2.1 Homologically trivial cyclic group actions on closed orientable surfaces of positive genus

We begin with background results on periodic surface diffeomorphisms, derived from the Nielsen-Thurston classification of surface homeomorphisms [Reference Thurston44]. We state a special case sufficient for our purposes (see [Reference Thurston44] for the full classification).

Proof. Let $\Sigma $ be a closed orientable surface of genus at least two, and h a finite-order homeomorphism acting trivially on $H_{1}(\Sigma ,\mathbb {Z})$ . First, we rule out the possibility that h is orientation-reversing. Suppose otherwise. Then by Poincaré duality and the universal coefficients theorem,

$$ \begin{align*}h^{*}: H^{1}(\Sigma,\mathbb{Z}) \rightarrow H^{1}(\Sigma,\mathbb{Z})\end{align*} $$

is the identity map. Choose $\alpha ,\beta \in H^{1}(\Sigma ,\mathbb {Z}) $ such that $\alpha \cdot \beta \neq 0$ (e.g., the Poincaré dual of two simple closed curves intersecting at a single point). Then $h^{*}(\alpha \cdot \beta ) = \alpha \cdot \beta $ , contradicting the fact that $h^{*}$ must act as multiplication by $-1$ on $H^{2}(\Sigma ,\mathbb {Z})$ . Thus, h is orientation preserving.

By [Reference Farb and Margalit14, Theorem 6.8], h is isotopic to the identity. Therefore by Nielsen-Thurston classification of surface homeomorphisms [Reference Thurston44], $h=\mathrm {Id}_{\Sigma }$ (for a concise reference for the statement needed see [Reference Jaco and Shalen21, Lemma 4.4]).

Next, we consider the case where $\Sigma =T^2$ . In this setting, there exist nontrivial examples of cyclic group actions that act trivially on homology, as shown in Example 2.2. For convenience, we realize $T^2$ as the Clifford torus

$$ \begin{align*}T^2=\{(z_1,z_2) \in \mathbb{C}^2 | \, |z_1| = |z_2|=1\}.\end{align*} $$

Example 2.2. Let $p,q$ be coprime positive integers. Define the diffeomorphism $f_{p,q}$ by

$$ \begin{align*}f_{p,q}(z_1,z_2) = (e^{\frac{ 2\pi iq}{p}}z_1,z_2).\end{align*} $$

This map generates an action of $\mathbb {Z}_{p}$ which acts trivially on homology.

While there are nontrivial examples in the genus one case, cyclic homologically trivial actions on $T^2$ admit a complete classification. Before stating this, we recall a simpler fact, which follows directly from the Lefschetz fixed point formula.

Recall that a group action is called free if the only group element acting as the identity map is the identity element itself.

The lemma has the following consequence:

Lemma 2.4. Let G be a finite group acting effectively by homeomorphisms on $T^2$ and trivially on $H_{1}(T^2,\mathbb {Z})$ . There exist nonnegative integers $p,q$ such that

$$ \begin{align*}G \cong \mathbb{Z}_{p} \times \mathbb{Z}_q.\end{align*} $$

Proof. By Lemma 2.3 the action of G is free. Since it is orientation-preserving, the quotient map $\pi : T^2 \rightarrow T^2/G$ is a covering, with $T^2/G \cong T^2$ . Therefore by [Reference Hatcher20, Proposition 1.39],

$$ \begin{align*}G \cong \pi_{1}(T^2,\mathbb{Z})/ \pi_{*}(\pi_{1}(T^2,\mathbb{Z})),\end{align*} $$

and considering the possible quotients of $\mathbb {Z}^2$ by subgroups isomorphic to $\mathbb {Z}^2$ gives the result.

A classical theorem of Nielsen and Smith further asserts that cyclic actions on surfaces are classified, up to conjugacy in the homeomorphism group, by the induced map on integral homology of the quotient

$$ \begin{align*}\pi_{*} : H_{1}(\Sigma) \rightarrow H_{1}(\Sigma/\mathbb{Z}_{p}).\end{align*} $$

In the case of $T^2$ , both the surface $\Sigma $ and its quotient $\Sigma /\mathbb {Z}_{p}$ are homeomorphic to $T^2$ .

For the actions in Example 2.2, the image of $\pi _*$ depends only on p: it is the subgroup generated by $(p,0)$ and $(0,1)$ . However the induced homomorphism also depends on q.

To state the classification precisely, we recall the following definition.

Definition 2.5. Two actions of $\mathbb {Z}_{p}$ on $T^2$ , generated by homeomorphisms $f_1,f_2$ of order p, are said to be topologically conjugate if there exists a homeomorphism h such that

$$ \begin{align*}f_{2} = h f_1 h^{-1}.\end{align*} $$

A result of Baranov, Grines, Pochinka, and Chilina [Reference Baranov, Grines, Pochinka and Chilina3, Theorem 2], obtained via Nielsen theory, establishes that the actions in Example 2.2 account for all possible homologically trivial $\mathbb {Z}_{p}$ -actions on $T^2$ , up to topological conjugacy.

2.2 Almost complex structures

We now collect the necessary tools concerning almost complex structures and pseudoholomorphic curves. A key input is the following theorem from [Reference Zhang47], establishing the existence and behavior of J-holomorphic curves on irrational ruled symplectic $4$ -manifolds.

For an arbitrary tamed almost complex structure, the existence of a pseudoholomorphic curve in the fiber class passing through every point of M follows from [Reference Zhang47, Theorem 1.2] by applying the J-nef technique developed in [Reference Li and Zhang30]. The continuity of the resulting map to the moduli space of fibers $\Sigma $ is established in [Reference Zhang47, Corollary 3.9]. Moreover, the fact that M admits the structure of a smooth fiber bundle in a neighborhood of an irreducible fiber is explained immediately below the proof of that corollary.

We refer to the map f in Theorem 2.7 as the moduli map:

We will analyze the behavior of this fibration under cyclic group actions. The following lemma, used repeatedly in our arguments, shows that the moduli map induces an isomorphism on the first integral homology group. For a topological space X, let $\tilde {H}_i(X)$ denote the reduced integral homology of X.

Another important tool is the evaluation map associated with moduli spaces of pseudoholomorphic spheres, which will be used in the proof of Theorem 5.1. We recall the definition below:

The following useful lemma ensures the compatibility of almost complex structures with group actions, which is an almost complex version of [Reference Palais39, Theorem 4.3.1]:

Proof. Let $g_0$ be a Riemannian metric compatible with $\omega $ . Averaging over the group G, we define:

$$ \begin{align*}g:=\frac{1}{|G|}\sum_{h\in G}h^{*} g_0,\end{align*} $$

which yields a G-invariant Riemannian metric. Since each $h\in G$ is a symplectomorphism, the averaged metric g remains compatible with $\omega $ .

We now apply the map constructed in [Reference McDuff and Salamon36, Proposition 2.5.6 (i)] (or Proposition 2.50 in the second edition of [Reference McDuff and Salamon36]) to this invariant metric. The resulting almost complex structure is G-invariant and compatible with $\omega $ .

