Complex analysis and differential topology on complex surfaces

In the paper, the relationship between the theory of holomorphic functions on two-dimensional complex manifolds and their differential topology is described. The basic fact, established by using the Seiberg–Witten invariants, is that the topological characteristics of embedded real surfaces in Stein surfaces satisfy adjunction-type inequalities. A version of Gromov's $h$-principle for totally real embeddings shows that these topological inequalities are sharp. In some cases, these results can be used to describe the envelopes of holomorphy of embedded real surfaces in a given complex surface. Our examples include real surfaces in $\mathbb C^2$ and $\mathbb{CP}^2$ and in products of $\mathbb{CP}^1$ with non-compact Riemann surfaces. A similar technique can be applied to the study of geometric properties of strictly pseudoconvex domains in dimension two.