Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds (Q2701595)

The following three theorems are proved.NEWLINENEWLINENEWLINETheorem 1. Assume an oriented Riemannian 4-manifold \(X\) admits a \(\text{spin}^c\) structure with a non-zero spinor which is parallel with respect to a \(\text{spin}^c\) connection. Then the smooth manifold X admits a Kähler structure.NEWLINENEWLINENEWLINETheorem 2. Assume that an oriented Riemannian 4-manifold \(X\) has a \(\text{spin}^c\) structure for which there is a non-zero positive harmonic spinor \(\psi\) such that \(\langle \nabla _v \psi,\psi\rangle=0\) for any \(v \in TX ,\) where \(\langle\;,\;\rangle\) denotes the Hermitian metric for the spin bundle and \(\nabla\) is the \(\text{spin}^c\) connection. Then the smooth manifold X admits a symplectic structure.NEWLINENEWLINENEWLINETheorem 3. Let \(X\) be an oriented Riemannian 4-manifold which admits a \(\text{spin}^c\) structure. Assume there is a non-zero positive spinor \(\psi\) which is parallel with respect to a \(\text{spin}^c\) connection. Then the spin\(^c\) structure determined by \(\psi\) is equivalent to the original one.