We now explain why this follows from [Reference McDuff and Salamon36, Proposition 2.5.6] and sketch the definition of the map r. Let $(V,\omega )$ be a symplectic vector space, and $\mathcal {M}et(V)$ denote the set of metrics on V and $\mathcal {J}(V,\omega )$ of $\omega $ -compatible almost complex structures. Recall that Sp $(V)=\text {Sp}(V, \omega )$ is the set of linear automorphisms of V preserving the symplectic structure.

For any symplectic vector space $(V,\omega )$ , there is a map

$$ \begin{align*}r: \mathcal{M}et(V) \rightarrow \mathcal{J}(V,\omega)\end{align*} $$

assigning to each Riemannian metric on V an $\omega $ -compatible almost complex structure. The key properties of r are:

• ○ $r(g_{J}) = J$ for any $J \in \mathcal {J}(V,\omega )$ ,

• ○ $r (\Phi ^{*}(g)) = \Phi ^{*} (r(g)),$ for all $\Phi \in \text {Sp}(V)$ , $g \in \mathcal {M}et(V)$ .

Applying this to the G-invariant metric g, and using the fact that each group element acts symplectically, we find

$$ \begin{align*}a^{*}(r(g)) = r(a^{*}g) = r(g)\end{align*} $$

for any $a \in G$ . Hence, the resulting $\omega $ -compatible almost complex structure $J:=r(g)$ is G-invariant.

To briefly recall the construction of r: for $g \in \mathcal {M}et(V)$ , there exists a linear automorphism $A: V\rightarrow V$ such that

$$ \begin{align*}\omega(Au,v) = g(u,v) \quad \text{for all } u, v\in V.\end{align*} $$

Let $P = A^{*}A$ . Then there exists a linear automorphism Q such that $Q^{2}=P$ . The compatible almost complex structure is then given by

$$ \begin{align*}r(g)=Q^{-1}A.\end{align*} $$

Full details are provided in [Reference McDuff and Salamon36, Proposition 2.5.6]

For the final statement, we know that the fixed point set $\mathrm {Fix}(G)$ of a smooth action by a finite group G on a closed manifold is a disjoint union of smooth connected submanifolds. Let $x\in \mathrm {Fix}(G)$ . Since J is G-invariant, the tangent space $T_xM$ inherits a complex structure $J_x$ , and the fixed subspace $T_x(\mathrm {Fix}(G))\subset T_xM$ is preserved by $J_x$ . Therefore, $\mathrm {Fix}(G)$ is an almost complex submanifold of M.

2.3 Weights of cyclic group actions

In this section, we recall the standard notion of weights for cyclic group actions on almost complex manifolds.

Suppose the finite cyclic group $\mathbb {Z}_{k}$ acts on an almost complex manifold $(M,J)$ , preserving the almost complex structure J. If $p\in M$ is a fixed point of the action, then the tangent space $T_{p}M$ inherits the structure of a complex $\mathbb {Z}_k$ -representation.

Recall that the k irreducible complex representations of $\mathbb {Z}_k$ are indexed by integers $w \mod k$ , with generator $\alpha = e^{\frac {2 \pi i w}{k}}$ . For a given $m \in \mathbb {Z}/k \mathbb {Z}$ , the group action is given by

$$ \begin{align*}m. z = \alpha^{m}.z.\end{align*} $$

Let $F\subset M$ be a fixed point component of the $\mathbb Z_k$ -action. By Lemma 2.11, F is an almost complex submanifold. The only fact about weights we will need in this article is the following: For any $p \in F$ , the $\mathbb {Z}_k$ -representation on $T_{p}(M)$ is independent of the choice of $p $ . In particular, the weights encoding the decomposition of $T_{p}(M)$ are constant along F. The proof is elementary and left to the reader.

2.4 Automorphism groups of irrational ruled surfaces

Let S be a smooth projective surface over $\mathbb C$ . The surface S is called geometrically ruled if there exists a morphism $\pi : S \rightarrow C$ , where C is a smooth curve, such that for every point $p \in C$ , $\pi ^{-1}(p) \cong \mathbb {P}^1$ . Note that S is irrational if and only if the base curve C is irrational. It is known that every geometrically ruled surface is isomorphic to the projectivization $\mathbb {P}(E)$ of a rank two vector bundle E over C [Reference Maruyama31].

In this section, we recall several results concerning the automorphism groups of such irrational ruled algebraic surfaces. The automorphism groups of geometrically ruled surfaces were classified by Maruyama in [Reference Maruyama31]. We summarize below the aspects of this classification most relevant to our setting, particularly for irrational geometrically ruled surfaces. We recall the following important definition from [Reference Maruyama31]:

The subgroup $\mathrm {Aut}_{C}(S) $ is particularly important in the paper, as any homologically trivial automorphism of S must lie in this subgroup when the base curve C has genus at least two. It is known that the quantity $N(E)$ is an invariant of the surface S and plays a key role in the classification of automorphism groups [Reference Maruyama31, Theorem 2]. The subgroup $\Delta \subset \mathrm {Jac}(C)$ defined in Definition 2.12(d) is a subgroup of the $2$ -torsion part of the Jacobian variety of C, and is therefore finite.

The following short exact sequence constructed in [Reference Grothendieck18] will be important in our analysis.

Lemma 2.13. Let $(S ,\pi ,C)$ be a geometrically ruled surface with $S \cong \mathbb {P}(E)$ . Then, there is a short exact sequence

$$ \begin{align*}1 \rightarrow \mathrm{Aut}(E)/\mathbb{C}^* \rightarrow \mathrm{Aut}_{C}(S) \rightarrow \Delta \rightarrow 1,\end{align*} $$

where $\mathbb {C}^*$ acts on E by scaling.

Proof. This result is stated in [Reference Grothendieck18, Proposition 2-Corollary, Page 202] for vector bundles of arbitrary rank n. Let $E_{0}$ be a vector space of rank n. Then there is an exact sequence of groups

$$ \begin{align*}1 \rightarrow \mathbb{C}^* \xrightarrow{i} \mathrm{Aut}(E_0) \xrightarrow{p} \mathrm{Aut}(\mathbb{P}(E_0)) \rightarrow 1,\end{align*} $$

where $i(\lambda ) = \lambda \cdot \mathrm {Id}$ and $p(T) = [T]$ is the induced projective transformation.

Passing to the corresponding locally free sheaves over the base curve C, one obtains a long exact sequence in cohomology:

$$ \begin{align*}1 \rightarrow H^{0}(C, \mathcal{O}^*) \xrightarrow{\varphi_1} H^{0}(C, \mathrm{Aut}(E)) \xrightarrow{\varphi_2} \mathrm{Aut}_{C}(X)\rightarrow H^{1}(C, \mathcal{O}^*) \rightarrow \ldots\end{align*} $$

Here, $ H^{0}(C, \mathcal {O}^*)\cong \mathbb {C}^*$ , and $ H^{1}(C, \mathcal {O}^*)\cong \mathrm {Jac}(C)$ , the group of line bundles over C. In [Reference Grothendieck18, Proposition 2-Corollary, Page 202], it is shown that the image of $\phi _2$ is exactly $\Delta $ , and that each $L \in \Delta $ satisfies $L^{\otimes n} \cong \mathcal {O}$ . Applying this in the case $n=2$ yields the claimed exact sequence.

