Symplectic rational $G$-surfaces and equivariant symplectic cones
Weimin Chen Tian-Jun Li Weiwei Wu




Weimin Chen, Department of Mathematics, Unviersity of Massachusetts at Amherst, Amherst, MA 01003 E-mail address:  

Tian-Jun Li, School of Mathematics,
      University of Minnesota,
      Minneapolis, MN 55455 E-mail address:  

Weiwei Wu, Department of Mathematics,
      University of Georgia,
      Athens, GA 30606 E-mail address:  






[2000]



Symplectic rational G-surfaces and equivariant symplectic cones
    Weimin Chen, Tian-Jun Li, and Weiwei Wu
    December 30, 2023
===============================================================




We give characterizations of a finite group G acting symplectically on a rational surface 
(^2 blown up at two or more points).  In particular, we obtain a symplectic 
version of the dichotomy of G-conic bundles versus G-del Pezzo surfaces for the corresponding 
G-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations 
of the group G (which is completely determined for the case of ^2# N^2, 
N=2,3,4), we also investigate the equivariant symplectic minimality and equivariant symplectic 
cone of a given G-rational surface.






§ INTRODUCTION


In this paper we study symplectic 4-manifolds (X,ω) equipped with a finite
symplectomorphism group G, where X is diffeomorphic to a rational surface.  
We shall call such a pair, i.e., ((X,ω),G), a symplectic rational G-surface. 


They are the symplectic analog of (complex) rational G-surfaces studied in algebraic geometry, which are rational surfaces equipped with a holomorphic G-action.  These rational 
G-surfaces played a central role in the classification of finite subgroups of the plane 
Cremona group, a problem dating back to the early 1880s, see <cit.>. 

Note that any rational G-surface can be regarded
as a symplectic rational G-surface – simply endowing it with a G-invariant Kähler form
which always exists. Our work shows that a large part of the story regarding the classification of
rational G-surfaces can be recovered by techniques from 4-manifold theory and symplectic topology. Furthermore, we also add some new, interesting symplectic geometry aspect to the
study of rational G-surfaces; in particular, in regard to the equivariant symplectic minimality and 
equivariant symplectic cone of the underlying smooth action of a rational G-surface.
In addition, we also obtain some result which does not seem previously known in the algebraic geometry literature (cf. Theorem <ref>). 


We begin with a discussion on the notion of minimality (i.e., G-minimality) in the equivariant context. 
Let (X,ω) be a symplectic 4-manifold with a finite symplectomorphism group G. Suppose there
exists a G-invariant set of disjoint union of symplectic (-1)-spheres in X. Then blowing
down X along the (-1)-spheres gives rise to a symplectic 4-manifold 
(X^',ω^'), which can be arranged so that G is natually isomorphic to
a finite symplectomorphism group of (X^',ω^'). The symplectic G-manifold
X is called minimal if no such set of (-1)-spheres exists. It was shown in <cit.> that when 
X is neither rational nor ruled, the symplectic G-manifold is minimal if and only if the underlying 
smooth manifold is minimal. However, in the case considered in the present paper, the underlying 
rational surface is often not minimal even though the corresponding symplectic rational G-surface 
is minimal. Furthermore, it is not known whether the notion of G-minimality is the same for the various
different categories, i.e., the holomorphic, symplectic, or smooth categories. In general it is a difficult problem
to establish the equivalence of G-minimality in the different categories, and we refer the reader to <cit.>
for a more thorough discussion on this topic. For our purpose in this paper, it suffices to only study 
minimal symplectic rational G-surfaces.

The most fundamental problem in our study is to classify symplectic rational G-surfaces up to equivariant 
symplectomorphisms. However, work in <cit.> showed that even in the simple case where X 
is ^2 or a Hirzebruch surface and G is a cyclic or meta-cyclic group, such a classification is already 
quite involved. In fact, in one circumstance where G is meta-cyclic,  a weaker classification, i.e., classification up to equivariant diffeomorphisms, still remains open. 

With the preceding understood, the main objectives of this paper are more basic: for symplectic rational G-surfaces X in general, we would like to


  
  * classify the possible symplectic structures;
  
  * describe the induced action of G on H_2(X);
  
  * give a list of possible finite groups for G;
  
  * understand the equivariant minimality and equivariant symplectic cones.


These problems, however, are still highly non-trivial and not completely settled.  In particular, 
part of our determination of G and the induced action on H_2(X) relies on the 
Dolgachev-Iskovskikh's solution of the corresponding problems in algebraic geometry, with new inputs from Gromov-Witten theory and a detailed analysis of the symplectic structures.




 §.§ The setup

In this paper, we shall be focusing on the case where the rational surface, denoted by X, is 
^2 blown up at 2 or more points. More concretely, we shall consider minimal symplectic rational G-surfaces (X,ω) where X=^2# N^2, for N≥ 2. (Note that the minimality assumption implies in particular that G is a nontrivial group.) The case where the rational surface is ^2 or a Hirzebruch surface had been previously studied, cf. <cit.>; we point out that the J-holomorphic curve 
techniques employed in this paper are drastically different in flavor from those developed in these previous works.

For convenience, we shall fix some notations and terminology, which will be frequently used throughout the paper.
We will denote by H, E_1,E_2,⋯, E_N a basis of H^2(X,), under which the intersection matrix takes its
standard form, i.e., H^2=1, E_i^2=-1, H· E_i=0, ∀ i, and E_i· E_j=0, ∀ i≠ j.
The canonical class of (X,ω) will be denoted by K_ω∈ H^2(X), in order to emphasize its
dependence on the symplectic form ω. Another frequently used notation is H^2(X)^G, which denotes the
subset of H^2(X,) consisting of elements fixed under the induced action of G, and is called the invariant lattice. 

Recall that a symplectic rational surface (X,ω) is called monotone if K_ω=λ [ω] is satisfied 
in H^2(X;) for some λ∈. In this case, we have λ<0, and N must be in the range N≤ 8. Such
a symplectic rational surface is the symplectic analog of Del Pezzo surface in algebraic geometry. Another important notion,
given in the following definition and called a symplectic G-conic bundle, corresponds to a conic bundle structure 
on a rational G-surface in algebraic geometry.


Let (X,ω) be a symplectic 4-manifold equipped with a finite symplectomorphism 
group G. A symplectic G-conic bundle structure on (X,ω) is a genus-0 smooth Lefschetz fibration 
π: X→ B which obeys
the following conditions:


  * each singular fiber of π contains exactly one critical point;

  * there exists a G-invariant, ω-compatible almost complex structure J such that the fibers of π are J-holomorphic;

  * the group action of G preserves the Lefschetz fibration.



    Although the above definition looks more rigid than it should be (in particular, it is always an almost complex fibration), Theorem <ref> shows that this is a purely symplectic notion in the case of minimal symplectic rational G-surfaces.


A symplectic G-conic bundle is called minimal if for any singular fiber there is an element of
G whose action switches the two components of the singular fiber.


Here are some immediate consequences from the definition:


  * X is a rational surface if and only if B=^2; in this case, note that the number of singular fibers of π equals
N-1, where X=^2# N^2;

  * the Lefschetz fibration is symplectic with respect to ω;

  * the fiber class lies in the invariant lattice H^2(X)^G as G preserves the Lefschetz fibration;

  * if the underlying symplectic G-manifold is minimal, then the symplectic G-conic bundle must be also minimal.















 §.§ The symplectic structures



Our first theorem is concerned with the symplectic structure of a minimal symplectic rational G-surface.


Let (X,ω) be a minimal symplectic rational G-surface, where 
X=^2# N^2 for some N≥ 2.
Then N≠ 2, and one of the following holds true:


  (1) The invariant lattice H^2(X)^G has rank 1. In this case, 3≤ N≤ 8 and 
(X,ω) must
be monotone.

  (2) The invariant lattice H^2(X)^G has rank 2.  In this case,  N=5  or N≥ 7, and   there exists a 
symplectic G-conic bundle structure on ((X,ω),G).






  (a) An analog of Theorem <ref>  for minimal rational G-surfaces is a classical theorem
in algebraic geometry (see e.g. Theorem 3.8 in <cit.>), a proof of which can be given using the 
equivariant Mori theory (see 4 of <cit.>). With this understood, we remark that our proof of
Theorem <ref> gives an independent proof of the corresponding result in algebraic
geometry by taking ω to be a G-invariant Kähler form. Consequently, a significant
portion of the theory of rational G-surfaces (e.g. as described in <cit.>) can be recovered 
(see Theorems <ref> <ref>).


  (b) In case (1) of Theorem <ref> the invariant lattice H^2(X)^G is spanned by K_ω, and in case (2), H^2(X)^G is spanned by K_ω and the fiber class of the symplectic G-conic bundle. We should point out that the analysis on the type of the group G,
the induced action on H^2(X), as well as the structure of the equivariant symplectic cone, 
depends on the rank of the invariant lattice H^2(X)^G.


  (c) In case (2), the proof of Theorem <ref> reveals the following additional information 
about the symplectic G-conic bundle structure: there exists a basis 
H, E_1,E_2,⋯, E_N of H^2(X) with standard intersection matrix such that 


  (i) the fiber class is given by H-E_1 and the pair of (-1)-spheres in a singular fiber 
are given by the classes E_j and H-E_1-E_j, where j=2,⋯,N;

  (ii) the symplectic areas satisfy ω(E_j)=1/2ω(H-E_1), 
ω(E_1)≥ω(E_j) for j=2,⋯, N;

  (iii) the canonical class K_ω=-3H + E_1+E_2+⋯ +E_N.




  (d)  In complex geometry, it was known that a minimal (complex) rational G-surface which is diffeomorphic to ^2 blown up at 6 points must be Del Pezzo  (cf. <cit.>, Theorem 3.8, Proposition 5.2). However, it seemed new that the invariant Picard group Pic(X)^G must be 
of rank 1.


    



Comparing the three minimality assumptions:




The reader should be noted that a minimal symplectic G-conic bundle is different from a minimal symplectic G-surface with G-conic bundle structure: a minimal 
symplectic G-conic bundle may still contain a G-invariant disjoint union of symplectic 
(-1)-spheres.  However, Lemma <ref> implies this cannot happen when N≥5 and N≠6.

The minimal G-conic bundles is an intermediate notion between minimal symplectic G-surfaces and minimal complex G-surfaces.  There is always a G-invariant symplectic form compatible with the given complex structure on a minimal complex G-surface.  Although we do not know whether this symplectic form is always G-minimal, if we assume a symplectic G-conic bundle structure underlies this action, this conic bundle structure is always minimal.  Therefore, proving 
our results in the more general minimal G-conic bundle context plays an important role in 
bridging the complex and symplectic G-surfaces as well as in the study of the equivariant symplectic cones.



 §.§ The homological action and the groups


Our next task is to describe possible candidates for G. 

We begin with case (1) in Theorem <ref> where (X,ω) is a minimal symplectic rational G-surface such that the invariant lattice H^2(X)^G has rank 1. 
In this case (X,ω) is monotone, H^2(X)^G is spanned by K_ω, and 
3≤ N≤ 8. With this understood, we note that the orthogonal complement of K_ω 
in H^2(X) (with respect to the intersection product), denoted by R_N, is a G-invariant root lattice of type E_N (N=6,7,8), D_5 (N=5), A_4(N=4), and A_2+A_1 (N=3) respectively. We denote by W_N the corresponding Weyl group.



Let (X,ω) be a minimal symplectic rational G-surface such that H^2(X)^G
has rank 1. There are two cases:


  (1) Suppose 4≤ N≤ 8. Then the induced action of G on H^2(X) is faithful, which gives rise to a monomorphism 
ρ: G→ W_N. Moreover, the image ρ(G) in W_N satisfies 

    ∑_g∈ G trace{ρ(g):R_N→ R_N}=0.


  (2) Suppose N=3. Let Γ be the subgroup of G which acts trivially on
H^2(X) and let K:=G/Γ be the quotient group. Then Γ is isomorphic to
a subgroup of the 2-dimensional torus and K is isomorphic to _6 or the dihedral 
group D_12. Furthermore, G is a semi-direct product of Γ by K. As a corollary,
G can be written as a semi-direct product of an imprimitive finite subgroup of PGL(3) by
_2.




       1. For N=4,5, the subgroups of the Weyl group W_N which satisfy
the condition ∑_g∈ G trace{ρ(g):R_N→ R_N}=0 are determined,
see Theorem 6.4 and Theorem 6.9 in <cit.> respectively. All such groups can be realized
by a minimal G-Del Pezzo surface, which is also minimal as a symplectic rational G-surface
with respect to any G-invariant Kähler form (cf. Theorem 1.10(2)). 

