Automorphisms of 3-folds of general type acting trivially on cohomology
Hang Zhao

	
	
	
	
	
	
	School of Mathematical Sciences 
 
		Peking University   
 
		Beijing

		China
	
	math_hang@pku.edu.cn
	
	
	
	
	
	
	
	
	
	
	
	
	[2010]14J50, 14J30
	
	
	
	
	
	
	
	
	Automorphisms of threefolds of general type acting trivially in cohomology
    Hang Zhao
    December 30, 2023
==========================================================================


		Let X be a minimal projective threefold of general type over ℂ with only Gorenstein quotient singularities, and let Aut_ℚ(X) be the subgroup of automorphisms acting trivially on H^*(X,ℚ). In this paper, we show that if X is of maximal Albanese dimension, then |Aut_ℚ(X)|≤ 6. Moreover, if X is nonsingular and K_X is ample, then |Aut_ℚ(X)|≤ 5.
		
		Seeking for higher-dimensional examples of varieties with nontrivial Aut_ℚ(X), we concern d-folds X isogenous to an unmixed product of curves. If d=3, we show that Aut_ℚ(X) is a 2-elementray abelian group whose order is at most 4 under some conditions on their minimal realizations. Moreover, each of the possible groups can be realized. If d≥ 3, we give a sufficient condition for Aut_ℚ(X) being trivial.
		
		Curiously, there exist examples of projective threefolds X with terminal singularities and maximal Albanese dimension whose Aut_ℚ(X) can have an arbitrarily large order.
	
	

§ INTRODUCTION

	
	Let X be a complex manifold, and Aut(X) be its group of holomorphic automorphisms. Consider the action of Aut(X) on the cohomology H^*(X,A) of X, where A=,; this gives a representation ρ_X,AAut(X)→GL(H^*(X,A)) defined by ρ_X,A(σ)(ω)=(σ^-1)^*ω for σ∈Aut(X) and ω∈ H^*(X,A). The interesting question is that is the representation ρ_X,A faithful?
	
	We say  that X is rationally cohomologically rigidified (resp. cohomologically rigidified) if ρ_X, (resp. ρ_X,) is faithful. The connected component of the identity (X)^0⊂(X) acts trivially on the cohomology, and is therefore contained in the kernel of ρ_X,. In general, those automomorphisms acting trivially on H^*(X,) are called numerically trivial and they form a subgroup of the (full) automorphism group, to be denoted by _(X) in this paper. Thus, the group Aut(X) splits into two basic parts: its neutral component Aut(X)^0, and its discrete image (X)^*⊂GL(H^*(X,)). The group of connected coponents (X)/(X)^0 is an extension of (X)^* by _(X)/(X)^0. It is interesting to study the structure of the group Aut_ℚ(X). In particular, when X is of general type, Aut_ℚ(X) is a finite group.
	
	Many authors have recently studied the numerically trivial automorophism group of surfaces whose Kodaira dimension ranging from 0 to 2.
	
		
  * for K3 surfaces <cit.>;
		
  * for Enriques surfaces <cit.>;
		
  * for properly elliptic surfaces <cit.>;
		
  * for surfaces of general type <cit.>.
	 
	For surfaces of general type, it turns out that nontrivial _(S) occurs only for those with
	irregularity q(S)≤ 2 due to Cai, Liu, and Zhang, who prove the following theorem.
	
	
		Let S be a minimal surface of general type. Then we have the following results:
		
			
  * if q(S)≥ 3, then S is rationally cohomologically rigidified;
			
  * if q(S)=2, then |Aut_ℚ(S)|≤ 2, and the equality holds only if S is a surface isogenous to an unmixed product of curves.
		
	
	
	This paper aims to study the numerically trivial automorphism group of threefolds of general type with maximal Albanese dimension. First, we prove the following result.
	
		Let X be a minimal projective threefold of general type with only Gorenstein quotient singularities, assume that it is of maximal Albanese dimension. Then |Aut_ℚ(X)|≤ 6. Moreover, if X is smooth and K_X is ample, |Aut_ℚ(X)|≤ 5.
	
	
	Our proof is inspired by <cit.>. Set X̅:=X/Aut_ℚ(X). One can show that  the Albanese map a_X X→ A_X of X factors through the quotient map π X→X̅ and that χ(ω_X)=χ(ω_X̅), see Lemma <ref>, (1). Since X is a Gorenstein minimal threefold, the Bogomolov-Miyaoka-Yau inequality <cit.> implies that
	
    (K_X)≤ 72χ(ω_X).
 
	Let Y̅ be a suitable desingularization of X̅, we can show that Y̅ is of general type and of maximal Albanese dimension, see (2) of Lemma <ref>. By the generalized Severi inequality <cit.>, we have 
	
    12χ(ω_Y̅)≤(K_Y̅).
 Comparing volumes (K_X) and (K_Y̅), we obtain 
	
    |Aut_ℚ(X)|(K_Y̅)≤(K_X).

	Combining the inequalities above, we get |Aut_ℚ(X)|≤ 6. The assumption that X has Gorenstein singularities is necessary; we give a counterexample when X has terminal singularities of Cartier index 2  by constructing a series of threefolds {X_n}_n∈ℕ such that |Aut_ℚ(X_n)| can be arbitrarily large, see Example <ref>.
	
	It is well-known that compact Riemann surfaces of genus g≥2 are rationally cohomologically rigidified. In Section <ref>, we generalize this fact to varieties isogenous to an unmixed product of curves, introduced by Catanese in <cit.>, which is a quotient of a product of curves of genus at least 2 by a finite group acting freely and diagonally.
	
		Let X be d-fold isogenous to an unmixed product of curves with d≥ 3, and let (C_1×…× C_d)/G be its minimal realization. Suppose g(C_i/G)≥ 1 for all 1≤ i≤ d. Set K_i=Ker(G→Aut(C_i)). If there is a pair (i,j) with j≠ i such that g(C_i/G)≥ 2 and g(C_j/K_i)≥ 2, then Aut_ℚ(X) is trivial. 
	
	
	In general, we first consider the case that a nonsingular projective variety Y of dimsion d≥ 3 which admits a higher irrational pencil g Y→ D where D is a smooth curve. Let σ be a nontrivial automorphism of Y such that g∘σ=g. If σ induces a trivial action on H^0(Y,ω_Y), then its restriction ot F induces the identity on H^0(F,ω_F), where F is a general fibre of g. Let σ_F be the resctriction of σ on F. We can use lower dimension result on pair (F,σ_F) to get a higher dimension result on pair (Y,σ), in fact, we can show that o(σ)≤ o(σ_F), see Lemma <ref>. This result is a generalization of <cit.>. Then we can use a induction procedure to the case that there are a sequence of higher irrational pencils g_j F_j→ D_j such that each F_j+1 is a general fibre of g_j for 0≤ j≤ Y-2 where F_0=Y, and show that there is no nontrivial automorphism of F_d-2 which is a successive restriction of an automorphism σ̃_0 of Y acting trivially on H^0(Y,ω_Y), see Corollary <ref>. In particular, applying this result to the case that X is as in Theorem <ref> shows the theoerm.
	
