[Paper Review] The Space of Symplectic Structures on Closed 4-Manifolds

Tianjun Li|ArXiv.org|May 19, 2008

Geometric and Algebraic Topology36 references9 citations

TL;DR

This paper investigates the space of symplectic structures on closed 4-manifolds, focusing on the moduli space of symplectic forms up to diffeomorphism. It establishes that for rational or ruled 4-manifolds with $b^+ = 1$, symplectic forms are uniquely determined by their cohomology class up to diffeomorphism, providing a complete description of the moduli space. The work further explores the relationship between symplectic and Kähler structures, offering a construction of non-holomorphic Lefschetz fibrations on Kähler surfaces.

ABSTRACT

This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004

Motivation & Objective

• To understand the space of symplectic structures on closed 4-manifolds, particularly the moduli space of symplectic forms up to diffeomorphism.
• To determine when symplectic forms are uniquely determined by their cohomology class in 4-manifolds with $b^+ = 1$.
• To compare the space of symplectic forms with the space of Kähler forms on manifolds admitting Kähler structures.
• To investigate whether every symplectic form on a Kähler surface is cohomologous to a Kähler form, especially in general type surfaces.
• To explore the implications of Donaldson's program on symplectic forms and their equivalence under diffeomorphism.

Proposed method

• The paper uses the space $\Omega(X)$ of orientation-compatible symplectic forms on a closed 4-manifold $X$, which is an open subset of the space of closed 2-forms.
• It defines the moduli space $\mathcal{M}_X = \Omega(X)/\mathrm{Diff}^+(X)$, capturing symplectic structures up to diffeomorphism.
• For $b^+ = 1$, the paper applies the uniqueness result: a symplectic form is determined up to diffeomorphism by its cohomology class.
• It employs the symplectic canonical class $K_\omega = -c_1(X,\omega)$ and the symplectic cone $\mathcal{C}_X$ to analyze cohomological invariants.
• It applies the Kähler Nakai-Moishezon criterion and the refined period map to characterize Kähler classes in $H^2(X;\mathbb{R})$, particularly in K3 and torus cases.
• It constructs non-holomorphic Lefschetz fibrations by selecting symplectic forms outside the $H^1_1$ subspace of the unique complex structure on ball quotients.

Experimental results

Research questions

• RQ1 For a closed 4-manifold with $b^+ = 1$, is a symplectic form uniquely determined up to diffeomorphism by its cohomology class?
• RQ2 In which closed 4-manifolds is every symplectic form cohomologous to a Kähler form?
• RQ3 Can symplectic forms that are not in the $H^1_1$ subspace of any complex structure give rise to non-holomorphic Lefschetz fibrations?
• RQ4 Does the cohomology Kähler cone coincide with the symplectic cone for minimal elliptic surfaces $E(n)$?
• RQ5 Under what conditions does Donaldson's program imply that any two symplectic forms with the same Chern class and cohomology class are related by a diffeomorphism?

Key findings

• For rational or ruled 4-manifolds with $b^+ = 1$, a symplectic form is uniquely determined up to diffeomorphism by its cohomology class.
• The moduli space $\mathcal{M}_X$ for such manifolds admits a simple description via the cohomology class and diffeomorphism action.
• On the K3 surface and $T^4$, every integral class of positive square lies in the Kähler cone after possibly adjusting the complex structure, due to the surjectivity of the refined period map.
• For minimal Kähler surfaces of general type with $p_g \geq 1$, the cohomology Kähler cone is strictly smaller than the symplectic cone, as shown by constructing symplectic forms $\tau_s = \omega + s\mathrm{Re}(\phi)$ that lie outside the Kähler cone for large $s$.
• The construction of a Lefschetz pencil dual to a large multiple of a symplectic class outside the $H^1_1$ subspace yields a non-holomorphic Lefschetz fibration on a Kähler surface with $p_g > 0$.
• The paper provides evidence that the symplectic cone may contain classes not cohomologous to any Kähler form, particularly in general type surfaces, though for $E(n)$ surfaces the cohomology Kähler cone may coincide with the symplectic cone under certain conditions.