2. For N=3, Theorem <ref> (2) and Theorem <ref> completely determined all possible G that acts minimally on X.  The statement in Theorem <ref> implies the corresponding statement in <cit.>, Theorem 6.3. In fact, the semi-direct product structure of G in our statement is an improvement upon the corresponding theorem in algebraic geometry.
For a list of imprimitive finite subgroups of PGL(3) up to conjugacy, see Theorem 4.7 in <cit.>.
 

Now we consider case (2) of Theorem <ref>. In fact, we will work in a slightly more general situation, where the symplectic rational G-surface (X,ω) is not assumed to be minimal, but only admits a minimal symplectic G-conic bundle π:(X,ω)→^2.  Furthermore, we assume N≥ 4 (instead of the fact that N≥ 5 when (X,ω) is a
minimal symplectic rational G-surface). 

We make the following definition.


  
  * Q G is the subgroup that acts trivially on the base ^2 of the G-conic bundle;
  
  * G_0◃ Q is the subgroup that acts trivially on H^2(X) (in the case we consider, there are N-1≥ 3 critical values on the base that was fixed, hence G_0◃ Q);
  
  * P=G/Q, so that G decomposes as
  
    1→ Q→ G→ P→ 1.


 

 We denote by Σ the subset of ^2 which parametrizes the
singular fibers of the symplectic G-conic bundle π. Note that #Σ=N-1, 
and the induced action of P on 
^2 leaves the subset Σ invariant. The action of P on ^2 is effective,
so P is isomorphic to a polyhedral group, i.e., a finite subgroup of SO(3).

Therefore, up to an extension problem, the description of G boils down to the following theorem 
which describes the subgroups G_0 and Q.


Let (X,ω) be a symplectic rational G-surface equipped with a minimal symplectic 
G-conic bundle structure with at least three singular fibers (i.e, N≥ 4). Let G_0,Q 
be given as in Definition <ref>.
Then one of the following is true.


  1. G_0=_m, m>1, and Q is either the dihedral group D_2m 
containing G_0 as an index 2 subgroup,
or Q=G_0 and m is even. Moreover, N must be odd, and any element 
τ∈ Q∖ G_0 switches the two (-1)-spheres in each singular fiber.

  2. G_0 is trivial and Q=_2 or (_2)^2. In the latter case, let 
τ_1,τ_2,τ_3 be the
distinct involutions in Q. Then Σ is partitioned into subsets Σ_1, Σ_2,
Σ_3, where Σ_i parametrizes those singular fibers of which τ_i
leaves each (-1)-sphere invariant, and #Σ_i≡ N-1 2,  for i=1,2,3.






  (a)
Note that when G_0 is trivial and Q=(_2)^2, each element of P acts on Q as an 
automorphism, permuting the three involutions τ_1,τ_2,τ_3. Consequently, 
the action of P on the base ^2 preserves the partition Σ=Σ_1⊔Σ_2⊔Σ_3. 
For the corresponding result in algebraic geometry, see <cit.>,
Theorem 5.7.


  (b) When the fiber class of the symplectic G-conic bundle is unique, Q is uniquely
determined as a subgroup of G, see Proposition 4.5 for more details. 





 §.§ Minimality and equivariant symplectic cones


In this subsection, we are concerned with the underlying smooth action of a minimal symplectic rational G-surface. In particular, our consideration here offers an interesting symplectic geometry
perspective to the study of rational G-surfaces in algebraic geometry.

We begin with the setup of our study here. Let X=^2# N^2, N≥ 2,
which is equipped with a smooth action of a finite group G. Suppose there is a G-invariant
symplectic form ω_0 on X such that the corresponding symplectic rational 
G-surface (X,ω_0) is minimal. With this understood, we denote

    Ω(X,G):={: is symplectic on X, g^*=, for any g∈ G}.


Note that Ω(X,G)
is non-empty as ω_0∈Ω(X,G). The part (2) of the following theorem shows 
that the underlying smooth action of a minimal (complex) rational G-surface satisfies the above 
assumption, where we can take ω_0 to be any G-invariant Kähler form.


(1) Let X=^2# N^2, N≥ 2, which is equipped with a smooth action 
of a finite group G. Suppose there is a G-invariant symplectic form ω_0 on X 
such that ((X,ω_0),G) is minimal. Then for any ω∈Ω(X,G), 
the canonical class K_ω=K_ω_0 or -K_ω_0, and the symplectic rational G-surface 
(X,ω) is minimal. 

(2) Let X be any minimal (complex) rational G-surface which is ^2 blown up at 2 or
more points. Then for any symplectic form ω which is invariant under the underlying
smooth action of G (e.g., any G-invariant Kähler form), the corresponding symplectic 
rational G-surface (X,ω) is minimal.


At the time of writing, it is not known whether
the symplectic minimality would imply the smooth minimality of the underlying group action.  The symplectic minimality in Theorem <ref> is a weaker statement that symplectic minimality 
is determined by the underlying smooth action, but the proof is still quite non-trivial.
For related (stronger) results in the case of G-Hirzebruch surfaces, see <cit.>.


5mm
With the minimality in Theorem <ref>(1) in place, we now turn our attention to the equivariant 
symplectic cone of the G-manifold (X,G).

 The equivariant symplectic cone of (X,G) is defined as
    
    C̃(X,G)={Ω:Ω=[], is a G-invariant symplectic form}⊂ H^2(X;)^G.






Note that K_-=-K_. Therefore it suffices to
consider the subset 

    C(X,G):={[ω]| ω∈Ω(X,G), K_ω=K_ω_0}⊂ H^2(X;)^G.

Furthermore, we observe that if H^2(X)^G has rank 1, 
C(X,G)={λ K_ω_0|λ∈, λ<0}.
In what follows, we shall assume that H^2(X)^G has rank 2. Note that under this assumption, 
N≠ 6 by part (2) of Theorem <ref>.


In order to describe C(X,G), it is helpful to introduce the following terminology. A class
F∈ H^2(X)^G is called a fiber class if there exists an ω∈Ω(X,G)
such that F is the class of the regular fibers of a symplectic G-conic bundle on
((X,ω),G). Since we will focus on the set C(X,G), we shall assume further that
[ω]∈ C(X,G), i.e., K_ω=K_ω_0. 

We observe that since rank H^2(X)^G=2, the class of such an ω can be written
as 

    [ω]=-a K_ω_0+bF, a>0.
 

With this understood, we consider the following subset of C(X,G) and its projective classes

    C(X,G,F)={[]∈ C(X,G): =-aK__0+bF, a>0,b≥0},
       Ĉ(X,G,F)={[]∈ C(X,G,F): (F)=2} (equivalently, a=1).











Now []∈Ĉ(X,G,F) can be written as [ω]=-K_ω_0+
δ_ω,F F. 
Then the one to one correspondence 
[ω]↦δ_ω,F identifies Ĉ(X,G,F) with a subset of . With this
understood, we introduce 

    δ_X,G,F:=inf_[ω]∈Ĉ(X,G,F)δ_ω,F∈ [0,∞).

Note that (X,ω) is monotone if and only if δ_ω,F=0, so δ_ω,F
may be thought of as a sort of gap function which measures how far away (X,ω) is from
being monotone. 

  
Let X=^2# N^2, N≥ 2, be equipped with a smooth finite G-action
which is symplectic and minimal with respect to some symplectic form ω_0. Furthermore,
assume rank H^2(X)^G=2.


  
  * If N≥ 9 or G_0 is nontrivial, there is a unique fiber class F, and C(X,G)=C(X,G,F). 
  
  * For N=5,7,8, either there is a unique fiber class F, or there are two distinct fiber 
classes F,F^'. In the former case, C(X,G)=C(X,G,F), and in the latter case,
C(X,G)=C(X,G,F)∪ C(X,G,F^'), with C(X,G,F)∩ C(X,G,F^') being either 
empty or consisting of [ω] such that (X,ω) is monotone.


  * For any fiber class F, Ĉ(X,G,F) is identified with either
[0,∞) or (δ_X,G,F,∞) under [ω]↦δ_ω,F. 
(In particular, δ_X,G,F can not be attained unless it equals 0.)




We conjecture that when there are two distinct fiber classes, the equivariant symplectic cone 
must contain the class of a monotone form. Furthermore, it is an interesting problem to determine
the gap functions δ_X,G,F for a given minimal rational G-surface X with 
Pic(X)^G=^2. We shall leave these studies for a future occasion. 




The organization of this paper is as follows. Section 2 is concerned with the proof of the structural
theorem, Theorem 1.3. In section 2.1 we collect some preliminary lemmas on minimal symplectic 
G-conic bundles. In section 2.2 we review a reduction process for the exceptional classes 
of a rational surface which plays an essential role in the proof of Theorem 1.3. Sections 2.3
and 2.4 are devoted to the proof of Theorem 1.3. Section 3 is concerned with the analysis of
the structure of group G. Proofs of Theorems 1.5 and 1.8 are presented here. Finally,
section 4 is devoted to the discussion on equivariant symplectic minimality and equivariant symplectic cones. In particular, we prove Theorems 1.10 and 1.12. At the end of section 4,
we also include a uniqueness result on the subgroup Q in Definition 1.7. 




Acknowledgments:

This long overdue project started from the Workshop and Conference on Holomorphic Curves and Low Dimensional Topology in Stanford, 2012.
We are grateful to Igor Dolgachev for inspiring conversations during the FRG conference on symplectic birational geometry 2014 in University of Michigan.  This work was partially supported by NSF Focused Research Grants DMS-0244663 and DMS-1065784, under FRG: Collaborative Research: Topology and Invariants of Smooth 4-Manifolds.








§ THE STRUCTURE OF (X,Ω)




 §.§ Preliminary lemmas on symplectic G-conic bundles


Let (X,ω) be a symplectic rational G-surface, where 
X=^2# N^2, and let π: X→^2 be
a symplectic G-conic bundle on (X,ω). We first observe that 
each singular fiber of π consists of a pair of (-1)-spheres. To see this,
let C_1,C_2 be the components of a singular fiber, which are embedded 
J-holomorphic spheres. Since C_1,C_2 are isolated J-curves, their self-intersection
must be negative. On the other hand, C_1· C_2=1, so it follows easily from
(C_1+C_2)^2=0 that both of C_1,C_2 are (-1)-spheres.

There are N-1 singular fibers. We shall pick a (-1)-sphere from each singular fiber
and name the homology classes by E_2,⋯, E_N. Then there is a unique pair of line class and exceptional class H and  E_1
such that 


  
  * H-E_1 is the fiber class of π,
  
  * H,E_1, E_2,⋯, E_N form a basis
of H_2(X) with standard intersection matrix.


 With this understood, observe that

 
   
  * for each (-1)-sphere E_j, j=2,⋯, N, the other (-1)-sphere lying in the same
singular fiber has homology class H-E_1-E_j,
   
  * the canonical class K_ω=-3H+
E_1+⋯ +E_N.
 


There are further consequences if the symplectic G-conic bundle is minimal. In this case, 
for each (-1)-sphere E_j, j=2,⋯,N, there exists a g∈ G such that 
g· E_j=H-E_1-E_j. This implies

    ω(E_j)=1/2ω(H-E_1),    j=2,⋯, N.


Thus a minimal symplectic G-conic bundle falls into three cases:

    (i)  ω(E_1)=ω(E_j),   (ii)  ω(E_1)>ω(E_j),   (iii) ω(E_1)<ω(E_j).

Case (i) occurs iff (X,ω) is monotone. Since E_1 is a section class, we shall call
case (ii) (resp. case (iii)) a symplectic G-conic bundle with small fiber area (resp. large
fiber area).

The following lemma is the J-holomorphic analog of Lemma 5.1 in <cit.>. 


Let π:X→^2 be a symplectic G-conic bundle on (X,ω),
which comes with a G-invariant, ω-compatible almost complex structure J.
Suppose E, E^' are two distinct J-holomorphic sections of self-intersection 
-m,-m^', and let r be the number of singular fibers where E,E^' intersect 
the same component. Then

    N-1=r+m+m^'+2E· E^'.

Moreover, if the symplectic G-manifold (X,ω) is minimal, then N≥ 5 must be true.




Since E_1,F=H-E_1,E_s,s>1 generate H^2(X), both E and E' have the form 

    E_1+cF+∑_t>1c_tE_t,

where c∈, and c_t=0 or 1 depending on which component they intersect on each singular fiber. 