	It is worth pointing out that Theorem <ref> is not valid for the case that q(X)≥ d+1, which is different from that of irregular surfaces of general type <cit.>. In these papers <cit.>, both authors construct a series of surfaces X of general type isogenous to a product with q(X)=2 such that Aut_ℚ(X)≅_2. In their example, X is a quotient of C× D by a finite group G, where C and D are curves with faithful group actions of G. 
	
	Seeking for higher-dimensional examples of varieties with nontrivial Aut_ℚ(X), we consider threefolds  X isogenous to a product of curves. A new phenomenon occurs in this case: let (C_1× C_2× C_3)/G be the minimal realization of X, then the group G probably does not act faithfully on each curve C_i for i=1,2,3. Denote the subgroup of G acting trivially on C_i by K_i; the appearance of K_i is a difficulty for studying the structure of Aut_ℚ(X). Suppose G is abelian, and all K_i are cyclic groups, then we can show that Aut_ℚ(X) is a 2-elementary abelian group. Concretely, we have the following theorem.
	
		Let X be threefold isogenous to an unmixed product of curves, and let (C_1× C_2× C_3)/G be its minimal realization. Suppose g(C_i/G)≥ 1 for all 1≤ i≤ 3. Then we have
		 
			
  * If there is a pair (i,j) with j≠ i such that g(C_i/G)≥ 2 and g(C_j/K_i)≥ 2, then Aut_ℚ(X) is trivial;
			
  * if for any 1≤ i≤ 3 with g(C_i/G)≥ 2, we have g(C_j/K_i)=1 for all j≠ i, then Aut_ℚ(X)≅(ℤ_2)^k with k=0,1;
			
  * if for all 1≤ i≤ 3, we have g(C_i/G)=1, and suppose that the group G is an abelian group, and K_i is a cyclic group for all 1≤ i≤ 3, then we have Aut_ℚ(X)≅ (ℤ_2)^k with k=0,1,2.
		 
	
	
	We give only the main ideas of the proof. Theorem <ref>, (1) follows directly from Theorem <ref>. We apply the induction procedure mentioned above to derive Theorem <ref>, (2), see Corollary <ref>. To prove Theorem <ref>, (3), we first show that Aut_ℚ(X) is determined by the algebraic data associated with the minimal realization of X, see Definition <ref> for the definition of algebraic data and Lemma <ref>. We next show that the group Aut_ℚ(X) can be embedded into an abstract 2-elementary abelian group, see Theorem <ref>. Finally, our assertion follows from the bound of |Aut_ℚ(X)| in Theorem <ref>.
	
	In Section <ref>, we construct some examples of threefolds X isogenous to an unmixed product of curves with Aut_ℚ(X)≅_2 and _2×_2. 
	
	

§ NOTATIONS AND PRELIMINARIES

	
	We work over the complex numbers throughout the paper.
	Varieties are always assumed to be normal and quasi-projective; a threefold X is a projective variety of dimension 3.
	
	Let (X) be the holomorphic automorphism group of X. Let f X→ Y be a surjective morphism to a variety Y with connected fibres, we set 
	
		
  Aut(X/Y)={σ∈Aut(X)|f∘σ=f}, the relative automorphism group over Y.
	
	
	For a finite group G, we denote
	
		
  G^* the set of irreducible characters on G.
	 When G is abelian, G^* is isomorphic to G, and is called the dual group of G.  For an element g of G, we denote the order of g by o(g). 
	
	For a variety X with a faithful group action of G and a cohomology group H of X, we set 
	
		
  X^g={x∈ X|g(x)=x}, the fixed loci of an element g∈ G;
		
  H^G={ω∈ H| g^*ω=ω}, the G-invariant subgroup of H.
	 
	
	

 §.§ Gorenstein quotient singularity

	
	For a normal variety X of dimension n with singularities, we denote its nonsingular part by X_0, and then the canonical sheaf ω_X_0 make sense. Let j X_0→ X be the inclusion morphism, and we define the sheaves Ω̃^i_X=j_*(Ω^i_X_0) of X for 1≤ i≤ n. The canonical divisor K_X on X is a Weil divisor such that 
    𝒪_X(K_X)=ω_X:=Ω̃^n_X,
 which is well-defined up to linear equivalence. We say that X has Gorenstein singularity if its canonical divisor K_X is a Cartier divisor. For the definitions of rational singularity, we recommend references <cit.> and <cit.>. 
	 
		If the variety X has rational singularities, for example, quotient singularities, for any nonsingular resolution f  Y→ X, we have R^if_*_Y=0 for i>0 and f_*_Y=_X. It follows that χ(X,_X)=χ(X,f_*_Y)=χ(Y,_Y).  Since rational singularities are Cohen-Macaulay (see <cit.>), using the Serre duality, we get χ(X,ω_X)=χ(Y,ω_Y).
	
	
	

 §.§ Volume of divisors

	
		Let X be a projective variety of dimension n, and let D be an integral divisor on X. The volume of D is defined to be the non-negative real number 
    _X(D)=lim sup_m→∞n!h^0(X,_X(mD))/m^n

		If D is a -divisor, the volume of D is defined as _X(D)=1/a^n_X(aD) for some a∈ such that aD is integral.
	
	
	<cit.>  
		Let X be a normal projective variety of dimension n.
		If X has canonical singularities and ν X'→ X is a nonsingular resolution of X, then 
    _X'(K_X')=_X(K_X)

	
	
		Note that _X(D)>0 if and only if D is big. If D is nef, then it follows from the asymptotic Riemann-Roch that _X(D)=D^n. If F is an effective divisor on X, we have _X(D)≤_X(D+F).
	
	

 §.§ Galois covers of curves

	We recall some facts about the Galois covers of algebraic curves, and refer the reader to <cit.>, <cit.> and <cit.> for more details.
	