Recall that a section of a geometrically ruled surface $\pi : S\rightarrow C$ is a morphism $f: C \rightarrow S$ such that $\pi \circ f = \mathrm {Id}_{C}$ . A section is called minimal if it has the smallest possible self-intersection among all sections of S. The following lemma describes the relationship between minimal sections and the vector bundle structures:

Finally, we highlight the special case where the geometrically ruled surface is a product $C\times \mathbb {P}^1$ . This case is distinguished in [Reference Maruyama31, Theorem 2] and, in a sense, admits a more restrictive automorphism group than a general projective bundle. Since the following is claimed without proof in [Reference Maruyama31] we give a proof below for completeness.

Lemma 2.15. Let C be a smooth projective curve of genus $g\ge 1$ , and let $S = C \times \mathbb {P}^1$ . Then:

$$ \begin{align*}\mathrm{Aut}(S) \cong \mathrm{Aut}(C) \times \mathrm{Aut}(\mathbb{P}^1).\end{align*} $$

Proof. Let $\pi _1: S \rightarrow C$ and $\pi _2: S \rightarrow \mathbb {P}^1$ denote the projections. Let $F_{C}$ denote the homology classes of $C \times \{p\}$ , and $F_{{\mathbb {P}}^1}$ the class of $\{q\} \times \mathbb {P}^1$ , for $p \in \mathbb P^1$ , $q \in C$ . Then the intersection pairing satisfies:

(1) $$ \begin{align} (a F_{C} + b F_{{\mathbb{P}}^1})^2 = 2ab. \end{align} $$

Let $\psi : S \rightarrow S$ be an automorphism. Consider the image $C_1 = \psi (\{p\} \times \mathbb {P}^1)$ for a fixed $p \in C$ . This is a smooth rational curve in $S=C \times \mathbb {P}^1$ with self-intersection zero. Since $C_1^2=0$ , and from Equation (1), $C_{1}$ must be of the form $\{p'\} \times \mathbb {P}^1 $ for some $p'\in C$ . That is, $\pi _{1}|_{C_1}$ is constant.

Similarly, for any $q \in \mathbb {P}^1$ , the image $C_2 = \psi (C \times \{q\})$ is a smooth curve of genus g with self-intersection zero. By the above, $C_2\cdot F_{{\mathbb {P}}^1}=1$ . Since $C_{2}^2 = 0$ and $C_2\cdot F_{{\mathbb {P}}^1}=1$ , Equation (1) implies that $[C_2] = [F_C]$ . In particular, $(\pi _{2})_{*}[C_2] = 0$ , and so $\pi _{2}|_{C_2}$ is constant. That proves that $C_2 = C \times \{q'\}$ for some $q' \in \mathbb {P}^1$ .

Thus $\psi $ preserves the product structure and must be of the form $\psi = \psi _{1} \times \psi _{2}$ , where $\psi _{1} \in \mathrm {Aut}(C) $ and $\psi _{2} \in \mathrm {Aut}(\mathbb {P}^1)$ .

The following lemma follows as an immediate consequence.

Lemma 2.16. Let C be a smooth projective curve of genus $g\ge 1$ , and let $S = C \times \mathbb {P}^1$ . Suppose $\psi \in \mathrm {Aut}(S)$ is a finite order automorphism of order k, acting trivially on $H_{1}(S,\mathbb {Z})$ . Then:

$$ \begin{align*}\psi = \mathrm{Id}_{C} \times \tilde{\psi},\end{align*} $$

where $\tilde {\psi }$ is an elliptic Möbius transformation of order k. In particular, the fixed point set of $\psi $ consists of two disjoint sections.

2.5 Background on Question 1.1 and related work

For surfaces, Question 1.1 has an affirmative answer. The analysis naturally divides into two cases, depending on whether the surface M is a sphere or has higher genus.

Case 1: $M=S^2$

We present two arguments for this case.

First Argument (Topological Conjugacy):

A classical result of Kérékjartò and Eilenberg [Reference Constantin and Kolev12] provides a topological classification of periodic homeomorphisms of the $2$ -sphere:

Theorem 2.17. Let $f: S^2\rightarrow S^2$ be a periodic homeomorphism. Then there exists $r\in \mathrm {O}(3)$ and a homeomorphism $h:S^2\rightarrow S^2$ such that

$$ \begin{align*}f=h\circ r\circ h^{-1}.\end{align*} $$

A few remarks follow. First, since Hamiltonian diffeomorphisms preserve area, we must have $r\in \mathrm {SO}(3)$ . Second, if f is a diffeomorphism, then the conjugating map h can also be taken to be a diffeomorphism (see, e.g., [Reference Guelman and Liousse19]).

Since every element of $\mathrm {SO}(3)$ is a rotation, it lies in a circle subgroup of $\mathrm {SO}(3)$ . Thus, the diffeomorphism f, being conjugate to r, is also contained in a circle action. This circle action is clearly Hamiltonian and, up to conjugation, essentially unique.

Second Argument (Holomorphic Realization):

This approach, which forms the base case for later discussions of irrational ruled $4$ -manifolds, uses almost complex geometry. By Lemma 2.11, any periodic symplectomorphism f preserves some $\omega $ -compatible almost complex structure J. On $S^2$ , this J is biholomorphic to the standard complex structure by the uniformization theorem. The group of biholomorphisms of $\mathbb {CP}^1$ is the Möbius group, and every Möbius transformation is conjugate to an element of SO $(3)$ . Hence, f again lies in a Hamiltonian circle action.

Case 2: M is a surface of genus $g\ge 1$

In this case, we show that no nontrivial periodic Hamiltonian diffeomorphisms can exist. To this end, we apply [Reference Franks and Handel15, Theorem 1.4]:

This theorem implies that any nontrivial Hamiltonian diffeomorphism on a higher genus surface cannot be of finite order, and thus no nontrivial periodic Hamiltonian diffeomorphisms exist on such surfaces.

Thus, we obtain the following consequence:

In higher dimensions, the analysis naturally divides into two cases, analogous to the surface case: uniruled and nonuniruled symplectic manifolds. For the nonuniruled manifolds, there are results analogous in spirit to Theorem 2.18, many of which are inspired by the Conley conjecture [Reference Ginzburg and Gürel16].

In general, the following inclusions hold among groups of symplectomorphisms:

$$ \begin{align*}\mathrm{Ham}(M, \omega)\subset \mathrm{Symp}_0(M, \omega)\subset \mathrm{Symp}_h(M, \omega),\end{align*} $$

where:

• ○ Ham $(M, \omega )$ is the group of Hamiltonian diffeomorphisms,

• ○ Symp $_0(M, \omega )$ is the identity component of the symplectomorphism group, and

• ○ Symp $_h(M, \omega )$ denotes the group of symplectomorphisms acting trivially on homology.

Questions 1.1 and 1 ask whether a finite subgroup of Ham $(M, \omega )$ or Symp $_h(M, \omega )$ can be extended to a circle subgroup of $\mathrm {Ham}(M, \omega )$ .

A natural variant of the question arises for Symp $_0(M, \omega )$ . When $H^1(M, \mathbb R)=0$ , Ham $(M, \omega )= \mathrm {Symp}_0(M, \omega )$ . In many other situations, finite subgroups of Symp $_0(M, \omega )$ are known to lie inside Ham $(M, \omega )$ . This is the case, for example, for irrational ruled $4$ -manifolds with base genus at least two. For more general results, see [Reference Atallah1].