Since E,E^' are sections, we have
 
    E^'-E=bF +∑_s b_s E_s,

 where b∈ and b_s=± 1 with s running over the set of singular fibers at where
 E^', E intersect different components. Note that the number of s is exactly N-1-r.
 Now 
 
    (E^'-E)^2= ∑_s b_s^2 E_s^2=-(N-1-r),

 which gives
 
    N-1=r+m+m^'+2E· E^'.


Now suppose (X,ω) is minimal. We will show that N≥ 5 in this case.
First, we claim that the sections E,E^' in the lemma do exist. This is because 
the class E_1 can be represented by a J-holomorphic stable curve.  By checking the intersection with H-E_1, the E_1 stable representative contains a unique  J-holomorphic section E.  Note that E,F=H-E_1,E_s,s>1 also form a basis of H_2(X), and 
E_1 can be written as
 
    E_1=a(H-E_1)+∑_s>1 a_s E_s+E,
 

 where a≥ 0, and a_s=0 or 1 depending on which component that E intersects at each singular fiber. 
 Note that 
 
    E^2=-2a-1-∑_s>1 a_s.

 We take E^'=g· E for some g∈ G such that E^'≠ E. Such a g∈ G exists 
 because E must intersect a singular fiber and there is a g∈ G which switches the two 
 (-1)-spheres in that singular fiber. Note that E^' is a section, as the fiber class is G-invariant so that E^'· (H-E_1)=E·(H-E_1)=1. Furthermore, note that (E^')^2=E^2. Consequently, if E^2≤ -2, we must have
 
    N=1+r+m+m^'+2 E· E^'≥ 1+0+2+2+0=5.

 If E^2>-2, then a=a_s=0 and E=E_1 must be a (-1)-sphere. The minimality
 assumption then implies that E^'=g· E for some g∈ G can be chosen such that 
 E^' intersects E. In this case,  we have 
 
    N= 1+r+m+m^'+2 E· E^'≥ 1+0+1+1+2=5.

 This finishes off the proof of Lemma <ref>.
 
 

 
 Let (X,ω) be a symplectic rational G-surface which admits a minimal
 symplectic G-conic bundle structure. Then the invariant lattice H^2(X)^G has
 rank 2 which is spanned by K_ω and the fiber class of the  
 symplectic G-conic bundle.
  
 
 
 
 First, we show that H^2(X;)^G is 2-dimensional.
To see this, we first note that  K_ω, H-E_1, E_2,⋯,E_N form a basis of 
H^2(X,).  We set 
 
    V=H^2(X;)/Span _(K_ω,H-E_1).

 Then it suffices to show that V^G={0} because K_ω,H-E_1∈ H^2(X)^G.
 
 We let e_2,⋯,e_N be the image of E_2,⋯,E_N under the quotient map, which
 form a basis of V. Suppose to the contrary, there is a x≠ 0 in V^G. We write
 x=∑_k=2^N a_k e_k. Then there exists a k_0 such that a_k_0≠ 0.
 With this understood, we note that by the minimality assumption,
 there is a g∈ G such that g· E_k_0=H-E_1-E_k_0,
 which means that g· e_k_0=-e_k_0. Now we set I={k| g· e_k=-e_k}. Then
 k_0∈ I; in particular, I≠∅. We let J be the complement of I in the set
 {2,⋯,N}. Then it follows easily that if k∈ J, then g· e_k=± e_l for some 
 l∈ J (g· E_k=E_l or H-E_1-E_l for some l).
 
 With this understood, we write x=y+z, where y=∑_k∈ I a_k e_k and 
 z=∑_k∈ J a_ke_k. Then g· x=-y+z^' for some z^'∈Span _(e_k|k∈ J). Since g· x=x, we have 
 2y=z^'-z∈Span _(e_k|k∈ J). Since e_2,⋯,e_N form a basis of V,
 this clearly contradicts the fact that y∈Span _(e_k|k∈ I) and y≠ 0. Hence the 
 claim that H^2(X;)^G is 2-dimensional.

 Now for any α∈ H^2(X)^G, we write α=aK_+b (H-E_1) for some a,b∈.
 Then α· E_N=-a, implying a∈. On the other hand, α· E_1=-a+b,
 we have b∈ also. Hence H^2(X)^G is spanned by K_ and H-E_1.




Let π: X→^2 be a minimal symplectic G-conic bundle where X=^2
# N^2 with N≥ 6 and even. Let J be any G-invariant, compatible
almost complex structure, and let 

    m_J=max{m∈: there is a J-holomorphic section of π of self-intersection -m}.


 Then m_J≤ (N-4)/2.  
In particular, when N=6, m_J≤ 1.
 
 
 
 First, note that if E is a J-holomorphic section of self-intersection -m, then there is
 a g∈ G such that g· E≠ E, where g· E is also of self-intersection -m.
 This is because E must intersect a singular fiber, and by the minimality assumption there
 is a g∈ G which switches the two components of that singular fiber. Clearly for this g,
 g· E≠ E. Now
 apply Lemma  <ref> to E and E^'=g· E, we see that m≤ (N-2)/2
 as N is even. Hence we reduced the lemma to showing m≠N-2/2.
 
 Suppose to the contrary that m=(N-2)/2, and let E,E^' be a pair of sections 
 whose self-intersection equals -m. Then by Lemma <ref>, we see that E,E^' are disjoint and r=1. Let F be the singular fiber where E,E^' intersect the
 same (-1)-sphere. Then again by the minimality assumption there is a h∈ G which 
 switches the two (-1)-spheres in F. It follows easily that h· E, E,
 and E^' are distinct. Let r,r^' be the number of singular fibers where 
 h· E, E and h· E, E^' intersect the same (-1)-sphere, respectively. 
 Then it follows easily that r+r^'=N-2. On the other hand, Lemma <ref> implies that r=r^'=1,
 contradicting the fact that N≥ 6. Hence the lemma.
 
  
 




 §.§ Reduction of exceptional classes
 




Momentarily we let ω be any symplectic structure on X=^2# N^2, 
and denote 
 
    _ω={e∈ H^2(X)| e^2=-1, K_ω· e= -1, ω(e)>0}

 which may depend on ω. The following fact is crucial in our considerations. 



(cf. <cit.>)
Assume N≥ 2. Then for any ω-compatible almost complex structure J on X, each class E∈_ω 
with minimal
area, i.e., ω(E)=min_e∈_ωω(e), is represented by an embedded
J-holomorphic sphere.


The key ingredient in the proof of Theorem <ref> is a reduction procedure which involves a
certain type of standard basis of H^2(X), called a reduced basis. We shall begin by
a brief digression and refer the reader to <cit.> or independently <cit.> for more details.



Recall that a reduced basis is a basis H, E_1,⋯, E_N of H^2(X) with standard
intersection matrix, where E_i∈_ω, such that ω(E_N)=min_e∈_ωω(e), and for any i<N, E_i satisfies the following inductive condition: let 
_i={e∈_ω| e· E_j=0   ∀ i<j}, then ω(E_i)=min_e∈_iω(e). Furthermore, the canonical class K_ω=-3H+E_1⋯+E_N. Reduced basis
always exists. 

If N=2, then _ω={E_1,E_2, H-E_1-E_2}. For N≥ 3, a reduction procedure can be
introduced, which requires the following discussions.




1. Introduce H_ijk=H-E_i-E_j-E_k for i<j<k and H_ij=H-E_i-E_j for i<j. Then
H_ij∈_j, which implies that ω(H_ijk)≥ 0. 




2. For any E∈_ω, write E=aH-∑_s b_s E_s. Then


  * a≥ 0;

  * if a>0, then b_s≥ 0 for all s;

  * if a=0, then E=E_l for some l;

  * assume a>0, and let b_i,b_j,b_k be the largest three coefficients (here we use the assumption N≥ 3), 
then b_i≤ a<b_i+b_j+b_k, which is equivalent to E· H_ijk<0 and E·(H-E_i)≥0.





3. The classes H_ijk are represented by embedded (-2)-spheres, hence for each H_ijk
there is a diffeomorphism of X inducing an automorphism R(H_ijk) on H^2(X):

    R(H_ijk)α =α + (α· H_ijk)· H_ijk,   ∀α∈ H^2(X).

Moreover, each R(H_ijk) has the following properties: (1) R(H_ijk)K_ω=K_ω,
(2) R(H_ijk)E∈_ω,  ∀ E∈_ω. 




With the preceding understood, consider any E∈_ω, where E=aH-∑_s b_s E_s 
with a>0. Let b_i,b_j,b_k be the largest three coefficients for some i<j<k. Then
it follows easily from the last bullet of item 2 that 


  * R(H_ijk)E=a^' H-∑_s b_s^' E_s for some a^'<a, and 

  * ω (R(H_ijk)E)≤ω(E) with “=" iff ω(H_ijk)=0.

Set E^':= R(H_ijk)E. We say that E is reduced to E^' by H_ijk. 

The operations R(H_ijk) has the following properties of our interests:


   
  * (finite termination) By <cit.>, one may find a sequence of finitely many H_ijk for any E∈_ω, such that after performing R(H_ijk), E is reduced to E_l for some l.
   
  * (monotonicity) The symplectic area is monotonically decreasing during the above reduction procedure.
 
  

Therefore, after the reduction procedure, ω(E)≥ω (E_l), with “=" iff 
ω(H_ijk)=0 for all the
H_ijk's involved. In particular, when E has the minimal area in _ω, i.e., 
ω(E)=ω(E_N), then we have


    ω(E_l)=ω(E_l+1)=⋯ =ω(E_N),
    ω(H_ijk)=0  H_ijk's



We first rule out the case of N=2.


There are no minimal symplectic rational G-surface with N=2. 



Suppose ((X, ω), G) is a minimal symplectic rational G-surface with  N=2. 
Let {H,E_1,E_2} be a reduced basis. Then
_ω={E_1,E_2,H-E_1-E_2}. 

Fix a G-invariant J, and let C be the J-holomorphic (-1)-sphere representing E_2
(cf. Lemma <ref>). We set Λ:=∪_g∈ G g· C. Then Λ is a union of
finitely many J-holomorphic (-1)-spheres, containing at least two distinct (-1)-spheres 
intersecting each other because of the minimality assumption. Since 
_={E_1,E_2,H-E_1-E_2}, 
there are only two possibilities: (1) Λ is a union of three (-1)-spheres, 
representing the classes
E_1,E_2 and H-E_1-E_2, and (2) Λ is a union of two (-1)-spheres representing
the classes E_2 and H-E_1-E_2. In either case, H-E_1-E_2 is represented by
a (-1)-sphere. Since H-E_1-E_2 is the only characteristic element in _, it
must be fixed by the G-action, which contradicts the minimality of the symplectic 
G-manifold (X,ω). Hence   there are no minimal symplectic rational G-surface with N=2. 





 §.§ Reduced basis for symplectic rational G-surfaces
 







Due to Lemma <ref>, in what follows we assume that (X,ω) is a minimal symplectic rational G-surface, 
where X=^2# N^2 with N≥ 3. 


Suppose N≥ 3. Then one of the following must be true.


  (i) (X,ω) is monotone. 

  (ii) The reduced basis of X satisfies ω(E_1)>ω(E_2)=⋯ =ω(E_N), (H-E_1)=2(E_j). 
Moreover, if E∈_ω has
minimal area, then either E=E_j or E=H_1j=H-E_1-E_j  for some j>1.




Assume (X,ω) is not monotone. We shall first prove a slightly weaker statement that is independent of the G-action.
5mm

Claim: If E∈_ω has 
minimal area
in _ω and E≠ E_s for any s, then E=H-E_1-E_j for some j>1, and furthermore,
if such an E exists and E=H-E_1-E_j for some j>2, then we must have

    ω(E_2)=⋯ =ω(E_N).


 
Let E∈_ω be such a class, i.e., E has minimal area in _ω and 
E≠ E_s for any s,
We reduce E to E_l for some l by a sequence of H_ijk's.
Since E has minimal area in _ω, it follows that ω(E_l)=⋯=ω(E_N).
Note that l>1.  Otherwise, (E_i)=(E_j) and (H_ijk)=0 for some i,j,k from property (3) in the reduction process in Section <ref>.  This violates that (X,ω) is not monotone. 

Suppose the reduction from E to E_l takes n steps, let E^'∈ be the class 
obtained at the (n-1)^th step.
Then E^'=aH -∑_s b_s E_s, where a>0 and b_s≥ 0. 
The equation E_l=E^'+ (E^'· H_ijk)· H_ijk reads 

    E_l=(2a-b_i-b_j-b_k)H -∑_s=i,j,k(b_s+a-b_i-b_j-b_k)E_s-∑_s≠ i,j,k b_sE_s,

which implies that l must be one of i,j or k, and without loss
of generality assuming that l=k, then 

    2a=b_i+b_j+b_k,  a=b_j+b_k=b_i+b_k,  a-b_i-b_j=-1.