		Let G be a finite group and let 
    g'≥ 0,   m_r≥ m_r-1≥…≥ m_1≥2
 be integers. A generating vector for G of type [g';m_1,…,m_r] is
		a sequence of elements 
    :=(η_1,…,η_2g';σ_1,…,σ_r )
 such that the following conditions are satisfied,
		
			
  * G is generated by the entries of the sequence ;
			
  * o(σ_i)=m_i for all 1≤ i≤ r;
			
  * σ_1 ⋯σ_r·∏_i=1^g'[η_i ,η_g'+i ]=1_G;	
		 Moreover, if the following Riemann-Hurwitz relation holds:
    2g-2=|G|(2g'-2+∑_i=1^r(1-1/m_i)),

		then there exists a compact Riemann surface C of genus g and a G-cover C→ D branched in r points P_1,…,P_r with ramification indexes m_1,…,m_r respectively, where g' is the genus of D. The subgroups g⟨σ_i⟩ g^-1 provide all nontrivial stabilizers of the G-action on C for g∈ G. Set Σ:=⋃_i=1^r⋃_g∈ Gg⟨σ_i⟩ g^-1. 
	
	In the terminology of representations, let φ G→(H^1,0(C)) be the representation given by the action of G on C. For any irreducible representation ρ G→(V) with V, a finite-dimensional vector space over . We denote by N_i,k(ρ) the multiplicity of μ_m_i^k as an eigenvalue of ρ(σ_i), where μ_m_i:=exp(2π√(-1)/m_i) and 0≤ k≤ m_i-1. For the character χ_φ afforded from φ, we have the following formula.
	 
		In the notations above, let 1_G be the trivial character of degree 1 on G. For each χ∈ G^* and ρ is its corresponding representation, it holds 
    ⟨χ,χ_φ⟩ = χ(1_G)(g'-1) + ∑_i=1^r∑_k=1^m_i-1k· N_i,k(ρ)/m_i+⟨χ,1_G⟩.
 
	
	 
		Under the assumption above, suppose G is abelian and g'≥1. Then for each χ∈ G^*∖{1_G}, _H^1,0(C)^χ>0 if and only if the following holds 
    σ∉(χ) for some σ∈Σ, i.e., σ=σ^k_i  for some  1≤ i≤ r,1≤ k≤ m_i-1.

	
	
	

 §.§ Threefolds isogenous to a product of curves

	According to Catanese <cit.>, a vareity X is said to isogenous to a higher porduct of curves if it admits a finite unramified covering which is isomorphic to  a product of curves of genus ≥ 2. There is a product of curves C_1×…× C_d together with finite group G such that C acts freely on C_1×…× C_d. The pair (C_1×…× C_d, G) or the quotient (C_1×…× C_d)/G is called a minimal realization of X if K_i∩ K_j={1} for all 1≤ i<j≤ d, where K_i is the normal subgroup of G which acts trivially on C_i. X is said to be of unmixed type if G acts diagonally on C_1×…× C_d, and is also said to isogenous to an unmixed product of curves.
	
	
		Let G be a finite abelian group with K_1,K_2,K_3 three normal subgroups, and let _i be a generating vector for the quotient group G/K_i for 1≤ i≤3. The 7-tuple 𝒜=(G,K_1,K_2,K_3,_1,_2,_3) is called an algebraic datum for G if the following conditions are satisfied:
		
			
  * K_i∩ K_j={1_G} for each 1≤ i<j≤3,
			
  * Σ_1∩Σ_2∩Σ_3={1_G}.
		
		Where Σ_i is the union of nontrivial stabilizers of the G-action on each factor C_i for 1≤ i≤ 3.
	
	
	
		A threefold isogenous to an unmixed product of curves is determined by an algebraic datum 𝒜 described above.  For each algebraic data (G/K_i,_i), by the Riemann existence theorem, there exists an algebraic curve C_i with a faithful group action of G/K_i. We denote by ψ_i G→(C_i), the action of G on C_i; and we have K_i=(ψ_i). The homomorphisms ψ_1,ψ_2 and ψ_3 induce a G-action on the product :
		
    g(x_1,x_2,x_3)=(ψ_1(g)x_1,ψ_2(g)x_2,ψ_3(g)x_3),

		where g∈ G and (x_1,x_2,x_3)∈ C_1× C_2× C_3. The second condition, which is called the freeness condition, ensures that the action of G on C_1× C_2× C_3 is free. Therefore, the quotient X=(C_1× C_2× C_3)/G is a threefold isogenous to a product of curves.
	
	
	

 §.§ Characters of finite abelian groups
 
	For basic definitions of the representation theory of groups, we refer to the books <cit.>.
	
	Let G be a finite abelian group with the identity 1, and let H be a subgroup of G. The restriction map G^*→ H^* is a surjective group homomorphism, for χ∈ G^*, we denote the restriction of χ on H  by χ_H. Since (G/H)^*≅(G^*→ H^*), we may identify (G/H)^* with the subset of characters of G whose restriction on H is trivial. If G is a cyclic group with a generator e, we say that a character χ of G is primitive if χ(e) is a o(e)-th primitive root of unit, so a primitive character χ is a generator of the dual group G^*. For a primitive character χ of G and g∈ G, we have χ(g)≠ 1 iff g≠1. Moreover, χ(g) is a o(g)-th primitive root of unit. 
	
	Let ρ G→(V) be a linear representation of G over , and let χ be the character of the representation ρ given by χ(s)=Tr(ρ(s)) for each s∈ G. Then V decomposes into a direct sum of irreducible representations:
	
    V=V^χ_1⊕…⊕ V^χ_k,

	where V^χ_i is sum of irrducible representation with character χ_i for all 1≤ i≤ k. Set n_i= V^χ_i, then we can write χ=∑_i=1^kn_iχ_i.
	
	Let G=H× K be a direct product of finite groups and let φ and θ be characters on H and K, respectively. We define a character χ=φ×θ of G by χ(hk)=φ(h)θ(k) for h∈ H and k∈ K. Since we have H≅ G/K, there is a corresponding character φ̂ of G such that K⊆(φ̂) and φ̂(hk)=φ(h). Similarly, there is a corresponding character θ̂ of G such that H⊆(θ̂) and θ̂(hk)=θ(k). It follows that φ×θ=φ̂θ̂. Moreover, the characters φ×θ for which φ and θ being irreducible are exactly the irreducible characters of G. Let V and W be linear representations of H and K, respectively. Let V=⊕_1≤ i≤ kV^χ_i and W=⊕_1≤ j≤ lW^ψ_j be corresponding decompositions. Then V⊗ W is a linear representation of G with decomposition:
	
    V⊗ W   =⊕_1≤ i≤ k⊕_1≤ j≤ lV^χ_i⊗ W^ψ_j
       =⊕_i,j(V⊗ W)^χ_i×ψ_j

	
	Let G be a finite group, not necessarily abelian, let H⊆ G be a subgroup, and let φ be a character of H. We define the induced character φ^G of G by
	
    φ^G(g)=1/|H|∑_x∈ Gφ^0(xgx^-1),

	where φ^0 is defined by φ^0(h)=φ(h) if h∈ H and φ^0(y)=0 if y∉ H. If G is abelian and g∉ H, then we have φ^G(g)=0. On the other hand, we may write φ^G=∑_i=1^k n_iχ_i with χ_i is an irreducible character of G, we call each χ_i a constituent of φ^G.  
	
