Twistor holomorphic immersions of real surfaces into Kähler surface (Q1093191)

Recently Th. Friedrich introduced the notion of a twistor holomorphic immersion \(f: M^ 2\to X^ 4\) of an oriented surface \(M^ 2\) into a 4- dimensional oriented Riemannian manifold \(X^ 4\). These are immersions which naturally can be lifted to complex curves in the twistor space of \(X^ 4.\) In this paper we consider twistor holomorphic immersions of closed surfaces into self-dual Einstein spaces \(X^ 4\) with vanishing scalar curvature, such that \(X^ 4\) with the opposite orientation has a Kähler structure. We prove that in general there are only finitely many complex points on the immersed surface, that every complex point is elliptic and establish the equation \[ \chi (M^ 2)-e(M^ 2)=2n(M^ 2)+2\sum_{i\geq 1}(i+1)n_ i(M^ 2). \] Here \(\chi\) denotes the Euler characteristic, e the Euler number of the normal bundle, n the geometric number of complex points on \(M^ 2\) which are not superminimal and \(n_ i\) the number of complex points which are superminimal points of order i.