It follows easily that E^'=H_ij=H-E_i-E_j. Here we assume  i<j, but do not require 
any condition on k=l>1. Note that E^' has minimal area in _ω, which implies that
ω(H)=ω(E_i)+ω(E_j)+ω(E_l) from (<ref>). 

Next we prove that i=1. Suppose to the contrary that i>1. Then
ω(H_1ij)=ω(H)-ω(E_1)-ω(E_i)-ω(E_j)≥ 0. On the 
other hand, ω(H)=ω(E_i)+ω(E_j)+ω(E_l) and ω(E_1)≥ω(E_l), which implies that ω(E_1)=⋯ =ω(E_N)=1/3(H). This is a contradiction
because we assume (X,ω) is not monotone. Hence i=1 and E^'=H-E_1-E_j.

We claim that E=E^'=H-E_1-E_j. Suppose this is not true. Then there must be a class 
Ẽ which is reduced to E^' by some H_vrt for v<r<t. We assert v>1. To see this,
write Ẽ=aH -∑_s b_s E_s, where a>0 and b_s≥ 0. Then similarly we have

    H-E_1-E_j=(2a-b_v-b_r-b_t)H -∑_s=v,r,t(b_s+a-b_v-b_r-b_t)E_s-∑_s≠ v,r,t b_sE_s.

If v=1, then 2a-b_v-b_r-b_t=1 and a-b_r-b_t=1, implying a=b_v. Now for s=r,t,
the coefficient of E_s on the right hand side is b_t, b_r respectively, which are 
non-negative. It follows that b_r=b_t=0 from the property 2 in Section <ref>, which contradicts the fact that a<b_v+b_r+b_t.
Hence v>1. 






We now get a contradiction as follows. Note that ω(H_vrt)=0 and
ω(H_1vr)≥ 0 implies that 

    ω(E_1)=⋯=ω(E_t).

Then with ω(H_vrt)=0 again, we have ω(H)=3ω(E_1). On the other
hand, ω(E_l)=ω(E^')=ω(H)-ω(E_1)-ω(E_j), from which it follows that

    ω(E_1)=⋯=ω(E_l)=⋯=ω(E_N).

This contradicts the assumption that (X,ω) is not monotone. Hence E=H-E_1-E_j
is proved. Finally, if j>2, then ω(H_12j)≥ 0 and ω(H)-ω(E_1)-ω(E_j)=ω(E_l) implies that ω(E_2)=ω(E_l), hence 
ω(E_2)=⋯=ω(E_N). This concludes the claim.




To obtain Lemma <ref>, it remains to show that if (X,ω) is not monotone, then there exists some j>2, such that E=H-E_1-E_j
attains the minimal area within ℰ_. To this end, we fix a G-invariant ω-compatible J. 
Then by Lemma <ref>, there exists a J-holomorphic (-1)-sphere C representing the class 
E_N. Note that for any g∈ G, g· C∈_ω. Now by the assumption that 
(X,ω) is minimal, there must be a g∈ G such that g· C≠ C and g· C intersects with C. 
The class g· C has minimal area in _ω and g· C≠ E_s for any s. 
Let g· C=H-E_1-E_j. Then j=N≥ 3 must be true because g· C intersects 
with C. This finishes the proof of the lemma.





 §.§ Proof of Theorem <ref>
 









In what follows, we still assume that N≥ 3. We will discuss according to the following three possibilities:




i) (X,ω) is not monotone: By Lemma <ref>, there is a reduced basis 
{H, E_1,⋯,E_N} such that 

    ω(E_1)>ω(E_2)=⋯=ω(E_N),

and moreover, if E∈_ω has
minimal area, then either E=E_j or E=H_1j=H-E_1-E_j  for some j>1.

Let J be any G-invariant ω-compatible almost complex structure. For each fixed j>1, E_j has minimal area so
by Lemma <ref>, so there is an embedded J-holomorphic sphere C representing E_j. Since 
(X,ω) is minimal,
there must be a g∈ G such that g· C≠ C and g· C∩ C≠∅. It follows easily that g· C must
be the J-holomorphic (-1)-sphere representing the class H-E_1-E_j, since it also has the minimal area. Furthermore,
g· C and C intersect transversely and positively at a single point. Standard gluing construction in J-holomorphic curve
theory yields a J-holomorphic sphere Ĉ, carrying the class C+g· C=H-E_1. 
It follows that Ĉ has self-intersection
0, and by the adjunction formula it must be embedded. By a standard Gromov-Witten index computation, the moduli space of J-spheres in class [Ĉ] gives
rise to a fibration (which contains singular fibers) structure on X, where each fiber is homologous to Ĉ. Since X is a rational surface,
the base of the fibration must be ^2. We denote the fibration by π_j: X→^2. 

Next we show that π_j is G-invariant. To see this, note that for any h∈ G, h· (C∪ g· C) must be a pair of
J-holomorphic (-1)-spheres representing E_k and H-E_1-E_k for some k 
(see Lemma <ref>). It follows that the class
of Ĉ, which is H-E_1, is invariant under the G-action. This implies that the fibration 
π_j is G-invariant. 

Finally, we note that the fibration π_j is independent of j, because the fiber class, which is 
H-E_1, is independent of j.
We will denote the fibration by π: X→^2. 
Note that the same argument shows also that π contains at least N-1 singular fibers, 
consisting of a pair of (-1)-spheres whose classes are E_j, H-E_1-E_j for j=2,⋯, N. 
There are no other singular fibers by an Euler number count.  Note that Lemma <ref> asserts rank H^2(X)^G=2 in this case.




ii) (X,ω) is monotone and H^2(X)^G has rank 1: In this case, note that 
K_ω∈ H^2(X)^G and is a primitive class, hence H^2(X)^G is spanned by 
K_ω. The constraint N≤ 8 is an easy consequence of (X,ω) being monotone.  




iii) (X,ω) is monotone and rank  H^2(X)^G>1: Note that the de Rham class 
[ω]=λ K_ω∈ H^2(X;)^G for some λ∈. 
Since rank  H^2(X)^G>1, we may pick a G-invariant closed 2-form η such that [η] lies in a different direction
in H^2(X;)^G. Let ω^':=ω+ϵη for some very small ϵ. Then ω^' is a 
G-invariant symplectic structure such that (X,ω^') is not monotone. 


Claim:  (X,ω^') is minimal as a symplectic G-manifold for sufficiently small ϵ. 


Suppose there is a disjoint union of ω^'-symplectic (-1)-spheres 
{C_i}, such that for any g∈ G, g· C_i=C_j for some j. Note that for sufficiently
small ϵ, we have _=_' because K_ω=K_ω^'.
Let e_i be the class of C_i.






It follows easily that for each i, e_i∈_. Now pick a G-invariant J compatible with .
Since (X,ω) is monotone, each e_i is represented by an embedded J-holomorphic (-1)-sphere Ĉ_i.
Notice that {e_i} has the following properties:
 e_i· e_j=0 for i≠ j, and for any g∈ G, g· e_i=e_j for some j. It follows that
{Ĉ_i} is a disjoint union of ω-symplectic (-1)-spheres which is invariant under the G-action. This
contradicts the minimality assumption on (X,ω), hence the claim. 

We apply the argument for case i) to (X,ω^'). 
Consequently there is a reduced basis {H, E_1,⋯,E_N}
(w.r.t. ω^'), and a G-invariant fibration π^':X→^2, whose fiber class is H-E_1 and
the singular fibers are pairs of (-1)-spheres representing E_j,H-E_1-E_j for j>1. Note that by taking ϵ small,
we have E_i∈_'=_.

Finally, observe the following
crucial property (∑_g∈ G g· E_j)^2=0 for any j>1 because E_k and H-E_1-E_k must appear in pairs in the sum.

Now let J be any G-invariant ω-compatible almost complex structure.  
For each fixed j>1, 
since (X,ω) is monotone and E_j∈_, E_j is represented 
by an embedded J-holomorphic (-1)-sphere C. Let Λ:=∪_g∈ G g· C. 


Then the set Λ is a union of finitely many distinct J-holomorphic
(-1)-spheres. Since (∑_g∈ G g· E_j)^2=0 for any j>1, it follows easily that 
Λ^2=0.

Let {Λ_i} be the set of connected components of Λ. Since G acts on the connected components of Λ transitively, 
it follows that for any i≠ k, Λ_i^2=Λ_k^2. Clearly Λ^2=∑_i Λ_i^2, which implies
that Λ_i^2=0 for any i. Now Λ_i^2=0, together with 
the fact that Λ_i is a union of finitely many distinct J-holomorphic (-1)-spheres and Λ_i is connected,
implies that Λ_i consists of two (-1)-spheres intersecting transversely at a single point. 
We claim that the pair of (-1)-spheres in each Λ_i have classes E_k,H-E_1-E_k for some k>1. 
This is because for each g∈ G, g· E_j is either E_k or H-E_1-E_k for some k>1, and for each k>1,
there is a g∈ G such that g· E_k=H-E_1-E_k. 
By the same argument as in case i), we obtain a G-invariant fibration on X as desired, independent of the choice of j. 

The statement that N≥ 5 was proved in Lemma <ref>. The statement that N 6 follows from Lemma 
<ref> below. 
This completes the proof of Theorem <ref>.









Let π:X→^2 be a minimal symplectic G-conic bundle where 
X=^2#6^2. Then for any G-invariant, compatible almost complex
structure J, there is a G-invariant J-holomorphic (-1)-sphere. 



Let H,E_1,⋯,E_6 be a basis of H^2(X) with standard interaction matrix such that
H-E_1 is the fiber class and E_2,⋯, E_6 are (-1)-spheres contained in singular
fibers. Note that the canonical class K_X=-3H+E_1+⋯ +E_6.

With this understood, note that C=-K_X-(H-E_1)=2H-∑_6≥ i≥2E_i∈ H^2(X)^G,
and C is an exceptional class. Hence for any given G-invariant, compatible almost complex
structure J, C has a J-holomorphic representative, which admits a decomposition of irreducible components

    C=F̂+Ê,

where F̂ is the sum of components contained in the fibers of the G-conic bundle (the vertical class), 
and Ê is the sum of other components (the horizontal class).  

By Lemma <ref>, E_1 must have a J-holomorphic representative: otherwise, it has a stable 
curve representative where one of the component has to be a section since E_1· F=1.  
Such a section has self-intersection less than -1, a contradiction.

To continue with our proof, we first show that Ê does not contain E_1-components.  Suppose the multiplicity of the E_1-component is k, since all irreducible components pairs with H-E_1 non-negatively, (C-kE_1)· (H-E_1)≥0, hence k≤2.  

Furthermore, if the multiplicity is 2 (meaning either there is a doubly covered component or two components), Ê=2E_1.  Otherwise, Ê· (H-E_1)≥3>C·(H-E_1), violating the positivity of intersection with H-E_1.

For the multiplicity 2 case, F̂=2H-2E_1-∑_6≥ i≥2 E_i, while all possible irreducible components are of the form H-E_1-E_i, E_i or H-E_1 for 6≥ i≥2.  A simple check shows this cannot be consistent with (<ref>).

For the multiplicity 1 case, Ê=E_1+E'. E' has E'·(H-E_1)=1 hence a section.  Therefore, E'≥-1 from Lemma <ref>.  Since E' E_1, its coefficient of H under the reduced basis must be positive.  One may again easily check F̂ cannot be represented as a combination of classes of forms H-E_1-E_i, E_i or H-E_1.

Now since the equivariant J-holomorphic representative of C does not contain E_1-components, it is disjoint from the E_1-section.  Therefore, F̂· E_1=0, which implies the sum of vertical components F̂=∑ m_sE_s for some s 1.  However, there is always an element g∈ G which sends E_s to H-E_1-E_s for any s 1.  This forces F̂=0, and C=Ê.

At last, again note that Ê has at most 2 components by positivity of intersection with H-E_1.  We assume Ê has two components, which are both sections S_1 and S_2.  


    C^2=(S_1+S_2)^2=-1=S_1^2+S_2^2+2S_1S_2≥ -2+2S_1· S_2

from Lemma <ref>.  Therefore, 2S_1· S_2≤ 1, which implies S_1· S_2=0.  This implies these are disjoint sections S_1^2=-1 and S_2^2=0.  A simple check on class C shows there are no such decomposition of section classes (recall that [S_1] E_1).

Summarizing, we have showed that the J-representative of C is indeed irreducible, 
which is a G-invariant J-holomorphic (-1)-sphere. Hence the lemma.