	Next, we present some technical results required in the proof of Theorem <ref>.
	
	
		Let G be a finite abelian group, H a proper subgroup of G, and let φ a character of H. If g∉ H, then for any root of unit c there is a constituent χ of φ^G such that cχ(g)≠ 1.
	
	
	 Since g∉ H, by formula (<ref>), we have φ^G(g)=0. Write φ^G=∑_i=1^k n_iχ_i. Suppose cχ_i(g)=1 for all χ_i constituent of φ^G, then we have 
    φ^G(g)=∑_i=1^k n_iχ_i(g)=c̅[G:H]≠0,
 which is a contradiction. 
		
	
	 
		Let K_1 and K_2 be two cyclic groups, let H=K_1× K_2 be their product, and let K_3 be a cyclic subgroup of H such that K_1∩ K_3={1} and K_2∩ K_3={1}. Then there are nontrivial characters α_1,α_2 of H satisfying 
    K_1⊆(α_2),K_2⊆(α_1) and K_3⊆(α_1α_2).
 In particular, we can choose α_i such that its restriction on K_i is a primitive character for each i=1,2. 
	
	 Choose a generator k_i of sugroup K_i for each i=1,2,3. We can write k_3=k_1^ak_2^b for some integers a and b. Set 
    n_1=o(k_1),n_2=o(k_2),m_1=o(k_1^a) and m_2=o(k_2^b).
 By assumption K_3∩ K_1=K_3∩ K_2={1}, we conclude that o(k_3)=m_1=m_2, and set m=o(k_3). We can choose two characters β_1,β_2 of H satisfying 
    β_i(k_j)=e^2δ_ijπ√(-1)/n_j
 for each 1≤ i< j≤ 2. Choose integers s_1,s_2 and set α_1=β_1^s_1,α_2=β_2^s_2, to require K_3⊆(α_1α_2), we note that α_1(k_3)α_2(k_3)=e^2(s_1+s_2)π√(-1)/m=1. Therefore it is sufficient to take α_1=β_1^s and α_2=β_2^-s for some integer s that is relatively prime to m. To require (α_i)_K_i being primitive, it is sufficient to take s=1, which is due to the fact that the restriction of β_i on K_i is primitive. It is clear that K_1⊆(α_2) and K_2⊆(α_1), which completes the proof.
		
	
	 
		Let K_1 and K_2 be two cyclic groups, and let H=K_1× K_2 be their product. Fix nontrivial element g_i∈ K_i for each i=1,2, and an element h∈ H. Write h=h_1h_2 for h_i∈ K_i. Let α_1,α_2∈ H^* be two characters satisfying the following conditions:
		
			
  * K_1⊆(α_2),K_2⊆(α_1);
			
  * the restriciton of α_i on K_i is primitive for each i=1,2.
		
		Set I_i:={χ∈ H^*|χ(g_i)≠ 1} for each i=1,2 and J:={χ∈ H^*|χ(h)≠1}.

		If max{o(g_1),o(g_2),o(h_1),o(h_2)}≥ 3, then we have
		
    α_1^s∈ I_1 and α_1^sα_2∈ J     if o(g_1)≥ 3 or o(h_1)≥ 3
    α_2^s∈ I_2 and α_1α_2^s∈ J     if o(g_2)≥ 3 or o(h_2)≥ 3

		for some s=1,2,3,5. 
	
	
		If o(g_1)≥ 3, then α_1^s∈ I_1 for s=1,2. Suppose 
    α_1(h)α_2(h)=α_1^2(h)α_2(h)=1,
 we have α_1(h_1)=α_1(h)=α_2(h)=α_2(h_2)=1, which contradicts to condition (2). Therefore, α_1^sα_2∈ J for some s=1,2. For the same reason, if o(g_2)≥ 3, we have α_1α_2^s∈ J for some s=1,2 with α_2^s∈ I_2. 
		
		Now we suppose that o(g_1)=o(g_2)=2 and o(h_1)≥ 3. If o(h_1)=3, we have α_1^sα_2∈ J and α_1^s∈ I_1 for some s=1,5. Otherwise, we have 
    α_1(h)α_2(h)=α_1^5(h)α_2(h)=1,
 and it follows that α_1^4(h)=1. Note that  α_1^4(h)=α_1(h_1)≠ 1 by condition (2); we have a contradiction. If o(h_1)≥ 4, for the same reason, we have α_1^sα_2∈ J and α_1^s∈ I_1 for some s=1,3. Similarly, if o(g_1)=o(g_2)=2 and o(h_2)≥ 3, then α_1α_2^s∈ J and α_2^s∈ I_2 for some s=1,3,5. 
		
	
	 
		Let G be a finite abelian group with the identity 1, and let g_1,g_2,h be three nontrivial elements in G. Set 
		
    J:={ϕ∈ G^*|ϕ(g_1)≠1 and ϕ(g_2)≠ 1} and I:={χ∈ G^*|χ(h)≠1}.
 
		If o(h)≥ 3, then we have ϕχ∈ J for some ϕ∈ J and χ∈ I.
	
	 Let H_i=⟨ g_i⟩ for i=1,2 and H=H_1H_2. If h∉ H, then there is a character χ∈ (G/H)^* such that χ(h)≠ 1. Choose a character α∈ H^* such that α(g_1)≠ 1 and α(g_2)≠ 1. Since the restriction map G^*→ H^* is surjective, there is a character ϕ∈ G^* such that ϕ_H=α.  Therefore, we have χ(g_i)ϕ(g_i)=ϕ(g_i)≠ 1 for all i=1,2. Hence χ∈ I, ϕ∈ J and ϕχ∈ J.
		
		Now we suppose that h∈ H. The inclusion relation between subgroups H_1,H_2 and H is one of the following forms:
		
			
  * H_2≤ H_1;
			
  * H_1 and H_2 are distinct and their intersection H_1∩ H_2 is a proper subgroup in both H_1 and H_2.
		
		
		In the former case, we have H=H_1. Let m be the order of g_1. Since o(h)≥ 3 and h∈ H, we have m≥ 3. Let α∈ H^* be the character such that α(g_1) is a m-th primitive root of unit. If o(g_2)≠ 2, we can take ϕ=χ=α such that ϕ∈ J,χ∈ I and that ϕχ=α^2∈ J; if o(g_2)=2, then m≥4, we can take ϕ=α,χ=α^2 such that ϕ∈ J,χ∈ I and that ϕχ=α^3∈ J.
		
