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...

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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 $\Sigma$.

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In the first part, one can also apply Li's theorem to a Fermat hypersurface $X$ (the zero set of polynomials $\sum x_i^d$ in $\mathbb{CP}^3$) of degree $d>4$: $X$ is homeomorphic to a connected sum of $\mathbb{CP}^2$s and $\overline{\mathbb{CP}}^2$s, has $b_2^+>2$, and can't be homeomorphic to a blow-up of $\mathbb{CP}^2$ (because of $b_2^+$) or any $S^2$-bundle over a surface (because of fundamental group the base has to be $S^2$, and $b_2^+$ rules this out again). – Marco Golla Dec 13 '11 at 12:13

This is regarding question 1. There is a much earlier approach through Atiyah-Singer G-signature theorem that works for certain divisible classes.

So if $g$ is the genus of an embedded surface $S$ representing $a$ then (under certain assumptions) one gets the following inequality $$\beta_2+2g\ge\left|\frac{1}{2}a\cdot a-\sigma(M)\right|$$ where $\sigma(M)$ is the signature of the intersection form on the second homology and $\beta_2$ is the second Betti number. (I am not sure about the coefficient 1/2 on the right, it is more complicated actually, but I think morally it's ok.)

This inequality applies to even classes $2nx$ in $\mathbb{CP}^2$ and $2nx+2my$ in $S^2\times S^2$ resulting in $$g\ge n^2-1,\;\;\; and\;\; g\ge 2nm-1$$ respectively. (Also it applies to other divisible classes.)

Assume $a=kb$, where $k$ is an integer. One has to look at $k$-fold cover $\tilde M$ that branches over $S$ and apply the G-signature theorem to this cover. This gives a formula for $\sigma(\tilde M)$ $$\sigma(\tilde M)=k\sigma(M)-\frac{(k^2-1)a\cdot a}{3k}.$$ Then the final estimate comes from comparing Euler characteristics, signatures and second Betti numbers.

I think the proof can be found in Rohlin's "Two dimensional submanifolds of 4 dimensional manifolds" or in Hsiang, Szczarba, "On embedding surfaces in four-manifolds".

It is interesting that Rohlin remarks that he had the above display formula from "a corresponding version of cobordism theory" without using Atiyah-Singer.

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Oh,yes. Thank you. It's founded on the idea that $ka$ with $k>>0$ can't be presented as sphere if $a\cdot a\ne 0$ (and additional estimations...). Does there exist other ideas which can be formulated such simply way? – Nikita Kalinin Dec 13 '11 at 20:33
Nikita, I don't know other approaches to the problem, but I am not an expert, I just happened to have read the Rohlin's paper. Does Seiberg-Witten give better estimates for even classes in $CP^2$? – Andrey Gogolev Dec 14 '11 at 3:31

To add a bit to Andrey's answer, you can improve the method of Hsiang and Szczarba (also Kotschick and Matic deserve credit here) a bit using Furuta's approach to Seiberg-Witten theory. The method can give genus bounds on four manifolds where the adjunction formula doesn't apply (for example on $\mathbb{CP}^2 \# \mathbb{CP}^2$). I wrote a short paper on this: Math. Res. Lett. 5 (1998), no. 1-2, 165–183. The arXiv version is here: http://arxiv.org/pdf/dg-ga/9704010v1. Theorem 1.6 gives a typical genus bound from this method.

The basic idea is the following. If your class has certain divisibility properties and it is represented by an embedded surface, then the four manifold obtained by taking a cyclic cover branched along the surface will be a new four manifold with certain properties which can be used to constrain the genus of the embedded surface. Under appropriate hypotheses, the cover will be a spin four manifold with a cyclic group action. Refining Furuta's proof of the 10/8ths theorem for spin four manifolds to include the presence of a cyclic action, one obtains a bound on the size of the signature in terms of the second Betti number which then translates into a genus bound on the original manifold.

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