To compare with known results in algebraic geometry, it seems worth to record the following easy consequence. 


Let (X,ω) be a minimal symplectic rational G-surface where 
X=^2# 6^2. Then the invariant lattice H^2(X)^G must be of rank
1. In particular, a minimal complex rational G-surface X which is ^2 blown up at 6 points
must be Del Pezzo with Pic(X)^G=.








Under the above assumptions, Lemma <ref> implies (X,) is not a G-conic bundle, hence rank(H^2(X)^G)=1 by Theorem <ref>.

For the second statement, notice that a conic bundle on a minimal complex rational G-surface defines a minimal symplectic G-conic
bundle with respect to any G-invariant Kähler form.
Theorem 3.8 of <cit.> asserts then X must be a del Pezzo surface if rankH^2(X)^G=1.












§ THE STRUCTURE OF G



 §.§ Proof of Theorem <ref>, 4≤ N≤ 8
 



We start with the following observation.


Suppose (X,ω) is monotone. If N≥ 4, then the representation of G on H^2(X) is faithful.
 
 
 
 Fix a G-invariant J.
 Let g∈ G be any element acting trivially on H^2(X). Then g fixes every element 
 E∈_, which implies that
 all the J-holomorphic (-1)-spheres are invariant under g. Now let C_1 be the J-holomorphic 
 (-1)-sphere representing E_1, and for each 1<j≤ N, let C_j be the J-holomorphic (-1)-sphere representing
 H-E_1-E_j. Then it is clear that C_1 intersects each C_j, j>1, transversely at one point and the C_j's are mutually
 disjoint. It follows that the cardinality of the set C_1∩ (∪_j C_j) is N-1. On the other hand, each point in
 C_1∩ (∪_j C_j) is fixed under g, hence the action of g on C_1 contains at least N-1 fixed points. When
 N≥ 4, it follows that C_1 must be fixed by g. A similar argument shows that every J-holomorphic (-1)-sphere
 is fixed by g. Since there are J-holomorphic (-1)-spheres intersecting transversely at a point, the action of g on the
 tangent space at the intersection point must be trivial, which shows that g must be trivial. Hence the lemma.
 
 


Assume (X,ω) is a symplectic rational G-surface with rank H^2(X)^G=1.

Since K_ω∈ H^2(X) is fixed under the action of G, there is an induced representation of
G on the orthogonal complement R_N, which is faithful by Lemma <ref>. This gives rise to a monomorphism 
ρ: G→ W_N. On the other hand, H^2(X)^G is spanned by K_ω, so that R_N^G={0}. This implies that

    1/|G|∑_g∈ G trace{ρ(g):R_N→ R_N}=rank R_N^G=0.

The proof for Case 1 is completed.






 §.§ Proof of Theorem <ref>, N=3
 



Our main objective in this case is to show that G contains an index 2
 subgroup which is isomorphic to an imprimitive finite subgroup of PGL(3).
 
 We first describe the list of subgroups of PGL(3) involved. For simplicity, we will adapt the
 following convention from <cit.>: we denote an element  T∈ PGL(3) by the image of
 [z_0,z_1,z_2]∈^2 under T. 
 
 Here is the list of imprimitive finite subgroups of PGL(3) up to conjugacy (see Theorem 4.7
 in <cit.>), where μ_r=exp(2π i/r) is the r-th root of unity.
 
 
 
  * G_n, generated by the following elements of PGL(3):
 
    [μ_n z_0,z_1,z_2],   [z_0,μ_nz_1,z_2],   [z_2,z_0,z_1]

 The group G_n is isomorphic to a semi-direct product of (_n)^2 and _3.
 
  * G̃_n, generated by the following elements of PGL(3):
 
    [μ_n z_0,z_1,z_2],   [z_0,μ_n z_1,z_2],   [z_0,z_2, z_1],   [z_2,z_0,z_1]

The group G̃_n is isomorphic to a semi-direct product of (_n)^2 and S_3.

  * G_n,k,s, where k>1, k|n, and s^2-s+1=0 k. It is generated by 
the following elements of PGL(3):
 
    [μ_n/k z_0,z_1,z_2],   [μ_n^sz_0,μ_nz_1,z_2],   [z_2,z_0,z_1]

 The group G_n,k,s is isomorphic to a semi-direct product of _n×_n/k and _3.
 
  * G̃_n,3,2, generated by the following elements of PGL(3):
 
    [μ_n/3 z_0,z_1,z_2],   [μ_n^2 z_0,μ_n z_1,z_2],   [z_0,z_2,z_1],   [z_1,z_0,z_2]

The group G̃_n,3,2 is isomorphic to a semi-direct product of _n×_n/3 
and S_3.
 
 
 




  §.§.§ Preliminaries: factorization of G by exceptional spheres

Let {H,E_1,E_2,E_3} be a reduced basis. Then

    _={E_1,E_2,E_3, H-E_1-E_2,H-E_1-E_3,H-E_2-E_3}.
 
Since (X,ω) is monotone, the classes in _ have the same area, and consequently, each class in _ is represented 
by a J-holomorphic (-1)-sphere for any fixed J, which we assume to be G-invariant. 
Let Λ be the union of 
these six (-1)-spheres. The intersection pattern of these curves can be described by a hexagon, where each edge represents 
a (-1)-sphere and each vertex represents an intersection point
(See Figure 1). For simplicity, for each E∈_ we shall use the same notation to denote the
corresponding (-1)-sphere.



Obviously there is an induced G-action on Λ.


The action of G on the components of Λ is transitive, and consequently, there is a short exact sequence 

    1→Γ→ G→ K→ 1,



where Γ is isomorphic to a subgroup of ^1×^1, and K is either D_12,
the full automorphism group of the hexagon, or the cyclic subgroup of order 6.



Let C be any of the (-1)-spheres. Then the class of the union ∪_g∈ G g· C must lie in
H^2(X)^G. Since H^2(X)^G=Span (K_ω), the class of ∪_g∈ G g· C must be a
multiple of K_ω. On the other hand, the class of Λ equals -K_, from which it follows 
easily that Λ=∪_g∈ G g· C. This proves that the action of G on components 
of Λ is transitive.

The action of G on Λ gives rise to a short exact sequence 

    1→Γ→ G→ K→ 1,

 where Γ is the normal subgroup of G consisting of elements which leave each (-1)-sphere
 invariant. The quotient group K=G/Γ has an effective, transitive action on the hexagon Λ,
 which must be either the automorphism group D_12 of the hexagon, or the cyclic subgroup of order 6.

 To see that Γ is a subgroup of ^1×^1, we look at
 the action of Γ on the tangent space of any intersection point of two adjacent (-1)-spheres in Λ.
 The action preserves a pair of complex lines intersecting transversely, giving a natural isomorphism of Γ to
 a subgroup of ^1×^1.
 
 
 
In fact, we may identify K geometrically as follows. There is a natural isomorphism D_12=_2× S_3, where the latter is the Weyl group W_N 
 of the corresponding root lattice R_N, which is generated by H-E_1-E_2-E_3, E_1-E_2 and E_2-E_3.
 In this sense, _2=⟨ s_1⟩ where s_1 is the rotation of 180 degrees of the
 hexagon, and S_3=⟨ s_2,s_3⟩ where s_2, s_3 are the reflections of the 
 hexagon which switches E_1 and E_2, E_2 and E_3 respectively. Note that s_2s_3,
 which is s_3 followed by s_2, is a counter-clockwise rotation of 120 degrees of the hexagon. 
 It follows that the short exact sequence 1→Γ→ G→ K→ 1
 is the same as the one obtained from the induced action of G on the root lattice R_N, 
 with Γ being the subgroup of G acting trivially on H^2(X). Under this identification, one of the following is true by Lemma <ref>

  
  * K is 
 generated by s_1, s_2 and s_3 if it is the cyclic subgroup of order 12.
  
  * K is 
 generated by s_1 and s_2s_3 if it is the cyclic subgroup of order 6.


 
 In order to understand the structure of G, we begin by getting more information about
 the subgroup Γ. To this end, we fix a monomorphism 
 Γ→^1×^1 induced from the action of Γ on the tangent
 space of the intersection point of E_1 and H-E_1-E_3, where the first ^1-factor
 is from the action on E_1. We summarize the main objects in consideration as follows for readers' convenience:


  
  * Γ G is the subgroup with trivial homological action,
  
  * K=G/Γ,   
  
  * ρ_i: Γ→^1, i=1,2, are the projection to the two ^1-factors,
  
  * Γ_i=ρ_i,
  
  * Γ_i^'=image ρ_i⊂^1. 
  
  * Γ̃_i is a subgroup of Γ such that 
 ρ_i: Γ̃_i→Γ_i^' is isomorphic. 



  
  * Γ_i, Γ_i' are cyclic,
  
  * ord (Γ_1)=ord (Γ_2) and ord (Γ_1')=ord (Γ_2')
  
  * ord (Γ_i)|ord (Γ_i')


We will hence denote n:=ord(Γ_i') and k:=ord(Γ_i')/ord(Γ_i)



 
 
 

It is clear that Γ_i^' is cyclic. Γ_i is also cyclic because both 
ρ_2|_Γ_1 and ρ_1|_Γ_2 are injective. 
 Note that this also shows that the order
of Γ_1^' (resp. Γ_2^') is divisible by the order of 
Γ_2 (resp. Γ_1). 

Finally, Γ_1 and Γ_2 have the same order. This is because if we
let g∈ G be an element whose action on the hexagon is a counter-clockwise 
rotation of 60 degrees, then gΓ_2g^-1=Γ_1. Consequently, the order of 
Γ_i^'=Γ/Γ_i is independent of i.




  
 
 
      The subgroup Γ̃_i<Γ exists.  In particular, since Γ is Abelian, 
      Γ≅Γ_i×Γ̃_i for i=1,2.
  
    
 
 Let h∈Γ be an element 
 such that ρ_1(h) is a generator of Γ_1^'. Since ρ_1(h)^n=1, 
 we have h^n∈Γ_1. We claim h^n=1. This is because ρ_2|_Γ_1 
 is injective, so that if h^n≠ 1 in Γ_1, then ρ_2(h)^n=ρ_2(h^n)≠ 1 in 
 Γ_2^'. But this contradicts the fact that the order of Γ_2^' 
 equals n. Hence h^n=1. With this understood, we simply take 
 Γ̃_1 to be the subgroup generated by h.
 
 
 


  §.§.§ Rotation numbers
 Next we recall some basic facts about rotation numbers. Let h∈Γ be any element and 
 E∈ be any (-1)-sphere which is invariant under h. Then either h fixes E or 
 h acts on E
 nontrivially. In the latter case, h fixes two points on E.  The following fact is a straightforward local computation:
5mm
 Fact: if we let (a,b)
  be the rotation numbers of h at one of the fixed point, where a is the tangential weight and
  b is the weight in the normal direction, then the rotation numbers at the other fixed point are
  (-a, b+a), with the second number being the weight in the normal direction. 

5mm

 Explicitly, the tangential weight being a means that the action of h is given by multiplication of 
 exp(2aπ√(-1)/ord(h)) and similarly for the normal weight.  Here we orient 
 the (-1)-sphere by the almost complex structure J and orient the normal direction accordingly,
 so there is no sign ambiguity on a, b. Note that even when E is fixed by h, this 
 continues to make sense, with the understanding that a=0 in this case. 
 
 

 With the preceding understood, we order the exceptional curves and their intersections according 
 to the counter-clockwise orientation of the hexagon in Figure <ref>, and ask E_1 to be the first exceptional curve.

 For example,  we say H-E_1-E_3 is before E_1 and H-E_1-E_2 is after E_1, 
 and the intersection point of H-E_1-E_3 and E_1 is the first fixed point on E_1, etc.




 Now for any h∈Γ, we denote by (a,b) the rotation
 numbers at the intersection point of H-E_1-E_3 and E_1, with a being the weight in the
 direction tangent to E_1. According to the orientation of the hexagon, this is the first fixed
 point of h on E_1. With this understood, the rotation numbers at the  fixed points of h
 on (-1)-spheres are given below according to the orientation of the hexagon,
 
    (a+b,-a), (b,-a-b), (-a,-b), (-a-b,a), (-b,a+b).

 Finally, we remark that h is completely determined by the rotation numbers at the six vertices of the hexagon.
 
 Throughout the rest of the proof of Theorem <ref>, 



g∈ G will denote those elements which act on the hexagon by a 

 counter-clockwise rotation of 60 degrees, 
  



 and we shall investigate the action of g on 
 Γ=Γ_1×Γ̃_1 given by conjugation, i.e., h↦ ghg^-1, 
 ∀ h∈Γ.