		In the latter case. Let m_i be the order of g_i. We can write h=g_1^ag_2^b for some integers a and b. Since o(h)≥ 3, we have max{o(g_1^a),o(g_2^b)}≥ 3. Now suppose o(g_1^a)≥ 3, so m_1≥ 3. If H_1∩ H_2={1}, then there are characters α_i∈ H^* such that α_i(g_j)=e^δ_ij2π√(-1)/m_j. We can take ϕ=α_1α_2 and χ=α_1, it is easy to verify that ϕ∈ J,χ∈ I and ϕχ∈ J. If H_1∩ H_2≠{1}. Let α be a character of H such that α(g_1)=e^2π√(-1)/m_1, and let n be the smallest integer such that g_2^n∈ H_1∩ H_2, by assumption n≥2. We can choose α such that α(g_2)^n=e^2π√(-1)/d, where d=m_2/n, by assumption d≥ 2, so we have m_2≥ 4 and α(g_2)≠ 1. Take ϕ=χ=α, it is easy to verify that ϕ∈ J,χ∈ I and ϕχ∈ J.
		
	
	

§ THE BOUND OF |AUT_ℚ(X)|

	 
		Let X be a minimal projective threefold of general type with only Gorenstein quotient singularities, assume that it is of maximal Albanese dimension. Then |Aut_ℚ(X)|≤ 6. Moreover, if X is smooth and K_X is ample, |Aut_ℚ(X)|≤ 5.
	
	
	For the proof of the above theorem we need the following lemma.
	  
		Let X be a threefold as in Theorem <ref>, and set G=Aut_ℚ(X). Then we have:
		
			
  * the Albanese map a_X X→ A_X of X factors through the quotient map π X→X̅ and that  χ(ω_X)=χ(ω_X̅);
			
  * the quotient X/G is of general type and of maximal Albanese dimension.
	
	
	To prove (1), let σ∈ G be a nontrivial automorphism. We claim that X^σ≠∅. Let k be the smallest integer such that X^σ^k≠∅, then the quotient map 
    π':X'=X/⟨σ^k⟩⟶ X”=X/⟨σ⟩
 is étale. It follows that 
		
    χ(ω_X')=|⟨σ⟩/⟨σ^k⟩|χ(ω_X”)
 
		Since X has only quotient singularities, the spectral sequence
		
    E_1^p,q:=H^q(X,Ω̃_X^p)⇒ H^p+q(X,ℂ)

		degenerates at E_1 page <cit.>. 
		And since σ induces trivial action on H^i(X,)≅ H^i(X,)⊗, it also induces identity action on H^i(X,ω_X) for all i≥ 0. It follows that
		
    H^i(X”,ω_X”)=H^i(X,ω_X)^⟨σ⟩=H^i(X,ω_X)
       H^i(X',ω_X')=H^i(X,ω_X)^⟨σ^k⟩=H^i(X,ω_X)
 for all i≥ 0, and that 
		
    χ(ω_X)=χ(ω_X')=χ(ω_X”).

		Since X is a Gorenstein minimal threefold of general type, by the Bogomolov-Miyaoka-Yau inequality <cit.>, 0<K_X^3≤ 72χ(ω_X), we have χ(ω_X)>0. Combining this with the two equalities (<ref>)  and (<ref>) we have |⟨σ⟩|=|⟨σ^k⟩|. Therefore, X^σ≠∅.
		
		Let a_X X→ A_X be the Albanese map of X. Notice that G induces identity on H^1(X,_X); for any σ∈ G, the induced map σ̅ on A is a translation. Since X^σ≠∅ and a(X^σ)⊆ A^σ̅, σ̅ must be the identity map. Then the quotient map X⟶X̅ factors through the Albanese map a_X of X.
		
		By (1), there is a commutative diagram
		
    X    [rr]^π[dr]_a_X      X̅[dl]^a'
        A_X
 According to the universal property of Albanese map a_X, the induced map a' is the Albanese map of X̅. As the map a_X is generically finite onto its image and the quotient map π is finite, we have that X̅ is of maximal Albanese dimension. By the generic vanishing theorem (see <cit.>), for a general α∈^0(A), h^i(X̅,ω_X̅⊗ a'^*α)=0 for all i>0, and hence χ(ω_X̅)=h^0(X̅,ω_X̅⊗ a'^*α). We can see that χ(ω_X)=χ(ω_X̅), thus χ(ω_X̅)>0, and so V^0_a'(ω_X̅) contains a dense open subset of ^0(A). It follows that  the cohomology support locus 
    V_a'^0(ω_X̅):={α∈^0(A)| H^i(X̅,ω_X̅ a'^*α)≠ 0}
 generates  ^0(A).   By Theorem 2.3 in the paper <cit.>,  we have X̅ is of general type.
		
	
	]
		Set X̅:=X/G; we perform a G-equivalent resolution of the quotient map π X→X̅ to obtain the following commutative diagram
		
    Y    [r]^π̅[d]     Y̅[d]
    
    				X [r]^π   X̅

		where Y̅→X̅ is any nonsingular resolution of X̅, then G acts on Y̅×_X̅X as g·(y,x)=(y,g· x) for any y∈Y̅,x∈ X; take Y to be the equivariant resolution of singularities on the component of Y̅×_X̅X which dominants X <cit.>, and we get a generically finite map π̅ Y→Y̅ of degree |G|. By <cit.>, we have 
		
    (K_Y)=|G|(K_Y̅+B)
 
		for some effective -divisor B on Y̅. Since X has only Gorenstein canonical singularities and K_X is nef, it is a minimal model of Y; by Proposition <ref>, we have 
		
    K_X^3=Vol(K_Y),

		and by  Remark <ref>, it follows that 
		
    K^3_Y_min=(K_Y̅)≤(K_Y̅+B)

		where Y_min is a minimal model of Y̅. Combining (<ref>),(<ref>),(<ref>), K_X^3≤ 72χ(ω_X) the Miyaoka-Yau inequality for X, and 12χ(ω_Y̅)≤ K_Y_min^3 the Clifford-Severi inequality <cit.> for Y_min (the smooth model Y̅ of Y_min is of general type and has maximal Albanese dimension by (2) of Lemma <ref>) shows that 
		
    12|G|χ(ω_Y̅)≤|G|K^3_Y_min≤ K_X^3≤72χ(ω_X)

		Since the map π is finite and X has Gorenstein canonical singularities, we can see that X̅ has rational singularities (see <cit.> or <cit.>). By Remark <ref> we obtain χ(ω_Y̅)=χ(ω_X̅). From formula (<ref>) and (<ref>) we have |G|≤6. Moreover if X is nonsingular and K_X is ample, replacing the inequality K_X^3≤ 72χ(ω_X) by Yau's inequality K_X^3≤ 64χ(ω_X) <cit.>, we get |G|≤ 5.
		
