Computation of the intersection index of a totally real and complex disk with common boundary in \(\mathbb{P}^ 2\) and \(\mathbb{P}^ 1\times\mathbb{P}^ 1\) (Q1326072)

Let \(X\) be a two-dimensional complex manifold, and \(\alpha \in H_ 2 (X, \mathbb{Z})\). The author shows that, if \(\alpha\) can be represented by an embedded \(S^ 2\) obtained by joining together an embedded totally real two-disc and an embedded holomorphic one-disc along their common boundary, then \(\alpha \cdot \alpha - c_ 1 (X) (\alpha) + 2 = 0\), where \(c_ 1 (X)\) is the first Chern class of \(X\). When \(X = P^ 1\) or \(P^ 1 \times P^ 1\), the above condition is also sufficient. When \(X = \mathbb{C}^ 2\), a related result has been proved by Forstneric (preprint). The main tool of both authors is a general formula due to \textit{H. Lai} [Trans. Am. Math. Soc. 172(1972), 1-33 (1973; Zbl 0247.32005)] which relates the number of complex points of an immersed compact oriented surface \(M\) in a complex two-manifold \(X\) to the topology of \(M\) and \(X\).