Mapping classes fixing an isotropic homology class of minimal genus $0$ in rational $4$-manifolds

Abstract

For any $N \geq 1$, let $M_N$ denote the rational $4$-manifold $\mathbb{CP}^2 \# N \overline{\mathbb{CP}^2}$. In this paper we study the stabilizer $\text{Stab}(w)$ of a primitive, isotropic class $w\in H_2(M_N; \mathbb Z)$ of minimal genus $0$ under the natural action of the topological mapping class group $\text{Mod}(M_N)$ on $H_2(M_N; \mathbb Z)$. Although most elements of $\text{Stab}(w)$ cannot be represented by homeomorphisms that preserve any Lefschetz fibration $M_N \to \Sigma$, we show that any element of $\text{Stab}(w)$ can be represented by a diffeomorphism that almost preserves a holomorphic, genus-$0$ Lefschetz fibration $\text{pr}: M_N \to \mathbb{CP}^1$ whose generic fibers represent the homology class $w$. We also answer the Nielsen realization problem for a certain maximal torsion-free, abelian subgroup $\Lambda_w$ of $\text{Mod}(M_N)$ by finding a lift of $\Lambda_w$ to $\text{Diff}^+(M_N) \leq \text{Homeo}^+(M_N)$ under the quotient map $q: \text{Homeo}^+(M_N)\to \text{Mod}(M_N)$ which can be made to almost preserve $\text{pr}: M_N \to \mathbb{CP}^1$. All results of this paper also hold for every primitive, isotropic class $w \in H_2(M_N; \mathbb Z)$ if $N \leq 8$ because any such class has minimal genus $0$.