	
	 
		If X is threefold isogenous to a product of curves, then its invariants satisfies K_X^3=48χ(ω_X). Thus in this case we have  |Aut_ℚ(X)|≤ 4.
	
	

§ RATIONALLY COHOMOLOGICALLY RIGIDITY FOR ALBANESE GENERAL TYPE VARIETIES

	
	We recall that a projective complex variety is said to be of maximal Albanese dimension if its Albanese map is generically finite onto its image. According to Catanese <cit.>, it is said to be of Albanese general type, if moreover, its Albanese map is not surjective. We say that a variety Y admits a higher irrational pencil if Y admits a surjective morphism with connected fibres onto a nonsingular curve D of genus g(D)≥2. Y is called Albanese primitive if it doesn't admit any higher irrational pencil.
	
	Based on the above definitions, we consider the following three classes of projective varieties.
	
	
		
  Class I Y is of Albanese general type, and there is some higher irrational pencil g Y→ D whose general fibre F is of Albanese general type.
		
  Class II Y is of Albanese general type, and for any higher irrational pencil g Y→ D, its general fibre F is of Albanese primitive.
		
  Class III Y is of Albanese primitive.	
	 
	
	If	 Y belongs to either class I or class II, let g Y→ D be a higher irrational pencil, and let F be its generic fibre.
	Let σ be an automorphism of Y such that g∘σ=g, we denote the restriction of σ on F by σ_F. The knowledge on the classification of pairs (F,σ_F) can help us to understand the classification of pairs (Y,σ). For example, σ_F=id implies σ=id. Therefore, we have an injective homomorphism
	
    Aut(Y/D)→Aut(F).

	
	The following result shows that if Y admits a fibration over a curve D with g(D)≥ 1, then its Aut_ℚ(Y) is controlled by the autmorphism group of its generic fibre.
	
		Let Y be a projective variety with χ(ω_Y)>0, and let g Y→ D be a surjective morphism with connected fibres, where D is a nonsingular curve. Assume that g(D)≥ 1. Then we have Aut_ℚ(Y)⊆Aut(Y/D). Moreover, if F is a general fibre of g, then the induced homomorphism Aut_ℚ(Y)→Aut(F) is injective.
	
	
	 To see Aut_ℚ(Y)⊆Aut(Y/D). Take σ∈_(X). Since σ acts trivially on H^2(Y,ℚ), it induces the trivial action on NS(Y)⊗ℚ, where NS(Y) is the Néron-Severi group of Y. We conclude that σ(F) is numerically equivalent to F, where F is any general fibre of g, hence σ(F) is a fibre of g. Let σ̅ D→ D be the map defined by σ̅(x) = g(σ(y)) for any y∈ g^-1(x) and x∈ D. It is sufficient to show that σ̅ is the identity map. Note that 
    g^* H^0(D,Ω^1_D)↪ H^0(Y,Ω^1_Y)
 is an injection, which implies that σ̅ induces the trivial action on H^0(D,Ω^1_D). Therefore, the quotient curve D/⟨σ̅⟩ has the same genus as D. By Hurwitz formula we obtain
		
    2g(D)-2=o(σ̅)(2g(D/⟨σ̅⟩)-2 + R)

		where R≥ 0. It follows that 2g(D)-2≥ o(σ̅)(2g(D)-2). If g(D)≥ 2, then we have o(σ̅)=1, so σ̅=id. If g(D)=1, then σ̅ is a translation on D. Since χ(ω_Y)>0, by the proof of Lemma <ref>, we have Y^σ≠∅, hence σ̅ has fixed pints and σ̅=id.
		
		Let F be a general fibre of g. Composing the inclusion Aut_ℚ(Y)↪Aut(Y/D) with the injective homomorphism Aut(Y/D)→Aut(F), we get an injective homomorphism Aut(Y/D)→Aut(F).
		
	
	Recall that for a nonsingular projective surface S which admits a fibration f S→ B of genus g≥ 2, if there is a nontrivial automorphism σ in (S/B) which induces a trivial action on H^0(S,ω_S), then g(B)≤ 1. This was proved by Cai <cit.>. The following result is a generalization of this to higher dimension. It shows that if Y admits a higher irrational pencil, then its Aut_ℚ(Y) is controlled by the subgroup of automorphisms of its general fibre F acting trivially on H^0(F,ω_F).
	
	 
		Let g Y→ D be a higher irrational pencil of a projective variety Y, and F be a general fibre of g. Let σ∈Aut(Y/D).
		
		If σ induces a trivial action on H^0(Y,ω_Y), then σ_F induces identity on H^0(F,ω_F). Moreover, if in addition χ(ω_Y)>0, then Im(Aut_ℚ(Y)→Aut(F)) consists of automorphisms of F acting trivially on H^0(F,ω_F).
	
	 Consider the induced action of σ on g_*ω_Y, we can decomposite it as g_*ω_Y=ℰ⊕ℱ with eignsubseaf ℰ of eignvalue =1 and direct sum of eignsubsheaves ℱ of eignvalues ≠ 1. Let ℰ'⊂ g_*ω_Y be the subsheaf generated by global sections of g_*ω_Y. Since σ acts trivially on H^0(Y,ω_Y), we have ℰ'⊆ℰ. Therefore, h^0(D,ℰ)=h^0(D,g_*ω_Y) and hence h^0(D,ℱ)=0. Applying the Riemann-Roch formula to ℱ we obtain
		
    (ℱ)+r(1-g(D))=-h^1(D,ℱ)≤ 0.

		The semi-positivity of g_*ω_Y⊗ω^-1_D imples that 
    (ℱ)-2r(g(D)-1)=(ℱ⊗ω^-1_D)≥ 0
 By the assumption that g Y→ D is a higher irrational pencil, we get g(D)≥ 2. Combining the two above inequalities, we have r=0 and hence ℱ=0. Note that the natrual map g_*ω_Y⊗ℂ(p)→ H^0(F,ω_F) is an isomorphism, where F=g^-1(p) for a general point p∈ D, it follows that σ_F induces a trivial action on H^0(F,ω_F). 
		
		By the proof of Lemma <ref>, g(D)≥ 2 implies that Aut_ℚ(Y)⊆Aut(Y/D). Hence for all σ∈Aut_ℚ(Y), σ_F induces the trivial action on H^0(F,ω_F).
		
	
	To apply induction, we now consider the following situation:
	
		
  Situation (*)Let Y be a projective variety of Y≥ 3. Suppose there are a seqence higher irrational pencils g_j F_j→ D_j such that each F_j+1 is a general fibre of g_j for 0≤ j≤ Y-2, set F_0=Y. We define
		
    𝒲_1:=Im(Aut(F_0/D_0)→Aut(F_1))∩Aut(F_1/D_1)

		and for 2≤ j≤ Y-2
		
    𝒲_k:=Im(𝒲_k-1→Aut(F_k))∩Aut(F_k/D_k).

