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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 assumptionassumptions) 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 characteristiccharacteristics, signatures and second Betti numbers.

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

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

2 added 339 characters in body

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 assumption) one gets the following inequality $$\beta_2+2g\ge|\frac{1}{2}a\cdot a-\sigma| beta_2+2g\ge\left|\frac{1}{2}a\cdot a-\sigma(M)\right|$$ where $\sigma$ \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.) One has to find a finite cyclic group of order$k$, Assume$k|a$acting on a=kb$, where $M$ so that the fixed point set k$is a surface$X$. Then an integer. One has to look at a$k$-fold cover$\tilde M$that branches over$X$S$ 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 characteristic, signatures and second Betti numbers.

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

It is interesting that Rohlin remarks that first he obtained the above display formula without using G-signature formula from a "corresponding version of cobordism theory."

1

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 assumption) one gets the following inequality $$\beta_2+2g\ge|\frac{1}{2}a\cdot a-\sigma|$$ where $\sigma$ 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.)

One has to find a finite cyclic group of order $k$, $k|a$ acting on $M$ so that the fixed point set is a surface $X$. Then look at a $k$-fold cover $\tilde M$ that branches over $X$ and apply the G-signature theorem to this cover.

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