Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$.
As I know there is the adjunction inequality for estimation of minimal genus via Seiberg-Witten theory.
Question 1: Does there exist other methods to estimate minimal genus ?
I heard that there are homeomorphic but not diffeomorphic 4-manifolds $M,N$ such that for some $a\in H^2$, $a$ has different minimal genus in $M,N$.
Question 2: Could you give me such examples? As I understand it should be some manipulations with Seiberg-Witten invariants…
Best Answer
Regarding Question 2, Corollary 2 of this paper by Li shows that any symplectic four-manifold which contains a smoothly embedded homologically essential sphere with nonnegative intersection is obtained by blowing up either $\mathbb{C}P^2$ or an $S^2$-bundle over a surface some nonnegative number of times. So you could let $M$ be $\mathbb{C}P^2$ # $k \overline{\mathbb{C}P^2}$ for suitable $k$ (by now $k\geq 2$ will do) and $N$ be any of the wide variety of examples of symplectic exotic rational surfaces. Then if $a\in H^2(M)\cong H^2(N)$ is Poincare dual to the pullback of the hyperplane class in $\mathbb{C}P^2$, $a$ will have minimal genus zero in $M$ but positive minimal genus in $N$.
There are also examples with $b^+>1$, in which case as you say one has the Seiberg-Witten adjunction formula $2g(\Sigma)-2\geq |K\cdot\Sigma|+\Sigma\cdot\Sigma$ for any surface $\Sigma$ of positive genus and nonnegative self-intersection. If you take for $M$ the $K3$ surface, viewed as an elliptic fibration with a section of square $-2$, and let $\Sigma$ be obtained from a fiber and the section by smoothing the intersection between them, then $\Sigma$ has square zero and genus 1 (and $\Sigma$ can't be represented by a sphere); this is consistent with the adjunction formula since the only basic class for the $K3$ surface is the zero class. But there are many exotic smooth structures on the $K3$ surface (for instance the ones here) for which some positive multiple of the fiber class is a basic class, and the minimal genus for the homology class of $\Sigma$ in one of these exotic $K3$ surfaces would be larger than one since there would be a basic class having positive intersection with $\Sigma$.