 Here is the first corollary of the analysis on rotation numbers: note that the action of g^3, 
 which is a rotation of
 180 degrees, sends every pair of rotation numbers to its negative, i.e., (a,b)↦ (-a,-b),
 (a+b,-a)↦ (-a-b,a) and (b,-a-b)↦ (-b,a+b). This implies that 
 
    g^3hg^-3=h^-1,   ∀ h∈Γ.

 Since g^3 is sent to s_1∈ K under the homomorphism G→ K in (<ref>), 
 we obtain the following lemma:


 (1) For any element of G which is sent to s_1∈ K under G→ K, its
 action on Γ by conjugation sends h∈Γ to h^-1. (2) g^6=1.
 
 
 
 It remains to show that g^6=1.
 First, note that g^6∈Γ. Secondly, the action of g^3 on g^6 by conjugation 
 is trivial, but the action of g^3 on Γ sends 
 h to h^-1 as we just showed, from which we see that either g^6=1 or g^6 
 is an involution.  We rule out the latter by showing that the action of g on any 
 involution of Γ is nontrivial, and G is a semi-direct product of Γ 
 and K follows. 
 
Let τ∈Γ be any involution. Without loss of generality, we assume τ acts 
nontrivially on E_1. Then in the rotation numbers of the two fixed points, (a,b) and 
(-a,b+a), a=1 must be true because τ has order 2, and furthermore, either b=0 or b=1. 
If b=0, the rotation numbers for the action of τ at the six vertices are 
 
    (1,0), (1,-1), (0,-1), (-1,0), (-1,1), (0,1).

 If b=1, the rotation numbers for the action of τ at the six vertices are 
 
    (1,1), (0,-1), (1,0), (-1,-1), (0,1), (-1,0).

In the former case, the rotation numbers for the action of gτ g^-1 are

    (0,1), (1,0), (1,-1), (0,-1), (-1,0), (-1,1),

which shows gτ g^-1≠τ. In the latter case, the rotation numbers for the action 
of gτ g^-1 are

    (-1,0), (1,1), (0,-1), (1,0), (-1,-1), (0,1),

which also shows gτ g^-1≠τ. This finishes the proof of the lemma.


  
 
With these preparation, we may now continue our proof according to the alternative by 
Lemma <ref> 

 
   
  * Case A: K=_6, and
   
  * Case B: K=D_12.
 

 
 

  §.§.§ Case A: K=_6
 We begin with the following observation which follows
 immediately from the fact that g^6=1.
 
 
G is a semi-direct product of Γ and K.
 
 
 Let ⟨Γ,g^2⟩ denote the index 2 subgroup of G 
generated by Γ and g^2.


If K=_6, then ⟨Γ,g^2⟩ is isomorphic to G_n when k=1, and is
isomorphic to G_n,k,s when k>1, where n=|Γ_1^'|=|Γ̃_1| 
and n/k=|Γ_1|.



 Let h_1∈Γ_1 be the generator whose rotation numbers at the six vertices of 
 the hexagon are
 
    (0,1), (1,0), (1,-1), (0,-1), (-1,0), (-1,1),

and let h̃_1∈Γ̃_1 be the generator whose rotation numbers 
at the six vertices of the hexagon are
 
    (1,b), (1+b,-1), (b,-1-b), (-1,-b), (-1-b,1), (-b,1+b).

This is can be shown by examining the weight on the first intersection (the one between 
H-E_1-E_3 and E_1), because on the base it should project to that of Γ_1'. Then g^2h_1g^-2 has rotation numbers

    (-1,0), (-1,1), (0,1), (1,0), (1,-1), (0,-1),

and g^2h̃_1 g^-2 has rotation numbers

    (-1-b,1), (-b,1+b), (1,b), (1+b,-1), (b,-1-b), (-1,-b).

Let  g^2h_1g^-2=h̃_1^kl h_1^u for some l and u (note that k· ord(Γ_i)=ord(Γ'_i)). 
Then comparing the rotation
numbers we have 

    (-1,0)= (l,bl)+(0,u),

which implies that l=-1 and u=b. Similarly, let g^2h̃_1 g^-2=h̃_1^m h_1^v
for some m and v, we have 

    (-1-b,1)=(m,mb)+(0,kv),

which implies that b^2+b+1=kv n and m=-1-b. Putting together, we have

    g^2h_1g^-2=h̃_1^-k h_1^b,     g^2h̃_1 g^-2=h̃_1^-b-1 h_1^v.

We will need a different presentation of ⟨Γ,g^2⟩. To this end, 
we let t_1=h_1, t_2=h̃_1^-1, and moreover, we set s=-b. Then we have

    g^2t_1g^-2=t_2^k t_1^-s,     g^2t_2 g^-2=t_2^s-1 t_1^-v,

 where s^2-s+1=kv n. With this presentation, one can identify the subgroup
 ⟨Γ,g^2⟩ with an imprimitive finite subgroup of PGL(3).
 More precisely, when k=1, i.e., Γ̃_1 and Γ_1 have the same 
 order, we can actually take Γ̃_1 to be Γ_2, which corresponds to 
 b=0. Then s=-b=0, and with k=1, we have v=1. In this case. ⟨Γ,g^2⟩ 
 is isomorphic to G_n by identifying t_1= [μ_n z_0,z_1,z_2], 
 t_2=[z_0,μ_n z_1,z_2], and g^2=[z_2,z_0,z_1].
 When k>1, ⟨Γ,g^2⟩ is isomorphic to the group
 G_n,k,s, by identifying t_1= [μ_n/k z_0,z_1,z_2], 
 t_2=[μ_n^s z_0,μ_nz_1,z_2], and g^2=[z_2,z_0,z_1].
 This finishes off the proof of Proposition <ref>.
 
 
 
 

  §.§.§ Case B: K=D_12
 We will first show that G is a semi-direct product of Γ 
 and K. 
 
 
 There exists an involution in G which is sent to the reflection s_3 under G→ K.
 Moreover, 
 
 
  * for any h∈Γ and any such involution τ, hτ h^-1=ĥτ, where
 h,ĥ are related as follows: if the rotation numbers of h at the first fixed point on E_1
 (according to the orientation of the hexagon which is counter-clockwise) are (a,b), then the 
 rotation numbers of ĥ at the first fixed point on E_1 are (2a,-a).
 
  * for any two such involutions τ,τ^', τ^'=hτ for some h∈Γ
 whose rotation numbers at the first fixed point on E_1 are (2a,-a) for some a.
 
 
 
 We note that analogous statements hold for the reflections s_2 and s_2s_3s_2 in
 K.
 
 
 Suppose τ∈ G is sent to s_3 under G→ K, then τ leaves the 
 (-1)-spheres E_1 and H-E_2-E_3 invariant. Let h_1∈Γ_1 be the generator 
 with rotation numbers (0,1) at the first fixed point. By examining the rotation numbers,
 one can check easily that τ h_1τ^-1=h_1. On the other hand, it is easily seen that
 τ^2∈Γ fixes 4 points (two fixed points from τ and two from intersections with other exceptional curves) hence the whole (-1)-sphere E_1.  Therefore, τ^2∈Γ_1.

  The
 key observation is that τ^2=h_1^2b is an even power of h_1. To see this, note that 
 τ has two fixed points on E_1 and their rotation numbers are
 (a,b) and (-a,b+a) for some a,b. Furthermore, the order of τ must be even, say 2m.
Since τ^2 fixes the (-1)-sphere E_1, we must have 2a=2m, and the rotation numbers
of τ^2 at the two fixed-points are (0, 2b) and (0,2b). Comparing with the rotation
numbers with h_1, we see easily that τ^2=h_1^2b. With τ h_1τ^-1=h_1, we 
see easily that τ h_1^-b is an involution which is sent to s_3 under G→ K.

Now we consider any involution τ which is sent to s_3. For any h∈Γ, let 
x,y be the first and second fixed points on E_1, and suppose the rotation numbers of h
at x are (a,b). Then τ(x)=y and τ: T_x E_1
→ T_y E_1. It is easily seen that hτ h^-1: T_x E_1→ T_y E_1 equals


(-a) τ (-a)=-2a τ (here (-a) means multiplication by exp(2(-a)π√(-1)/ord(h))).
Similarly, hτ h^-1= (a+b)τ (-b)=aτ in the normal direction.
Hence hτ h^-1=ĥτ for some ĥ whose rotation numbers at y are
(-2a,a). It follows easily that the rotation numbers of ĥ at x are (2a,-a).

Finally, let τ,τ^' be any two involutions sent to s_3. We let 

    f=dτ:T_xX→ T_yX


    g=dτ':T_yX→ T_xX.

 Then 


    g∘ f=d(τ^'τ): T_x X→ T_x X

 
    g^-1∘ f^-1=d(ττ'): T_y X→ T_y X

 
   Note that
τ^'τ=h∈Γ. Since g^-1∘ f^-1=(f∘ (g∘ f)∘ f^-1)^-1,
we see easily that the rotation numbers of h at y are the negative of the rotation numbers
at x. If the rotation numbers at x are (c,d), then the rotation numbers at y are
(-c,c+d) (the second number in each pair stands for the weight in the normal direction).
This gives rise to the relation c+d=-d, so that the rotation numbers of h at x are (-2d,d).
Setting d=-a we proved that τ^'=hτ for some h∈Γ whose rotation 
numbers at the first fixed point on E_1 are (2a,-a) for some a, as we claimed.





The group G is a semi-direct product of Γ and K.



Recall that we fixed an element g∈ G which is sent to a counter-clockwise rotation of 60 degrees under G→ K.
We set τ_1=g^3, which is an involution by Lemma <ref>. We pick another involution τ_3∈ G which is sent to s_3∈ K
from Lemma <ref>. We shall first show that we can always arrange to have τ_1τ_3τ_1=τ_3.

By Lemma <ref>, τ_1τ_3τ_1=h^'τ_3 for some h^'∈Γ whose rotation numbers at the first fixed point on E_1
are (2a_3,-a_3) for some a_3. On the other hand, if we replace τ_3 by hτ_3h^-1=ĥτ_3, then from Lemma <ref>

    τ_1(ĥτ_3)τ_1=ĥ^-1τ_1τ_3τ_1=ĥ^-2h^' (ĥτ_3).

Hence if there exists an ĥ∈Γ such that ĥ^2=h^', which is equivalent to a_3 being even, then we can
replace τ_3 by hτ_3h^-1 for an h∈Γ to achieve the commutativity property. 

To show that a_3 is even, we pick an involution τ_2 which is sent to s_2∈ K (by an 
analog of Lemma <ref>) and consider τ_1τ_2τ_1. By the corresponding version of
Lemma <ref>, we see that τ_1τ_2τ_1=h̃τ_2 for some h̃∈Γ whose rotation numbers at the
second fixed point on E_1 are (2a_2,-a_2) for some a_2. It follows easily that the rotation numbers of h̃ at the
first fixed point on E_1 are (a_2,a_2). 

Now τ_2τ_3 is sent to a counter-clockwise rotation of 120 degrees under 
G→ K, so that there exists an
h∈Γ such that τ_2τ_3=hg^2. Now 

    τ_1(τ_2τ_3)τ_1=h̃τ_2h^'τ_3=h̃h^' kτ_2τ_3

for some k∈Γ whose rotation numbers can be determined as follows: since the rotation 
numbers of h^' at the second fixed point on E_1 are (a_3,-2a_3), by the analog of
Lemma <ref> the rotation numbers of k at the second fixed point on E_1 are (-2a_3,a_3).
It follows that the rotation numbers of k at the first fixed point are (-a_3,-a_3).
With this understood, note that the rotation numbers of h̃h^' k at the first fixed point on E_1 are (a_2+a_3,a_2-2a_3). On the other hand,

    τ_1(τ_2τ_3)τ_1=τ_1(hg^2)τ_1=h^-1τ_1g^2τ_1=h^-1 g^2=h^-2τ_2τ_3,



which implies that both a_2+a_3, a_2-2a_3 are even. It follows that a_3 is even, and hence there is an involution
τ_3 sent to s_3∈ K with the property that τ_1τ_3τ_1=τ_3.

We set τ_2:=gτ_3g^-1. Then τ_2 is an involution sent to s_2∈ K which naturally satisfies τ_1τ_2τ_1=τ_2.
We will show that τ_2,τ_3 satisfy the relation τ_2τ_3τ_2=τ_3τ_2τ_3. 
Note that with this relation, the subgroup generated by
τ_2,τ_3 is isomorphic to S_3. Together with the involution τ_1, we obtain a lifting of K=_2× S_3 in G, proving that G is
a semi-direct product of Γ and K. 

