Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds

Abstract

Abstract. We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spin c structure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the spin c structure in which the non-zero parallel spinor lives is equivalent to the canonical spin c structure associated to the Kähler structure.

Nowhere-Zero Harmonic Spinors and Their Associated Self-Dual 2-Forms

“…Remark 3.10. This last result was obtained independently in [BLPR01]. Note also that this is only the spin C version of a similar statement holding for genuine spinstructures: If M admits a spin-structure with a parallel spinor field, then M is Kähler.…”

“…While this work was being completed, the paper [BLPR01] was electronically published. It contains two partial results in the direction of our work (see Remarks 3.10 and 3.19 for a discussion).…”

Kahler and symplectic structures on 4-manifolds and hyperKahler geometry

“…If u ∈ Γ(X, W + ) is a nonvanishing section, then σ • u defines a non-degenerate, self-dual 2-form on X. It was shown in [BLPR00,Sco02] independently that if ∇ A u = 0 for some U (1)connection A, then σ • u defines a Kähler structure on X, compatible with a metric g ′ X = c • g X , c ∈ R. On the other hand, under mild conditions, it was shown that if u is a harmonic spinor, then σ • u defines a symplectic structure on X. Scorpan [Sco02] gave a charaterization of the Kähler and symplectic 2-forms that lie in the image of the quadratic map.…”