The other inclusion $\mathrm {Symp}_0(M, \omega )\subset \mathrm {Symp}_h(M, \omega )$ is less well understood. In the rational $4$ -manifold case, there has been solid progress in computing the symplectic Torelli group

$$ \begin{align*}\mathrm{Symp}_h(M, \omega)/\mathrm{Symp}_0(M, \omega);\end{align*} $$

see [Reference Li, Li and Wu27] for effective computations. A central open question in this direction, attributed to Donaldson (see [Reference Sheridan and Smith42, Reference Li, Li and Wu27]), is the following:

In the case of a minimal irrational ruled $4$ -manifold, a positive answer to Question 2.20 would imply that the symplectic Torelli group is trivial, as such manifolds contain no Lagrangian spheres. Combining this with the results of [Reference Atallah1] mentioned above, one would conclude that the symplectic involutions constructed in Theorem 4.1 are in fact Hamiltonian diffeomorphisms when the base genus is at least two.Footnote ³

For nonuniruled symplectic manifolds, it is known that they do not admit Hamiltonian circle actions [Reference McDuff35]. In this setting, a positive answer to Question 1.1 is equivalent to the statement that such manifolds admit no finite order Hamiltonian diffeomorphisms. In particular, if a closed symplectic manifold $(M, \omega )$ admits a Hamiltonian diffeomorphism of finite order, then there must exist a spherical homology class $A\in H_2(M, \mathbb Z)$ such that

$$ \begin{align*}\langle c_1(TM), A\rangle>0 \text{ and } \langle [\omega], A\rangle>0.\end{align*} $$

In contrast, the standard rotational finite-order elements in Symp $_0(T^4, \omega _{\mathrm {std}})$ are clearly not Hamiltonian. For the Enriques surface, we have Ham $=\mathrm {Symp}_0$ , since $H^1(M, \mathbb R)=0$ . However, it remains unclear to the authors whether the homologically trivial involutions of Enriques surfaces classified in [Reference Mukai and Namikawa37] are Hamiltonian.

3.1 Construction of the fibration

In this subsection, we provide a complete description of the isomorphism types of finite groups which can act symplectically on an irrational ruled symplectic $4$ -manifold with base genus at least $2$ , under the assumption that the action is trivial on the first integral homology group. In Corollaries 3.2 and 3.3 of the next subsection, Proposition 3.1 will be applied to further describe the possible isomorphism types of such groups.

Proof. Firstly by Lemma 2.11 there is a compatible almost complex structure J invariant by the action of G. Theorem 2.7 provides a fibration $f: M\rightarrow \Sigma $ by J-holomorphic spheres.

Next, let $\varphi $ be the symplectomorphism corresponding to the action of any nontrivial element in G. Since $\pm [F]$ are the only homology classes with square zero that can be represented by embedded spheres, it follows that $\varphi _*[F]=\pm [F]$ . However, as $\varphi $ is a symplectomorphism preserving orientation, we must have $\varphi _*[F]=[F]$ .

Moreover, $\varphi $ preserves the almost complex structure J, and since it preserves the fiber class $[F]$ , it must map each fiber to another fiber. Therefore, the homeomorphism $\varphi : M\rightarrow M$ induces a homeomorphism $\tilde {\varphi } : \Sigma \rightarrow \Sigma $ of the base surface $\Sigma $ , fitting into the following commutative diagram:

By assumption, $\varphi _{*}=\mathrm {Id}$ . Since $f_{*} : H_{1}(M,\mathbb {Z}) \rightarrow H_{1}(\Sigma ,\mathbb {Z}) $ is an isomorphism by Lemma 2.9, it follows that $\tilde {\varphi }_{*} =\mathrm {Id}$ .

If $g(\Sigma )>1$ , since $\tilde {\varphi }$ is a periodic homeomorphism acting trivially on first homology, Theorem 2.1 implies that $\tilde {\varphi }=\mathrm {Id}_{\Sigma }$ . Thus $\varphi $ acts fiberwise.

Let n be the order of $\varphi $ , so $\varphi $ generates an effective $\mathbb {Z}_n$ -action on M. To complete the argument, let $H \subset \mathbb {Z}_{n}$ be any nontrivial subgroup. By Lemma 2.11, Fix $(H)$ is an almost complex submanifold of M, with respect to the H-invariant, $\omega $ -compatible almost complex structure J. Since the action is effective, Fix $(H) \neq M$ .

Observe that Fix $(H)$ intersects every fiber in at least two points. Therefore, Fix $(H)$ must contain a J-holomorphic curve that intersects the fiber positively. As Fix $(H)$ is an almost complex submanifold and contains such a curve, it cannot contain any of the fibers. Hence, the $\mathbb {Z}_{n}$ -action restricted to each fiber is effective. Applying this reasoning to each nontrivial element of G shows that the group acts effectively on the fibers.

If $g(\Sigma ) = 1$ , define

$$ \begin{align*}G_{1} : = \{g \in G | f \circ g = f\},\end{align*} $$

and set $G_{2} : = G/G_{1}$ . Then there is a short exact sequence

$$ \begin{align*}1 \rightarrow G_1 \rightarrow G \rightarrow G_{2} \rightarrow 1 .\end{align*} $$

The fact that $G_{1}$ acts effectively on the fibers follows by the same argument as in the case $g(\Sigma )>1$ . Furthermore, by the commutative diagram above, $G_{2}$ acts effectively on $\Sigma $ , and its induced action on $H_{1}(\Sigma ,\mathbb {Z})$ is trivial. Hence, by Lemma 2.4, the isomorphism type of $G_{2}$ satisfies the claimed property.

Example 6.1 illustrates that Proposition 3.1 applies to a broader class of symplectic group actions beyond the homologically trivial case. As we will see in Remark 3.8, for irrational ruled symplectic $4$ -manifolds, a finite group action is homologically trivial if and only if it acts trivially on both $H_{1}(M,\mathbb {Z})$ and $H_{2}(M,\mathbb {Z})$ .

3.2 Classification of finite groups acting on irrational ruled symplectic $4$ -manifolds

We now use Proposition 3.1 to classify finite subgroups of symplectomorphisms. We first restrict our attention to symplectomorphisms that act trivially on homology.

We remark that the classification of finite subgroups of symplectomorphisms is not known for rational $4$ -manifolds, even in the case of $\mathbb {CP}^2$ ; see [Reference Chen, Li and Wu9] for partial results and further references.

It follows from Corollary 3.2 that the only finite abelian groups that can act homologically trivially on an irrational ruled symplectic $4$ -manifold with base genus $g\ge 2$ are cyclic groups and the Klein four-group $\mathbb Z_2\times \mathbb Z_2$ . We will later show that any homologically trivial $\mathbb Z_n$ -action with $n\ge 3$ can be extended to a Hamiltonian $S^1$ -action, possibly after modifying the symplectic form. However, this result does not hold in general for homologically trivial $\mathbb Z_2$ -actions.

We also prove the following statement for base genus $1$ , which necessarily allows for nontrivial transformations on the base.