As we have seen earlier, τ_2τ_3=hg^2 for some h∈Γ, where h^2=1 from (<ref>) because both τ_2,τ_3 commute
with τ_1. The rotation numbers of h at the first fixed point on E_1 must be one of the following: (i) (0,1), (ii) (1,0),
(iii) (1,1). We claim that in case (i), we have τ_2τ_3τ_2=τ_3τ_2τ_3. To see this,

    τ_2τ_3τ_2=hg^2τ_2=hτ_3 g^2=τ_3hg^2=τ_3τ_2τ_3,

where we use the fact hτ_3=τ_3h because the rotation numbers of h are (0,1). 

It remains to rule out the cases (ii) and (iii). To this end, we compute

    gτ_2τ_3g^-1=g^2τ_3g^-2τ_2=(hτ_2τ_3)τ_3(hτ_2τ_3)^-1τ_2
    =hτ_2τ_3τ_2 hτ_2=hhkτ_2τ_3τ_2τ_2=kτ_2τ_3,

where the rotation numbers of k∈Γ can be determined as follows. Note that 
τ_2τ_3τ_2 is sent to the reflection s_2s_3s_2∈ K, so the rotation numbers of k
at the first fixed point on H-E_1-E_3 can be determined by an analog of Lemma <ref>. In case (ii), 
the rotation numbers of h at the first fixed point on H-E_1-E_3 are (0,1), so by an analog 
of Lemma <ref>, the rotation numbers of k at the first fixed point on H-E_1-E_3 are (0,0),
i.e., k is trivial in this case. In case (iii), the rotation numbers of h at the first fixed point on 
H-E_1-E_3 are (1,0), so by an analog of Lemma <ref>, the rotation numbers of k at the first fixed point on 
H-E_1-E_3 are (0,1). It follows that the rotation numbers of k at the first
fixed point on E_1 are (1,0) in this case.




On the other hand, g(hg^2)g^-1=ghg^-1g^2=h^' g^2 for some h^'∈Γ, 
where the rotation numbers of h^' at the first fixed point on E_1 are (0,1) in case (ii) 
and (1,0) in case (iii). Since in both cases, h^' h≠ k, so we reached a contradiction.
This ruled out the cases (ii) and (iii), and the proposition is proved.



Finally, we show that G contains an index 2 subgroup which is isomorphic to an imprimitive
subgroup of PGL(3). To this end, we fix a lifting K^' of K to G, and let g∈ K^'
be an element of order 6 and τ∈ K^' be the involution sent to s_3∈ K. We denote
by ⟨Γ,g^2,τ⟩ the subgroup generated by the elements in Γ,
g^2 and τ.


Suppose K=D_12. Then ⟨Γ,g^2,τ⟩ is isomorphic to the imprimitive
finite subgroup G̃_n of PGL(3) when k=1, and is isomorphic to G̃_n,3,2
when k>1, where n=|Γ_1^'|=|Γ̃_1| and n/k=|Γ_1|.



 Let h_1∈Γ_1 be the generator whose rotation numbers at the six vertices of 
 the hexagon are
 
    (0,1), (1,0), (1,-1), (0,-1), (-1,0), (-1,1),

and let h̃_1∈Γ̃_1 be the generator whose rotation numbers 
at the six vertices of the hexagon are
 
    (1,b), (1+b,-1), (b,-1-b), (-1,-b), (-1-b,1), (-b,1+b).

Then the rotation numbers of τh̃_1τ are 

    (-1,1+b), (b,1), (1+b,-b), (1,-1-b), (-b,-1), (-1-b,b).

Writing τh̃_1τ =h̃_1^l h_1^u for some l,u, we get

    (-1, 1+b)=(l,lb)+(0,ku),

which implies that l=-1 and 2b+1=ku n.  

(i) Assume Γ̃_1 and Γ_1 have the same order n, i.e. k=1. 
Recall from the proof of Proposition <ref> that in this case, 
b=0, so that u=1 and τh̃_1τ=h̃_1^-1 h_1. Renaming t_1=h_1, t_2=h̃_1^-1
as in the proof of Proposition <ref>, we get

    g^2t_1g^-2=t_2,     g^2t_2 g^-2=t_2^-1 t_1^-1,     τ t_1τ =t_1,
         τ t_2τ =t_2^-1 t_1^-1.

With this presentation, the subgroup ⟨Γ,g^2,τ⟩ can be identified with 
G̃_n by 

    g^2=[z_2,z_0,z_1],  τ=[z_0,z_2,z_1],   t_1=[μ_n z_0,z_1,z_2],   t_2=[z_0,μ_n z_1,z_2].

 
 (ii) Assume Γ̃_1 and Γ_1 have different orders, i.e., 
  |Γ̃_1|=n, |Γ_1|=n/k, with k>1. In this case, we first note that 2b+1=0k. 
  On the other hand, recall from the proof of Proposition <ref> that b^2+b+1=0 k. It follows that  b=1k and
k=3. With this understood, note that one can modify h̃_1 by a suitable power of h_1 to arrange so that b=-2. With this choice,
we then have s=-b=2 and v=1, where s,v appear in the relations (see the proof of Proposition <ref>)

    g^2t_1g^-2=t_2^kt_1^-s,     g^2t_2 g^-2=t_2^s-1 t_1^-v.

  Moreover, b=-2 implies u=-1, hence the presentation of ⟨Γ,g^2,τ⟩: 
 
    g^2t_1g^-2=t_2^3 t_1^-2,     g^2t_2 g^-2=t_2 t_1^-1,      τ t_1τ =t_1,
         τ t_2τ=t_2^-1 t_1

With this presentation, the subgroup ⟨Γ,g^2,τ⟩ can be identified
with G̃_n,3,2 by identifying 

    g^2=[z_2,z_0,z_1],  τ=[z_0,z_2,z_1],   t_1=[μ_n/3 z_0,z_1,z_2],   
    t_2=[μ_n^2 z_0,μ_n z_1,z_2].



It is clear that G is a semi-direct product of the imprimitive subgroup of PGL(3) and _2. 
The proof of Theorem <ref> is completed.



 
The rest of this section is occupied by the proof of Theorem <ref>, where we assume (X,ω) 
admits a minimal symplectic G-conic bundle structure π:X→^2. Furthermore,
we assume N≥ 4. 
Recall that Q is the subgroup of G which leaves each fiber of 
π invariant, and G_0 is the subgroup of Q which acts trivially on H^2(X).






 §.§ Proof of Theorem <ref>
 







We begin with some useful observations about the rotation numbers of an element of Q
at a fixed point. 

3mm
Rotation numbers of fixed points.
3mm

Suppose q∈ X is fixed by a nontrivial element g∈ Q and q is not 
the singular point of a singular fiber. Then g must fix the symplectic orthogonal direction of the fiber at q, 
because g induces a trivial action on the base. It follows that the rotation numbers of g 
at q are (a,0) for some a≠ 0. 

Furthermore, if q lies in a singular fiber, then g must leave the (-1)-sphere containing q invariant, 
and the rotation numbers of g at the other 
fixed point, which is the singular point of the singular fiber, must be (a,-a). (See Section <ref>)

3mm
Structure of fixed point sets.
3mm

Note that the fixed-point set of a nontrivial element g∈ Q consists of embedded J-holomorphic
curves and isolated points. Since each regular fiber contains two fixed points of g, it follows that the 
fixed-point set of g consists of a bisection and isolated points which are the singular points
of those singular fibers of which g leaves each component invariant. 

Clearly, one of the following must be true:


  
  * g leaves components of a singular fiber invariant.  

  Then there is a fixed point q as described above, and the singularity p on the fiber has rotation number (a,-a) hence is an isolated fixed point (and there is one more fixed point on either component of the fiber).
  
  * g switches the 2 components of the singular fiber.  

  Then

 
  
    
  * the singularity p on the fiber is contained in the fixed bisection,
    
  * p is a branched point of the double branched covering from the bisection
to the base,
    
  * g must be an involution, because if g^2≠ 1, then p
must be an isolated fixed point of g^2 from (i), a contradiction.
  
  

  











  

With the preceding understood, note that the subgroup G_0 can be identified with the subgroup
which leaves each (-1)-sphere in a singular fiber invariant. Since there are N-1≥ 3
singular fibers, the induced action of G_0 on the base ^2 has at least 3 fixed points.
It follows that the action of G_0 on the base must be trivial, and G_0 is a subgroup of Q.
It is clear that either G_0 is trivial, or it is finite cyclic. 

Furthermore, the fixed-point set of 
each nontrivial element of G_0 consists of N-1 isolated points with rotation numbers 
(a,-a) for some a≠ 0 (these are the singular points of the singular fibers) and two disjoint fixed sections from the two other fixed points other than the singular points on each fiber. 
The two fixed sections must have the same self-intersection number because there exists a g∈ G which switches 
the two sections.
Applying Lemma <ref>  to the fixed sections, we see that each has self-intersection (N-1)/2; in particular, N must be odd
if G_0 is nontrivial. 
Finally, we note that each element of Q∖ G_0 must be an involution (the square fixes 4 points in a general fiber: intersections with the bisection, and the intersections of the two disjoint sections as above).



The group Q contains an involution; in particular, it has an even order. 



Let F be the singular fiber which contains the exceptional sphere representing E_N. The minimality assumption implies that there is a g∈ G which 
switches the two (-1)-spheres in F. Clearly, g has an even order, say 2m. If g∈ Q, then as we have shown
earlier, g must be an involution, and we are done. Suppose m>1. We let h=g^m, which is an involution. We claim
that h∈ Q. 

Suppose h is not contained in Q. Then h induces a rotation on the base, so that the fixed-point set of h must be 
contained in the two fibers which h leaves invariant. With this understood, we claim that h must fix one of the (-1)-spheres
in F. Denote Σ as the two dimensional component of the fixed point set of h, which could be empty



If h does not fix either of the (-1)-spheres in F, we must have E_N·Σ=0. On the other hand, 
by Proposition 5.1 in <cit.>, we have

    (h· E_N)· E_N= E_N·Σ2.

This is a contradiction because h· E_N=E_N or H-E_1-E_N, so that (h· E_N)· E_N=± 1. Hence h fixes
one of the (-1)-spheres in F. Now since g commutes with h and g switches the two (-1)-spheres in F, h must
also fix the other (-1)-sphere. But this clearly contradicts the fact that h is nontrivial. Hence h∈ Q
and the lemma is proved.





If G_0 is nontrivial, say, of finite order m>1, and Q≠ G_0, then Q is the dihedral group D_2m.
Moreover, any involution τ∈ Q∖ G_0 switches the two fixed sections of G_0, hence the two (-1)-spheres in each singular fiber.



For any g∈ Q, since G_0 is normal in Q, g leaves the two fixed sections of G_0
invariant. Note that if g leaves each of the section invariant, then it must fix both of them because the
induced action of g on the base is trivial. Consequently, Q/G_0 is either trivial or _2,
depending on whether there is a g∈ Q switching the two fixed sections of G_0. It follows easily that 
every element in Q∖ G_0 switches the two (-1)-spheres in each singular fiber.
Finally, if Q≠ G_0, then Q must be the corresponding dihedral group because each element
in Q∖ G_0 is an involution. (g^2 preserves all exceptional spheres hence in G_0) This finishes the proof of the proposition.



Next we consider the case where G_0 is trivial. Let Σ be the set of singular fibers. 


Suppose G_0 is trivial. Then Q=_2 or (_2)^2. In the latter case, let τ_1,τ_2,τ_3 be the
distinct involutions in Q. Then Σ is partitioned into subsets Σ_1, Σ_2,
Σ_3, where Σ_i parametrizes the set of singular fibers of which τ_i
leaves each (-1)-sphere invariant, and #Σ_i≡ N-1 2,  for i=1,2,3.



First of all, Q consists of involutions as G_0 is trivial. Suppose Q≠_2, and let τ, τ^'∈ Q
be two distinct nontrivial elements. We claim that there is no singular fiber such that both τ,τ^' leave both of
the (-1)-spheres in this fiber invariant. This is because if there is such a singular fiber, then by examining the
action of ττ^' at the singular point of the fiber, we see ττ^' must be trivial (the rotation numbers at this signular point is
(0,0)). But this contradicts the assumption that τ≠τ^'. 

With the preceding understood, let τ_1,⋯,τ_n, n>1, be the distinct involutions in Q, and 
let Σ_i be the set of singular fibers of which τ_i leaves each (-1)-sphere invariant. Then the
previous paragraph shows that Σ_i∩Σ_j=∅ for i≠ j. On the other hand, suppose
τ_k=τ_iτ_j, then Σ∖ (Σ_i∪Σ_j)⊂Σ_k, implying 

    Σ∖ (Σ_i∪Σ_j)=Σ_k.

