Immersions of surfaces in almost-complex 4-manifolds (Q2781315)

The author studies the relation between double points and complex points of an immersed surface in an almost-complex 4-manifold. He uses estimates of the minimal genus of embedded surfaces to get inequalities between the number of double points and the number of complex points of the immersion. Let \(d_-,d_+\) give the numbers of negative and positive double points, and \(n^-,n^+\) represent the algebraic numbers of negative and positive complex points. His main result says that if \(X\) is an almost complex 4-manifold and \(F_0 \subset X\) is an embedded pseudoholomorphic curve with \(F_0 \cdot F_0 > 0,\) and \(F\) is an immersed surface contained in a tubular neighborhood of \(F_0\) with \(F\cdot F_0 > 0,\) then \(n^-(F) \leq d_-(F).\) The proof uses a result of \textit{H. F. Lai} [Trans. Am. Math. Soc. 172, 1-33 (1972; Zbl 0222.32002)] to first get an equality \(g(F) + n^- - d_- + d_+ = 1 + {1 \over 2} (F\cdot F + K \cdot F).\) The author then uses Seiberg-Witten techniques to prove a version of the local Thom conjecture for immersed surfaces, generalizing a result of the reviewer [Expo. Math. 15, No. 5, 385-431 (1997; Zbl 0894.57013)] for embedded surfaces. The main theorem follows from combining these results.