Corollary 3.3. Let $(M,\omega )$ be an irrational ruled symplectic $4$ -manifold whose base surface has genus $g =1$ . Suppose a finite group G acts effectively by symplectomorphisms and trivially on $H_1(M, \mathbb Z)$ . Then, there is an exact sequence

$$ \begin{align*}1 \rightarrow G_1 \rightarrow G \rightarrow G_{2} \rightarrow 1 ,\end{align*} $$

such that:

1. (1) $G_1$ is isomorphic to a finite subgroup of $\mathrm {SO}(3)$ ; that is, $G_1 \cong \mathbb {Z}_{p}, D_{p}$ , or one of the Platonic rotation groups (the symmetry group of a regular tetrahedron, octahedron, or icosahedron).

2. (2) There exist nonnegative integers $p,q$ such that

   $$ \begin{align*}G_{2} \cong \mathbb{Z}_{p} \times \mathbb{Z}_{q}.\end{align*} $$

We record a structural consequence of Corollary 3.2, describing the general form of any finite symplectomorphism group acting on an irrational ruled $4$ -manifold with base genus $g\ge 2$ .

Corollary 3.4. Let $(M,\omega )$ be an irrational ruled symplectic $4$ -manifold whose base surface has genus $g \geq 2$ , and let G be a finite group acting effectively by symplectomorphisms on M. Then there exists a short exact sequence

$$ \begin{align*}1 \rightarrow H_{1} \rightarrow G \rightarrow H_{2} \rightarrow 1,\end{align*} $$

where $H_{1}$ is a finite subgroup of $\mathrm {SO}(3)$ , and $H_{2}$ is a finite subgroup of $\mathrm {SL}_{2g}(\mathbb Z)$ .

We now state the corresponding structural result when the base has genus $1$ .

Corollary 3.5. Let $(M,\omega )$ be an irrational ruled symplectic $4$ -manifold whose base surface has genus $g =1$ , and let G be a finite group acting effectively by symplectomorphisms on M. Then there exist short exact sequences

$$ \begin{align*}1 \rightarrow H \rightarrow G \rightarrow H' \rightarrow 1,\end{align*} $$

$$ \begin{align*}1 \rightarrow H_1 \rightarrow H \rightarrow H_{2} \rightarrow 1,\end{align*} $$

such that:

1. (1) $H'\cong \mathbb {Z}_{k}$ , where $k = 1,2,3,4,6$ .

2. (2) $H_1$ is isomorphic to a finite subgroup of $\mathrm {SO}(3)$ . That is, $H_1 \cong \mathbb {Z}_{p}, D_{p}$ , or one of the Platonic rotation groups (the symmetry group of a regular tetrahedron, octahedron, or icosahedron).

3. (3) There exist nonnegative integers $p,q$ such that

   $$ \begin{align*}H_{2} \cong \mathbb{Z}_{p} \times \mathbb{Z}_{q}.\end{align*} $$

Proof. Define

$$ \begin{align*}H := \{g \in G |g_*= \mathrm{Id}: H_{1}(M,\mathbb{Z}) \rightarrow H_{1}(M,\mathbb{Z} )\} ,\end{align*} $$

and set $H' = G/H$ . Then H is finite and acts by symplectomorphisms on $(M,\omega )$ while trivially on $H_{1}(M,\mathbb {Z})$ . The claimed exact sequence for H then follows directly from Proposition 3.1b).

Finally, the finite group $H'$ acts effectively on $H_{1}(T^2,\mathbb {Z}) \cong \mathbb {Z}^2$ , and hence is isomorphic to a finite subgroup of $\mathrm {SL}_{2}(\mathbb {Z})$ . The finite subgroups of $\mathrm {SL}_{2}(\mathbb {Z})$ are well known to be cyclic of order $1, 2, 3,4, 6$ , so $H'\cong \mathbb {Z}_{k}$ for $k = 1,2,3,4,6$ .

In Example 6.2 an algebraic action of the symmetric group $S_3$ on a ruled surface S over an elliptic curve is given, showing that the first short exact sequence does not split in general.

In the preceding classification of finite subgroups of symplectomorphisms, we do not see how they would act on the second homology group $H_2(M, \mathbb Z)$ . The following classifies the induced action of $\mathrm {Symp}(M, \omega )$ on $H_2(M, \mathbb Z)$ .

Here $SG(l)=(\mathbb Z/2\mathbb Z)^l \rtimes S_l$ is the group of even permutations $\phi $ of the set $\{-l, \cdots , -1, 1, \cdots , l\}$ satisfying $\phi (-k)=-\phi (k)$ for all $1\le k\le l$ . This is the Weyl group of SO $(2l)$ and has rank $2^{l-1}l!$ .

Figure 1

The complete root system of Lagrangian spheres with the induced homological action.

The classifications given in Theorem 1.8 and in Proposition 3.6 are independent of each other. Indeed, for a finite group G acting effectively by symplectomorphisms, the induced action on $H_2(M, \mathbb Z)$ is a subgroup of $SG(l)$ and can contribute to any component of the triple $\{H', H_1, H_2\}$ . However, when we restrict to symplectomorphisms acting trivially on $H_{1}(M,\mathbb {Z})$ , the induced action on $H_2(M, \mathbb Z)$ can contribute only to the group $H_1$ . In Example 6.1, we have $H_1=\mathbb Z_k$ , which arises entirely from the induced action on $H_2(M, \mathbb Z)$ , while the other terms vanish.

3.3 Fixed point sets of symplectic cyclic actions

In this subsection, we analyze the fixed point sets of symplectic cyclic actions. Our first result concerns a $\mathbb Z_k$ -action with $k>2$ , which will be used in the construction of the circle action in Section 5.

Proof. By Proposition 3.1, the moduli map $f: M \rightarrow \Sigma $ is $\mathbb Z_k$ -equivariant, and each fiber is preserved by the $\mathbb {Z}_k$ -action.

Let $F = f^{-1}(p) \cong S^2$ be a smooth fiber. Since the almost complex structure J is integrable on F, the induced action on the fiber is by a Möbius transformation of finite order, in particular, an elliptic transformation. Such a transformation fixes exactly two points, with weights $a \mod k$ and $-a \mod k$ , where $\mathrm {gcd}(a,k)=1$ because the action is effective. Since $k>2$ , we must have $a \neq -a \mod k$ , as otherwise $2a = 0 \mod k$ , contradicting the condition gcd $(a,k)=1$ .

Therefore, the fixed point set $M^{\mathbb {Z}_k}$ intersects each smooth fiber in exactly two distinct points. Let $C\subset M^{\mathbb {Z}_k}$ be an irreducible component of the fixed point set that intersects the smooth fiber F. Then the intersection number $ C\cdot F =\int _{\Sigma } f_{*}([C])> 0$ . By Lemma 2.11, C is a smooth J-holomorphic curve. The curves C and F intersect transversely because the fixed subspace of the tangent space at a fixed point is disjoint from the tangent space of the fiber away from the origin, as the action is nontrivial on F. Therefore, $C\cdot F \leq 2$ .

We now show that $C\cdot F =1$ . Each fixed point on F lies in a connected component of the fixed point set $M^{\mathbb {Z}_k}$ of dimension $2$ . The normal bundle to the fiber at a fixed point is the tangent space $T_{p}F$ , and the $\mathbb Z_k$ -action on the normal bundle has weight $w \in \mathbb {Z}_k$ , constant along connected components. Since the weights at the two fixed points on F are distinct, they must lie in distinct connected components of $M^{\mathbb {Z}_k}$ . Hence, the two fixed points belong to two different $2$ -dimensional components $\Sigma _1,\Sigma _{2} \subset M^{\mathbb {Z}_k}$ , each intersecting F transversely at a single point. Therefore both $\Sigma _1$ and $\Sigma _2$ are sections of the fibration, proving part (1).