It follows easily that n=3 and Q=(_2)^2. 

It remains to see that for each i, #Σ_i=N-1 2.
Consider the fixed-point set S_i of τ_i. Then S_i is a bisection and the projection of S_i onto the base
is a double branched covering which ramifies exactly at the singular points of those singular fibers not parametrized by
the set Σ_i. Since the number of ramifications must be even, we have #Σ_i=N-1 2 as claimed.
 
 
 
The proof of Theorem <ref> is completed.




§ MINIMALITY AND EQUIVARIANT SYMPLECTIC CONES
 


Now let X=^2# N^2, N≥ 2,
which is equipped with a smooth action of a finite group G. Suppose there is a G-invariant
symplectic form ω_0 on X such that the corresponding symplectic rational 
G-surface (X,ω_0) is minimal. With this understood, we denote by 
Ω(X,G) the set of G-invariant symplectic forms on X. 


The following is a crucial observation.


For any ω∈Ω(X,G), the canonical class K_ω=K_ω_0 or -K_ω_0.




The lemma is obvious if H^2(X)^G has rank 1 because [_0],[] and K_ are proportional. 

Assume that  H^2(X)^G has rank 2. Recall from the proof of Theorem <ref>, there is a reduced basis 
H,E_1,⋯, E_N of (X,ω_0) such that K__0= -3H + E_1 +E_2 + ⋯ + 
E_N. Now after changing ω by sign and with a further scaling if necessary, we can write 
[]=-K__0+bF= (3+b)H - (1+b) E_1 -⋯ -E_N for some b∈. 
We claim that [] is always a reduced class in the sense of <cit.>.  
To see this, note that the condition []^2>0 implies 4b>N-9, which implies b≥-1
since N≥ 5. This gives 3+b>1+b>0 and 3+b=1+b+1+1.  Now the conclusion 
follows from <cit.>, which asserts that any symplectic class with reduced form has canonical class 
K__0.





Recall that we have shown that a minimal symplectic rational G-surface where X=^2# N^2
admits a minimal symplectic G-conic bundle only when  N≥ 5 and N≠ 6.
The following lemma deals with precisely these cases and gives the converse of this fact. 


Let (X,ω) be a symplectic rational G-surface where X=^2# N^2
with N≥ 5 and N≠ 6. Suppose (X,ω) admits a minimal symplectic G-conic bundle 
structure. Then for any ω^'∈Ω(X,G) such that K_ω^'=K_ω,


the symplectic rational G-surface (X,ω^')
is minimal. 



By Lemma <ref>, H^2(X)^G is of rank 2 spanned by K_ and the fiber class F of
the symplectic G-conic bundle. Now suppose to the contrary that (X,ω^')
is not minimal. Then there is a G-invariant, disjoint union of ω^'-symplectic
(-1)-spheres C_1,C_2,⋯, C_m. Let C=C_1+⋯ +C_m. Then since 
K_ω^'=K_ω, we have 

    -m=C^2=K_ω· C.

On the other hand, C∈ H^2(X)^G, so that 

    C=aK_ω+bF, 2mm a,b∈.





The key ingredient for deriving a contradiction is the fact that F· C≥ 0,  which 
as a corollary implies that a<0, and so that F· C>0. 
To see this, Note that F is represented by an embedded
J-holomorphic sphere S where J is ω-compatible. 
On the other hand, since K_ω^'=K_ω, ω and ω^' have the same set of
symplectic (-1)-classes. Hence since the class of each C_i is represented by a 
ω^'-symplectic (-1)-sphere, it is also represented by a ω-symplectic 
(-1)-sphere. Consequently, the class of C_i can be represented by ∪_j m_j D_j 
where D_j
is a J-holomorphic curve and m_j>0. Now since S is irreducible and S^2=0, we
have S· D_j≥ 0 for each j, which implies that F· C_i≥ 0 for each i.
Hence our claim that F· C≥ 0. 

With this understood, we consider the pairing of C=aK_ω+bF with K_ω 
and then square both sides, we have 


    aK_ω^2-2b=-m


    -m=a^2K_ω^2-4ab

which gives 


    m=-a^2K_ω^2/2a-1.

Notice that m>0, hence K_ω^2>0, which excludes all cases for N≥9.  
Moreover, a^2 and 2a-1 are co-prime for all a≤ -1, therefore, 2a-1|K_ω^2.  But this is not
possible because of the assumption N≥ 5 and N≠ 6.











Proof of Theorem <ref>




The claim regarding the canonical class is proved in Lemma <ref>. The minimality claims
are trivial if H^2(X)^G is of rank 1. For the case rank H^2(X)^G>1, they
follow from Lemma <ref>, Lemma <ref> for N≥5 and N≠ 6; and Lemma <ref> for N=6. 
Theorem <ref> implies these are the only cases to consider.  Combining these results with
Theorem 3.8 of <cit.>, we cover the case (2) of minimal complex rational G-surfaces.






In Theorem <ref>, the case of complex rational G-surface admits an alternative proof, which we sketch
it here. Let X be a minimal complex rational G-surface which is ^2 blown up at 6 points,
and assume X is a conic bundle. 
Then by Proposition 5.2 of <cit.>, X must be Del Pezzo, and hence has a G-invariant
monotone Kähler form ω.  Part (1) of Lemma <ref> below implies there are two distinct fiber classes in H^2(X)^G, but N=6 contradicts part (2) of the lemma.


5mm
We now turn to the proof of Theorem <ref>.  Fixing the canonical class K_ω_0, we recall that F∈ H^2(X)^G is called a fiber class if it is the class of the fibers of a symplectic
G-conic bundle on X for some G-invariant symplectic form  with K_=K_ω_0.

 
 Suppose H^2(X)^G has rank 2.
 

   
  * If X admits a G-invariant monotone symplectic form, then there are at least two distinct
 fiber classes in H^2(X)^G.
   
  * Suppose F,F^'∈ H^2(X)^G are distinct fiber classes. Then F+F'=-aK_ω_0 
 for some integer a>0, and N=5,7 or 8 where a=1, 2, 4 respectively. In particular, there
 are at most two distinct fiber classes in H^2(X)^G.

 

 

To prove (1), let ω be a G-invariant monotone form on X, and let 
F be the fiber class of the G-conic bundle structure obtained from Theorem 1.3. Pick
a G-invariant closed 2-form η representing F. Then for sufficiently small 
ϵ≠ 0, the G-invariant symplectic form ω^':=ω+ϵη 
is non-monotone, and the symplectic G-manifold (X,ω^') is minimal
(as shown in the proof of Theorem <ref>). 
By Theorem <ref>, (X,ω^') admits a symplectic G-conic bundle with small
fiber. An easy check with the symplectic areas shows that for ϵ>0, the small fiber class (whose area is twice the minimal exceptional spheres)
of the symplectic G-conic bundle equals F, but for ϵ<0, it is not F.



To prove (2), let ω,ω^' be the G-invariant symplectic forms associated with the
symplectic G-conic bundles whose fiber classes are F,F^' respectively. 
For simplicity, we set K=K_ω_0. 

Write 

    F^'=-aK +b F,

for some a,b∈. Note that K· F=K· F^'=-2, 
F^2=(F^')^2=0, and the assumption that
F≠ F^' implies a≠0.  Then by pairing (<ref>) with K, F and F', respectively, one has -aK^2=-4 and b=-1. Therefore, a and K^2
are both divisors of 4, and F+F'=-aK.

We claim 2a=F· F'≥ 0, which implies a>0 and N=5,7,8. The point is that F
can be represented by an embedded J-holomorphic sphere V with V^2=0, where 
J is ω-compatible. Since K_=K_^', this fact implies the Gromov-Witten invariant of F' is nontrivial, 
hence F^'
can be represented by a stable J-holomorphic curves, from which our claim F· F'≥ 0
follows easily.




Suppose G_0 is nontrivial. Then there is a unique fiber class.



We note first that the 2-dimensional fixed point set of G_0 consists of two embedded
2-spheres S_1,S_2 , each with self-intersection -(N-1)/2 (in particular, N must be odd).
Suppose to the contrary that there are two distinct fiber classes F,F^'. Then S_1
(and S_2) is a J-holomorphic section of the corresponding symplectic G-conic bundles
with fiber classes F,F^', for an appropriate ω or ω^'-compatible
G-invariant J. This implies that S_1· F=S_1· F^'=1. On the other hand,
by Lemma <ref>, F+F^'=-aK_ω_0 for some a>0, implying that
K_ω_0· S_1<0. This violates the adjunction formula because S_1^2=
-(N-1)/2≤ -2, and K_=K_^'=K_ω_0.



 
Proof of Theorem <ref>




Part (1) and part (2) follow immediately from Lemma <ref> and Lemma 4.4, and the proof of Theorem <ref>.  In the case (2) when there is a unique fiber class F for the G-conic bundle, any G-invariant symplectic form has the form (3H-E_1-⋯-E_m)+bF.  From Theorem <ref>, (E_1)≥(E_k), hence b≥0.  This also shows C(X,G,F)∪ C(X,G,F')=C(X,G).

To see that C(X,G,F)∩ C(X,G,F^') is either empty or consists of
classes of G-invariant monotone symplectic forms, let [ω]∈
C(X,G,F) and [ω^']∈ C(X,G,F^') such that [ω]=[ω^'].
Then [ω]=-a K_ω_0+bF, 
[ω^']=-a^' K_ω_0+b^' F^', where a,a^'>0 and
b,b^'≥ 0. If both b,b^' are non-zero, then [ω]=[ω^'],
together with the fact that F+F^' is a multiple of K_ω_0 (cf. Lemma <ref>),
would imply that F,F^' are linearly dependent, a contradiction.

To see (3), first we note that if [ω]∈Ĉ(X,G,F), then for any 
δ>δ_ω,F,
there is an ω^'∈Ω(X,G) such that [ω^']∈Ĉ(X,G,F)
with δ_ω^',F=δ. We can simply take ω^':=
ω+ (δ-δ_ω,F)π^∗η, where π is a symplectic G-conic 
bundle on (X,ω) with fiber class F, and η is an area form on the base of π
with total area 1. Secondly, if δ_X,G,F>0, then it can not be attained. Suppose to
the contrary that there is an ω such that δ_ω,F=δ_X,G,F.
Then take 0<ϵ<δ_X,G,F sufficiently small, the G-invariant form
ω^':=ω-ϵη, where η is a G-invariant closed 2-form representing F, is a symplectic form. The condition ϵ<δ_X,G,F implies
that [ω^']∈Ĉ(X,G,F), which contradicts the definition of δ_X,G,F. 




We end this section with a uniqueness result on the subgroup Q from Definition 1.7.
 
 
 Suppose (X,G) has a unique fiber class F. Then the normal subgroup Q of G is uniquely
 determined, i.e., it is independent of ω∈Ω(X,G), nor the symplectic G-conic
 bundle structure on (X,ω) involved in the definition of Q. 
 
 
 
 Let ω,ω^'∈Ω(X,G), and let π, π^' be symplectic 
 G-conic bundles with fiber class F, and let Q,Q^' be the subgroups of G
 defined using π,π^' respectively. Let H,E_1,⋯,E_N and 
 H^',E_1^',⋯,E_N^' be a reduced basis associated to π,π^'
 respectively.
 
  Let Q_(H,E_i)={g∈ G| g· E_j=E_j  H-E_1-E_j}. We claim Q=Q_(H,E_i).
 First, it is clear that Q⊂ Q_(H,E_i). Secondly, if g∈ Q_(H,E_i), then g leaves
 each singular fiber invariant. Since the number of singular fibers is N-1 which is greater
 than 3, the induced action of g on the base ^2 has at least 3 fixed points. This
 implies that the action of g on the base must be trivial, and g∈ Q. Hence Q=Q_(H,E_i).
 Similarly, Q^'=Q_(H^',E_i^').
 
 If we normalize so that ω(F)=ω^'(F)=2, then for each j=2,⋯,N,
 ω(E_j^')=ω^'(E_j)=1 also. In particular,  
 H^',E_1^',⋯,E_N^' is a reduced basis for (X,ω). By
 Lemma <ref>, and the fact that ω is not monotone, we have,
 for each j>1, E_j^'=E_k
 or H-E_1-E_k for some k>1. It follows that H^'-E_1^'-E_j^'=
 H-E_1-E_k or E_k for the same k. From these relations we see immediately that
 Q_(H,E_i)=Q_(H^',E_i^'). Hence Q=Q^'.
  
 
 
  
 


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