of 4-manifolds under specific conditions, especially when applied to symplectic 2-tori with trivial normal bundle.
• To construct generalized complex structures with an arbitrarily large number of connected components in the type-changing locus on known 4-manifolds.
• To show that the number of type-changing loci can be increased beyond the natural limit of two in generalized Kähler structures, highlighting a sharp contrast in behavior.

Proposed method

• The authors use $C^\infty$-logarithmic transformations—specifically multiplicity-0 and multiplicity-1 transformations—on symplectic 2-tori embedded in a generalized complex 4-manifold with trivial normal bundle.
• They apply techniques from broken Lefschetz fibrations to analyze the topological effects of these transformations and show that the diffeomorphism type of the underlying manifold remains unchanged.
• A key technical tool is Lemma 4.4, which proves that a multiplicity-1 logarithmic transformation on a 2-torus in a genus-1 Lefschetz fibration with a cusp neighborhood preserves the diffeomorphism type.
• The construction relies on the fact that the monodromy around the torus induces a trivial element in $\pi_1(\mathrm{Diff}^+(S^2)) \cong \mathbb{Z}/2\mathbb{Z}$, allowing extension of the diffeomorphism over the full neighborhood.
• The method generalizes earlier results by Cavalcanti and Gualtieri and Torres, extending them to arbitrary $n$ via iterative transformations.
• The authors use the equivalence between generalized Kähler structures and bihermitian structures to contrast their results with the known upper bound of two type-changing components in such settings.

Experimental results

Research questions

• RQ1 Can generalized complex structures with an arbitrary number of type-changing loci be constructed on 4-manifolds that do not admit complex or symplectic structures?
• RQ2 Does a logarithmic transformation of multiplicity 0 on a symplectic 2-torus with trivial normal bundle preserve the diffeomorphism type of the ambient 4-manifold?
• RQ3 Can the number of connected components of the type-changing locus be increased beyond the natural limit of two, as seen in generalized Kähler structures?
• RQ4 Is there a systematic way to generate generalized complex structures with $n$ type-changing loci for any $n$ on manifolds like $S^1\times S^3$ or connected sums of $S^2\times S^2$?
• RQ5 Can the Gluck twist on a 2-sphere fiber be realized via a multiplicity-1 logarithmic transformation on an associated 2-torus?

Key findings

• The manifold $(2m-1)\mathbb{C}P^2\#l\overline{\mathbb{C}P}^2$ admits a twisted generalized complex structure $\mathcal{J}_n$ with $n$ connected components of the type-changing locus for any $n \geq 0$, even when $m \neq 1$ and no complex or symplectic structure exists.
• The connected sum $(2m-1)S^2\times S^2$ admits a twisted generalized complex structure $\mathcal{J}_n$ with $n$ type-changing components for every $n \geq 0$, constructed via successive multiplicity-0 logarithmic transformations.
• The manifold $S^1\times S^3$ admits a twisted generalized complex structure $\mathcal{J}_n$ with $n$ type-changing components for every $n \geq 1$, obtained by a multiplicity-0 logarithmic transformation on $\{\ast\}\times T^2 \subset S^2\times T^2$.
• The construction shows that logarithmic transformations of multiplicity 1 on symplectic 2-tori with trivial normal bundle preserve the diffeomorphism type of the 4-manifold, as proven in Lemma 4.4.
• The number of type-changing loci can be increased indefinitely via iterative multiplicity-0 logarithmic transformations, even on manifolds that previously had only one or two such components.
• The result provides a sharp contrast to generalized Kähler structures, where the number of type-changing components is at most two due to the bihermitian structure and holomorphic Poisson structure constraints.