Now let S be an irreducible component of a singular fiber. By Theorem 2.7, such components are J-holomorphic spheres with self-intersection $S \cdot S < 0$ . Suppose, for contradiction, that $\varphi (S) \neq S$ . Since $\varphi $ acts trivially on $H_{2}(M,\mathbb {Z})$ , we have $[\varphi (S)]=[S]$ . Then by positivity of intersections

$$ \begin{align*}S \cdot S = \varphi(S) \cdot S \geq 0,\end{align*} $$

contradicting $S\cdot S<0$ . Hence $\varphi (S)=S$ , and each irreducible component of every fiber is preserved, proving part (2).

Upon reviewing the proof of Proposition 3.7, the reader will notice that we only used the assumption that the action is trivial on $H_{1}$ and $H_{2}$ . In the present context, this is equivalent to the action being homologically trivial.

Let N be an orientable, closed $4$ -manifold with torsion-free homology, and let $h: N \rightarrow N$ be an orientation-preserving homeomorphism acting trivially on $H_{1}(N,\mathbb {Z})$ . Let

$$ \begin{align*}\mathrm{PD}: H^{3}(N,\mathbb{Z}) \rightarrow H_{1}(N,\mathbb{Z})\end{align*} $$

denote the Poincaré duality isomorphism. By the naturality of Poincaré duality for orientation-preserving homeomorphisms, we have

$$ \begin{align*}\mathrm{PD} = h_{*}\circ \mathrm{PD}\circ h^{*},\end{align*} $$

which implies that if $h_*$ acts trivially on $H_1(N, \mathbb Z)$ , then $h^*$ acts trivially on $H^{3}(N,\mathbb {Z})$ . Since N has torsion-free homology, we have

$$ \begin{align*}H^{3}(N,\mathbb{Z}) \cong \mathrm{Hom}(H_{3}(N,\mathbb{Z}),\mathbb{Z}),\end{align*} $$

and therefore h must act trivially on $H_{3}(N,\mathbb {Z})$ . This proves the following:

We now turn to the case of involutions. The following amendment of Proposition 3.7 holds for $k=2$ , where an additional possibility arises: unlike the case $k>2$ , the fixed point set may be connected. In Theorem 4.1, we will construct an example where this occurs, providing a key ingredient for a counterexample to Question 1.1 in the case of involutions on minimal irrational ruled $4$ -manifolds.

Proof. As in Proposition 3.1, the involution $\tau $ preserves the fibration structure and acts fiberwise. Since $M^{\tau }$ intersects each fiber in two points, it forms an almost complex submanifold that intersects fibers transversely in two points, so

$$ \begin{align*}M^{\tau}\cdot F=2 .\end{align*} $$

In particular, $M^{\tau }$ cannot contain a fiber as an irreducible component. Decomposing $M^{\tau }$ into connected components, it follows that it must consist of either one or two components, which is a bisection or two disjoint sections.

The projection of each component to the base surface $\Sigma $ is either a homeomorphism (in the case of a section) or a two-to-one covering (in the case of a bisection). This gives the claimed genera:

• ○ For a section: genus g,

• ○ For a bisection: genus $2g-1$ .

Since M is an $S^2$ -bundle over a surface, the signature $\sigma (M) = 0 $ . By the main result of [Reference Jännich and Ossa22], the self-intersection of the fixed point set satisfies:

$$ \begin{align*}M^{\tau}\cdot M^{\tau} = \sigma(M) = 0,\end{align*} $$

which establishes the claims about the self-intersections.

From the above argument, if the condition $b_2=2$ is removed, the same topological classification of $M^{\tau }$ holds, though the self-intersection calculation no longer applies.

4.1 Exotic actions of the Klein 4-group on irrational ruled symplectic $4$ -manifolds

Using the symmetry among different embeddings of $S^1$ into $S^2$ relative to a pair of antipodal points, we can construct exotic, homologically trivial symplectic actions of the Klein $4$ -group on irrational ruled $4$ -manifolds diffeomorphic to $\Sigma _g\times S^2$ , for any genus $g\ge 1$ .

Recall that the Klein $4$ -group is the only noncyclic abelian subgroup that can arise as a group of homologically trivial symplectomorphisms on irrational ruled symplectic $4$ -manifolds. This makes it a natural next case to investigate beyond the cyclic examples constructed earlier.

Proof. The idea is to apply Theorem 4.1 to construct three involutions. For each $j=1,2,3$ , let $i_j: S^1 \rightarrow S^2$ be such that the associated embeddings $\mu _1, \mu _2, \mu _3$ have pairwise disjoint images. The resulting involutions have fixed point sets given by Im $(\mu _j)$ , respectively. To ensure that these involutions and the identity give an action of $\mathbb {Z}_2 \times \mathbb {Z}_2$ on $\Sigma _g\times S^2$ , we have to ensure that the images of $i_j$ form an orthonormal frame of $\mathbb {R}^3$ .

This may be done as follows. Let

$$ \begin{align*}i_1(\cos\theta, \sin\theta)&=(\cos\theta, \sin\theta, 0),\\i_2(\cos\theta, \sin\theta)&=(-\sin\theta, \cos\theta, 0)\end{align*} $$

and $i_{3}(\cos \theta , \sin \theta ) = (0,0,1)$ . The associated involutions $q_1$ and $q_2$ are exotic in the sense that they cannot be extended to symplectic circle actions, the proof is as in Corollary 4.2 using the fact that the fixed point set is connected. However, the third involution $q_3$ , generated by composing $q_1$ and $q_2$ , is not exotic. Its fixed point set in each $S^2$ -fiber is the standard pair $(0, 0, \pm 1)$ , and $q_3$ acts as the standard product involution $\mathrm {id_{\Sigma }}\times r$ of $\Sigma _g\times S^2$ , where r is the $180^{\circ }$ rotation about the z-axis on $S^2$ .

To construct a symplectic form invariant under this group action, we average the standard product symplectic form $\omega $ over the group:

$$ \begin{align*}\alpha = \frac{1}{4}(\omega + q_1^{*}\omega+ q_2^{*}\omega+ q_3^{*}\omega).\end{align*} $$

Since each $q_j$ is a symplectomorphism on fibers and acts trivially on the base, $\alpha $ is closed and nondegenerate on the fibers. Let $\beta $ be the pullback of an area form on $\Sigma _g$ . Then for sufficiently large $K\gg 0$ , the form

$$ \begin{align*}\omega'=\alpha+K\beta\end{align*} $$

is a symplectic form invariant under the Klein $4$ -group action.

Note that the fixed point sets of the involutions associated to $i_1$ and $i_{2}$ are bisections. It follows from Corollary 4.2 that involutions $q_1,q_2$ cannot extend to a symplectic circle action, even after changing the symplectic form.

