2001/10/31 by Alexandru Scorpan
Mathematics · #math.DG #math.GT #msc:53C57 #msc:57R57 #msc:53D35 #msc:53C55
published as Commun. Contemp. Math., vol. 4 (2002), no. 1, pp. 45--64 · 16 pages, 2 LaTeX figures. Infinitesimal revision
arxiv created 2003/04/11 · arxiv updated 2009/11/30
Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spinC structure induced by an almost-complex structure ω. Each connection A on the determinant line bundle induces a unique connection ∇A, and Dirac operator \DA on spinor fields. Let σ: W+ --> Λ+ be the natural squaring map, taking self-dual (= positive) spinors to self-dual 2-forms. In this paper, we characterize the self-dual 2-forms that are images of self-dual spinor fields through σ. They are those αfor which (off zeros) c1(α) = c1(ω), where c1(α) is a suitably defined Chern class. We also obtain the formula: || ϕ||2 DA ϕ= i (2 d^* σ(ϕ) + < ∇A ϕ, i ϕ>)* ϕ. Using these, we establish a bijective correspondence between: Kahler forms αcompatible with a metric scalar-multiple of g, and with c1(α) = c1(ω) and gauge classes of pairs (ϕ, A), with ∇A ϕ= 0, as well as a bijective correspondence between: Symplectic forms αcompatible with a metric conformal to g, and with c1(α) = c1(ω) and gauge classes of pairs (ϕ, A), with D^ A ϕ= 0, and < ∇A ϕ, i ϕ> = 0, and ϕnowhere-zero.