		For any σ∈𝒲_k, it uniquely determines a sequence of automorphisms σ̃_j∈Aut(F_j/D_j) for 0≤ j≤ k-1 such that σ̃_j+1=σ̃_j|_F_j+1, where σ̃_k=σ. We call σ the successive restriction of σ̃_0.
	
	
	
		Let Y be a projective variety as in Situation (*). Then there is no nontrivial automorphism σ∈𝒲_ Y-2 such that σ̃_0 induces trivial action on H^0(Y,ω_Y).
	
	
	 Set d:= Y. By the definition of 𝒲_ Y-2, there is a sequence of automorphisms σ̃_j∈Aut(F_j/D_j) for 0≤ j≤ d-3 such that σ̃_j+1=σ̃_j|_F_j+1, where σ̃_d-2=σ. Applying Lemma <ref> repeatly to fibration g_j, we see that σ̃_j+1 induces the trivial action on H^0(F_j+1,ω_F_j+1) for 0≤ j≤ d-3. In particular, σ̃_d-2 induces the trivial action on H^0(F_d-2,ω_F_d-2) and it belongs to Aut(F_d-2/D_d-2). Note that F_d-2 is a nonsingular surface and g(D_d-2)≥ 2, from Lemma 2.1 of <cit.>, we conclude that σ=σ̃_d-2=id. 
		
	
	
		Let X be a d-fold isogenous to an unmixed product of curves, and let 
    (C_1×…× C_d)/G
 be its minimal realization. Suppose X belongs to class I. Since X is of Albanese general type and q(X)=∑_i=1^dg(C_i), we have g(C̅_j)≥ 1 for all 1≤ j≤ d and g(C̅_i)≥ 2 for some 1≤ i≤ d. Without loss of generality, suppose g(C̅_d)≥ 2, then f_d X→C̅_d is a higher irrational pencil. Let U_d be the  complement of the branch points of the quotient map C_d→C̅_d, then for all x∈ U_d, F_x is of Albanese general type.
		
		Since F_x is isomorphic to (C_1×…× C_d-1)/K_d which is a d-1-fold isogenous to an unmxied product of curves.  For the same reason, F_x admits a higher irrational pencil. Without loss of generality, suppose g(C_d-1/K_d)≥ 2, hence g F_x→ C_d-1/K_d is the corresponding higher irrational pencil. Repeat this process, we get a sequence of higher irrational pencils as following:
		
    F_d-2[r] [d]^g_d-2   …[r]     F_2 [r] [d]^g_2    F_1 [r] [d]^g_1    X [d]^f_d
    
    				D_d-2   …    D_2      D_1     C̅_d

		where F_1≅ (C_1×…× C_d-1)/K_d and D_1=C_d-1/K_d. Each F_j+1 is a general fibre of the higher irrational pencil g_i. Therefore, F_2 is isomorphic to 
    (C_1×…× C_d-2)/K_d∩ K_d-1=C_1×…× C_d-2
 provided that K_d∩ K_d-1={1}, and we have F_j=C_1×…× C_d-j and D_j=C_d-j for 2≤ j≤ d-2. Therefore X is in Situation (*).
	
	
	
	
		Let X be d-fold isogenous to an unmixed product of curves with d≥ 3, and (C_1×…× C_d)/G be its minimal realization. Suppose g(C_i/G)≥ 1 for all 1≤ i≤ d.  If there is some 1≤ i≤ d with g(C_i/G)≥ 2 such that g(C_j/K_i)≥ 2 for some j≠ i, then Aut_ℚ(X) is trivial. 
	 
	
	 By Example <ref>, X is in Situation (*), we get a sequence of higher irrational pencils g_j F_j→ D_j for 1≤ j≤ d-2. Set F_0=X,D_0=C̅_d and g_0=f_d. We may assume 
    F_1≅ (C_1×…× C_d-1)/K_d,D_1=C_d-1/K_d  and 
       F_j≅ C_1×…× C_d-j, D_j=C_d-j for  j≥ 2,
 each F_j+1 is a general fibre of g_j. 
		
		Let τ∈Aut_ℚ(X), and let k be the maximal integer such that σ∈𝒲_k which is the successive restriction of τ. If k=d-2. Since τ acts trivially on H^0(X,ω_X), by Corollary <ref>, σ=id and hence τ=id.
		
		Now assume that  τ≠id and k<d-2. Since τ induces the trivial action on H^0(F_0,ω_F_0), by Lemma <ref>, we know that  σ̃_j induces identity on H^0(F_j,ω_F_j) and σ̃_j∈Aut(F_j/D_j) for 1≤ j≤ k. In particular, we have σ'=σ̃_k|_F_k+1 induces the trivial action on H^0(F_k+1,ω_F_k+1) and σ'∉Aut(F_k+1/D_k+1).
		
		Step 1. σ' induces an automorphism of D_k+1. 
		
		Let U_k+1' and U_k+1 be Zariski open subsets of D_k+1 and C̅_d-k-1=C_d-k-1/G, respectively, such that the corresponding map ρ U_k+1'→ U_k+1 is unramified. Set W_y:=g_k+1^-1(y) where y∈ U_k+1' and x=ρ(y). Let f_d-k-1 X→C̅_d-k-1 be the fibration induced by the natrual projection, then there is a commutative diagram
		
    F_k+1[r]^η[d]^g_k+1    X[d]^f_d-k-1
    
    				D_k+1[r]^ρ   C̅_d-k-1

		where η is the natrual embedding. By Lemma <ref>, f_d-k-1∘τ=f_d-k-1 and hence 
		
    f_d-k-1(η(σ'(W_y)))=f_d-k-1(τ(η(W_y)))=f_d-k-1(η(W_y))=ρ(y)=x

		It follows that σ'(W_y) is a fibre of g_k+1 for any y∈ U_k+1', then σ' induces an automorphism σ̅ of D_k+1. 
		
		Step 2. ρ factors through the quotient map D_k+1→ D_k+1/⟨σ̅⟩.
		
		By assumption σ̅≠id, otherwise g_k+1∘σ'=g_k+1. Consider the induced action of σ̅ on H^0(D_k+1,ω_D_k+1), we have a decomposition 
    H^0(D_k+1,ω_D_k+1)=V⊕ V',
 where V is the subspace with eignvalue =1, V' is the direct sum of subspaces with eignvalues ≠ 1. By the definition of σ̅ we know that: for any x∈ D_k+1, σ'(W_x)=W_σ̅(x) where W_x=g_k+1^-1(x). Since f_d-k-1∘τ = f_d-k-1 we have
		
    f_d-k-1(η(W_σ̅(x)))=f_d-k-1(η(σ'(W_x)))=f_d-k-1(τ(η(W_x))=f_d-k-1(η(W_x)).