For the last statement, suppose the Klein $4$ -group action could extend to a symplectic action of SO $(3)$ with some embedding $\mathbb Z_2\times \mathbb Z_2\subset \mathrm {SO}(3)$ . Then any nontrivial element $a\in \mathbb Z_2\times \mathbb Z_2\subset \mathrm {SO}(3)$ would lie on some maximal torus $S^1\subset \mathrm {SO}(3)$ . But as shown above, two such a cannot extend to a symplectic circle action. This contradiction shows the Klein $4$ -group action does not extend to SO $(3)$ symplectically.

In Section 6.3, some actions of a dihedral group $D_{2k}$ on minimal irrational ruled surfaces are exhibited, for which the fixed set of some of the elements is a bisection and which preserve integrable complex structures of a certain type.

6.1 Actions preserving $H_1$ but not $H_2$

Let $\Sigma $ be any Riemann surface, and set $\alpha = e^{\frac {2 \pi i }{k}}$ . Define a $\mathbb {Z}_{k}$ -action on $M = \Sigma \times \mathbb {CP}^1$ by

$$ \begin{align*}\alpha^{n}\cdot (p,[z_0:z_1]) = (p, [\alpha^{n} z_0:z_1]) .\end{align*} $$

This gives a homologically trivial (in fact, Hamiltonian) symplectic action with respect to the product symplectic form on M.

Now fix a point $p \in \Sigma $ , and consider the free orbit

$$ \begin{align*}\mathcal O =\{ (p,[\alpha^n:1])| n= 1, \cdots, k \}.\end{align*} $$

Blowing up the points in $\mathcal O$ using the standard equivariant local model yields a new symplectic $4$ -manifold $\tilde {M} = Bl_{\mathcal O}(M)$ , on which $\mathbb Z_k$ continues to act effectively. The resulting action is trivial on $H_{1}(\tilde {M},\mathbb {Z})$ but nontrivial on $H_{2}(\tilde {M},\mathbb {Z})$ .

Furthermore, since the original action is holomorphic and admits fixed points, the blowup inherits a natural $\mathbb {Z}_k$ -invariant Kähler form. This can be realized explicitly via an equivariant embedding into a projective space followed by equivariant resolution of singularities.

This gives the following example.

6.2 Short exact sequences of automorphisms

The following example illustrates the short exact sequences constructed in Section 3.2.

Example 6.2. Let $C \subset \mathbb {CP}^2$ be the smooth elliptic curve defined by

$$ \begin{align*}z_{1}^3 + z_{2}^3 + z_{3}^3=0.\end{align*} $$

Let $G = S_{3}$ act on C by permuting the homogeneous coordinates. Let $N_{C} \cong \mathcal {O}(3)|_{C}$ be the normal bundle of C. Then the action of G on C induces an action on $N_{C}$ , and thus on the projectivized bundle $\mathbb {P}(N_{C}\oplus \mathcal {O})$ , which is a nontrivial $S^2$ -bundle over $C\cong T^2$ .

Observe that every nontrivial element of $S_{3}$ induces a nontrivial automorphism of C, so $H_{1}$ is the trivial group. Moreover, the elements of order $3$ act on C without fixed points, and hence trivially on $H_{1}(C,\mathbb {Z})$ ; while the elements of order $2$ each fix a single point of C.

For example, the transformation $\tau :[z_1:z_2:z_3] \mapsto [z_2:z_1:z_3]$ has four fixed points:

$$ \begin{align*}[1:1 : -\sqrt[3]{2} \alpha] \in C,\end{align*} $$

where $\alpha ^3=1$ and $[-1:1:0]$ . That is, it is the Galois involution giving the elliptic curve as a branched double cover of the Riemann sphere. By the Lefschetz fixed point theorem, $\tau $ acts on $H_{1}(C,\mathbb {Z})$ with trace $-2$ and order $2$ , that is, $\tau _{*} = -Id$ .

Next, consider the matrix

$$\begin{align*}\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \end{align*}$$

which has three distinct eigenvalues. The corresponding transformation

$$ \begin{align*}\mu:[z_1:z_2:z_3] \mapsto [z_3:z_1:z_2]\end{align*} $$

on $\mathbb {CP}^2$ has three fixed points $[1:1:1], [1:\beta :\beta ^2], [1:\beta ^2:\beta ],$ where $\beta = e^{\frac {2 \pi i}{3}}$ . A direct check via the defining equation shows that none of these points lie on C. Therefore, the first exact sequence of Corollary 3.4 takes the form

$$ \begin{align*}1 \rightarrow \mathbb{Z}_{3} \rightarrow G \rightarrow \mathbb{Z}_2 \rightarrow 1.\end{align*} $$

In particular, this shows that the short exact sequences in Corollary 3.4 do not split in general. Moreover, since all of the transformations preserve both the $S^2$ -fibers and the section $\mathbb {P}(N_{C} \oplus 0)$ , they act trivially on $H_{2}(M,\mathbb {Z})$ . This conclusion also follows directly from Proposition 3.6, as minimal irrational ruled surfaces, such as $M=\mathbb {P}(N_{C}\oplus \mathcal {O})$ , contain no Lagrangian spheres.

6.3 Exotic dihedral actions preserving a complex structure

From the structure of their fixed point sets and by Lemma 2.16, it follows that the homologically trivial actions of $\mathbb {Z}_2$ and $\mathbb {Z}_2 \oplus \mathbb {Z}_2$ constructed in Theorem 4.1 and Theorem 4.5 respectively cannot preserve the product complex structure on $\Sigma \times S^2$ . Furthermore, as shown, these actions do not extend to Hamiltonian actions of $S^1$ or SO $(3)$ , respectively.

In this subsection, we observe that certain holomorphic involutions with respect to the integrable complex structures of a certain type on $\Sigma \times S^2$ can also have connected bisection fixed point sets. This provides an algebro-geometric model for the $\mathbb Z_2$ -action construction in Section 4.

In the following, we work over the field of complex numbers $\mathbb C$ , and we adopt the notation established in Section 2.4 of the preliminaries.

Proposition 6.3. Let C be a smooth algebraic curve, and L a nontrivial line bundle on C with $L \otimes L \cong \mathcal {O}$ . Define $E =\mathcal {O} \oplus L$ , and let $S = \mathbb {P}(E)$ be the associated ruled surface over C. Let $s_{1},s_{2}$ be the sections corresponding to $ \mathbb {P}(\mathcal {O} \oplus 0)$ , $ \mathbb {P}( 0 \oplus L)$ , respectively.

Then, there is a short exact sequence:

$$ \begin{align*}1 \rightarrow \mathbb{C}^* \xrightarrow[]{\varphi_1} \mathrm{Aut}_{C}(S) \xrightarrow[]{\varphi_2} \mathbb{Z}_2 \rightarrow 1 .\end{align*} $$

For an element $g \in \mathrm {Aut}_{C}(S)$ , the following hold:

1. a) $\varphi _{2}(g) =1, \;\; g \neq 1 \iff $ the fixed point set of g is the disjoint union $s_{1} \cup s_2$ .

2. b) $\varphi _{2}(g) \neq 1 \iff \, g^2 =\mathrm {Id}_{S}$ , the fixed point set of g is a smooth irreducible bisection of $\pi : S\rightarrow C$ , and $g(s_{i}) = s_{j}$ for $i \neq j$ .

Proof. By [Reference Maruyama31, Lemma 2 (2)], we have $M(E) = N(E)=0$ , and the only two maximal subbundles of E are $\mathcal {O} \oplus 0$ and $ 0 \oplus L$ . Thus, by Lemma 2.14, the only sections of $S=\mathbb P(E)$ with self-intersection zero are $s_{1}$ and $s_{2}$ , and all other sections have strictly positive self-intersection.

Consider the short exact sequence from Lemma 2.13:

$$ \begin{align*}1 \rightarrow \mathrm{Aut}(E)/\mathbb{C}^* \xrightarrow[]{\varphi_1} \mathrm{Aut}_{C}(S) \xrightarrow[]{\varphi_2} \Delta \rightarrow 1 .\end{align*} $$

By [Reference Maruyama31, Lemma 4(2)], $\Delta \cong \mathbb {Z}_2$ . Note that

$$ \begin{align*}\det(E)^{-1} \otimes L^2 \cong L^{-1} \cong L,\end{align*} $$

which admits no global sections. Hence, by [Reference Maruyama31, Theorem 1(3)], Aut $(E)/\mathbb {C}^* \cong \mathbb {C}^*$ (see the proof of [Reference Maruyama31, Theorem 2]). The induced $\mathbb {C}^*$ -action on S is given by

$$ \begin{align*}z\cdot [(x,y)] = [(zx,y)] ,\end{align*} $$

and fixes precisely the union $s_{1} \cup s_{2}$ . Therefore, all nontrivial elements of $\ker (\varphi _2) \cong \mathbb {C}^*$ have fixed point set $s_{1} \cup s_{2}$ .

Now let $g\in \mathrm {Aut}_{C}(S) $ be an automorphism with $\varphi _2(g)\ne 1$ . We claim that $g^2=\mathrm {id}_S$ . Since $\varphi _2(g^2)=\varphi _2(g)^2=1$ , it follows that $g^2\in \ker (\varphi _2)$ . Suppose for contradiction that $g^2\ne \mathrm {id}_S$ , so that $g^2\in \ker (\varphi _2)\setminus \{\mathrm {id}_S\}$ . Then Fix $(g^2)=s_1\cup s_2$ .

On each fiber, the action of g is a Möbius transformation of order $2$ , hence an involution with exactly two fixed points. Since Möbius transformations have the property that the fixed point set of g is contained in that of $g^2$ , and they are equal when $g^2$ is nontrivial, we conclude that Fix $(g)=\mathrm {Fix}(g^2)=s_{1} \cup s_2$ .

Now consider the action of g on $S \setminus s_{1}$ , which is isomorphic to the total space of the line bundle L. On this space, g must act fiberwise by Möbius transformations and must preserve the zero section, since g fixes both $s_1$ and $s_2$ . But this implies that $g\in \varphi _{1}(\mathbb {C}^*) = \ker (\varphi _2)$ , contradicting our assumption that $\varphi _2(g)\ne 1$ . Therefore, $g^2=\mathrm {id}_S$ , as claimed.

The same argument shows that any element $g\notin \ker (\varphi _2)$ cannot fix the union $s_{1} \cup s_{2}$ . Since $s_{1}$ and $s_{2}$ are the only sections of S with self-intersection zero, and g is an automorphism over C, the only possibility is that $g(s_{1}) = s_{2}$ and $g(s_{2}) = s_{1}$ .

Next, we examine the fixed point set Fix $(g)$ . Since g acts fiberwise as an involutive Möbius transformation, each fiber contains exactly two fixed points under g. Thus, Fix $(g)$ is a curve in S that intersects each fiber transversely in two points and is disjoint from $s_{1} \cup s_{2}$ .

Suppose for contradiction that Fix $(g)$ is reducible and consists of two disjoint irreducible components. Then each component must be a section, distinct from $s_1$ and $s_2$ , and hence must have strictly positive self-intersection. However, by the light cone lemma, such sections would necessarily intersect one of $s_{1}$ or $s_2$ , contradicting the assumption that Fix $(g)$ is disjoint from $s_{1} \cup s_{2}$ . Therefore, Fix $(g)$ must be a smooth irreducible bisection.

The group structure of Aut $_C(S)$ is described in [Reference Maruyama31, Theorem 2 (4)] and is isomorphic to the subgroup $H\subset \mathrm {PGL}(2, \mathbb {C})$ given by

$$ \begin{align*}H = \left\{ M_{\alpha}= \left[ {\begin{array}{cc} \alpha & 0 \\ 0 & 1 \\ \end{array} } \right] | \alpha \in \mathbb{C}^* \} \cup \{ M^{\prime}_{\beta}= \left[ {\begin{array}{cc} 0 & \beta \\ 1 & 0 \\ \end{array} } \right] | \beta \in \mathbb{C}^* \right\} .\end{align*} $$

An element of H lies in the kernel of $\varphi _2$ if and only if it is one of $M_{\alpha }$ .

For convenience, we record the group law as given in ([Reference Maruyama31, Page 96, footnote]):

$$ \begin{align*}M_{\alpha}M_{\alpha'} = M_{\alpha \alpha'} \;\;,\;\; M^{\prime}_{\beta}M^{\prime}_{\beta'} = M_{\beta \beta^{\prime -1}}\end{align*} $$

$$ \begin{align*}M_{\alpha}M^{\prime}_{\beta} = M^{\prime}_{\alpha \beta} \;\;,\;\; M^{\prime}_{\beta}M_{\alpha} = M^{\prime}_{\beta \alpha^{-1}}\end{align*} $$

We note that every element of the form $M^{\prime }_{\beta }$ is an involution. The group H is nonabelian; for instance, $M^{\prime }_{-1}$ and $M_{\alpha }$ commute in PGL $(2, \mathbb {C})$ if and only if $\alpha = \pm 1$ . However, the subgroup consisting of the four matrices

$$ \begin{align*}I_{2},\;\;\; M_{-1},\;\;\; M^{\prime}_1,\;\;\; M^{\prime}_{-1}\end{align*} $$

forms a Klein $4$ -group isomorphic to $\mathbb {Z}_2 \times \mathbb {Z}_2$ .

This $\mathbb {Z}_2 \times \mathbb {Z}_2$ action may be generalized as follows:

Example 6.4. Consider the set:

$$ \begin{align*}D =\left\{ M_{\alpha},M^{\prime}_{\beta} \mid \alpha^n=\beta^n=1\right\} \subset H.\end{align*} $$

It is not hard to check that D is closed under the group operation, and therefore is a subgroup. Moreover, D is isomorphic to a dihedral group of order $2n$ .

For the action of the above subgroup on S, the fixed point set of $n-1$ of its elements is a disjoint union of sections, while the fixed point set of the n involutive elements $M^{\prime }_{\beta }$ is an irreducible bisection.

Finally, we give an example of an action of the Klein $4$ -group where every nontrivial element in $G = \mathbb {Z}_2 \times \mathbb {Z}_2 $ fixes an irreducible bisection.

The exact sequence recalled in Lemma 2.13 will be important here. The following lemma gives a sufficient condition for an element $g \in \text {Aut}_{C}(\mathbb {P}(E))$ to be contained in the first term of this short exact sequence. Recall that a section of a geometrically ruled surface is a morphism $s: C \rightarrow \mathbb {P}(E)$ such that $\pi \circ s = Id_{C}$ .