		The commutative diagram <ref> shows that 
    for any  y∈ W_x we have f_d-k-1(η(y))=ρ(g_k+1(y))=ρ(x).
 From the formula (<ref>) we know that 
		
    f_d-k-1(η(y))=f_d-k-1(η(σ'(y)))=ρ(σ̅(x)),

		hence that ρ∘σ̅=ρ. Set D̅_k+1=D_k+1/⟨σ̅⟩, hence the morphism ρ factors through the quotient map D_k+1→D̅_k+1.
		
		Step 3.
		
		By step 2, g(D̅_k+1)≥ g(C̅_k+1)≥ 1 and V ≥ 1. Since σ̅≠id, V'≥ 1. By step 1, σ' induces the trivial action on 
		H^0(F_k+1,ω_k+1). Note that F_k+1≅ C_1×…× C_d-k-1 for k>1 and F_1≅ (C_1×…× C_d-1)/K_d. Let h_j F→ D_j be the fibration induced by the natrual projection, then we have an injection 
    h_1^*H^0(D_1,ω_D_1)∧…∧ g_k+1^*H^0(D_k+1,ω_D_k+1)→ H^0(F_k+1,ω_F_k+1)

		Therefore σ' acts trivially on h_1^*H^0(D_1,ω_D_1)∧…∧ g_k+1^*H^0(D_k+1,ω_D_k+1). Since the induced action of σ' on g_k+1^*H^0(D_k+1ω_D_k+1) can be identified as the induced action of σ̅ on H^0(D_k+1,ω_D_k+1), the induced action of σ' on g_k+1^*H^0(D_k+1,ω_D_k+1) has two different eign-subsapces, which contradicts to the fact that σ' induces identity on 
    h_1^*H^0(D_1,ω_D_1)∧…∧ g_k+1^*H^0(D_k+1,ω_D_k+1).
 Hence τ=id.
	
	

§  AUT_ℚ(X) FOR THREEFOLDS ISOGENOUS TO AN UNMIXED PRODUCT OF CURVES

	
	In this section, we focus on the threefolds isogenouse to an unmixed product of curves with maximal Albanese dimension. Our main main result is the following.
	
	
		Let X be a threefold isogenous to an unmixed product of curves, and let (C_1× C_2× C_3)/G be its minimal realization. Suppose g(C_i/G)≥ 1 for all 1≤ i≤ 3. The kernel of group homomorphism ψ_i G→Aut(C_i) will be denoted by K_i. Then the following cases occur
		 
			
  * if there is a pair (i,j) with j≠ i such that g(C_i/G)≥ 2 and g(C_j/K_i)≥ 2, then Aut_ℚ(X) is trivial;
			
  * if for any 1≤ i≤ 3 with g(C_i/G)≥ 2, g(C_j/K_i)=1 for all j≠ i, then Aut_ℚ(X)≅(ℤ_2)^k where k=0,1;
			
  * if for all 1≤ i≤ 3, g(C_i/G)=1, and suppose that the group G is an abelian group and K_i is a cyclic group for all 1≤ i≤ 3, then  Aut_ℚ(X)≅ (ℤ_2)^k where k=0,1,2.
	  
	
	

 §.§ The case of Albanese general type

	Let X be a threefold isogenous to an unmixed product of curves, and let (C_1× C_2× C_3)/G be its minimal realization. Suppose g(C_i/G)≥ 1 for all 1≤ i≤ 3 and there is some 1≤ i≤ 3 such that g(C_i/G)≥ 2. Let F be a general fibre of the fibration f_i X→C̅_i induced by the natrual projection. Fix an element σ∈Aut_ℚ(X). Then the following properties are satisfied:
	
		
  * X is of Albanese general type and f_i is a higher irrational pencil.
		
  * F is a surface isogenous to an unmixed product of curves with maximal Albanese dimension.
		
  * f_i∘σ=f_i ( Lemma <ref>).
		
  * σ induces trivial action on H^0(F,ω_F) (Lemma <ref>).
	
	
	According to the classification of projective varieties of general type of maximal Albanese dimension in Section <ref>. We can divide X into three classes.
	
	
		
  Class I There is a pair (i,j) with j≠ i such that g(C_i/G)≥ 2 and g(C_j/K_i)≥ 2.
		
  Class II  For any 1≤ i≤ 3 with g(C_i/G)≥ 2, g(C_j/K_i)=1 for all j≠ i
		
  Class III For all 1≤ i≤ 3, g(C_i/G)=1.
	 
	
	 	
		If X belongs to class I, then Aut_ℚ(X) is trivial.
	
	
	
		By Theorem <ref>, the conclusion holds.
	
	
		If X belongs to class II, then |Aut_ℚ(X)|≤ 2.
	
	
	
		By the definition of class II, we can assume g(C_3/G)≥ 2, then the fibration f X→C̅_3=C_3/G induced by the natrual projection is a higher irrational pencil, whose general fibre F≅(C_1× C_2)/K_3 is Albanese primitive. It follows that g(C_i/K_3)=1 for i=1,2. By Lemma <ref>, for any σ∈Aut_ℚ(X), f∘σ=f.
		
		Step 1. We claim that σ_F:=σ|_F induces trivial action on H^0(F,Ω^1_F).
		
		Note that 
		
    H^0(F,Ω^1_F)=H^0(C_1,Ω^1_C_1)^K_3⊕ H^0(C_2,Ω^1_C_2)^K_3
       H^0(X,Ω^1_X)=H^0(C_1,Ω^1_C_1)^G⊕ H^0(C_2,Ω^1_C_2)^G⊕ H^0(C_3,Ω^1_C_3)^G

		From 1=g(C_i/K_3)≥ g(C_i/G)≥ 1 we know g(C_i/K_3)=g(C_i/G)=1, hence 
    H^0(C_i,Ω^1_C_i)^K_3=  H^0(C_i,Ω^1_C_i)^G
 for i=1,2.  Let j F→ X be the natrual embedding. Based on the equalities of dimensions mentioned above, we know that 
    j^* H^0(X,Ω^1_X)→ H^0(F,Ω^1_F)
 is a surjective homomorphism. Consider the induced actions of σ and σ_F on H^0(X,Ω^1_X) and H^0(F,Ω^1_F), respectively, there is a commutative diagram
		
    H^0(X,Ω^1_X) [r]^j^*[d]^σ^*    H^0(F,Ω^1_F) [d]^σ_F^*
    
    			H^0(X,Ω^1_X) [r]^j^*    H^0(F,Ω^1_F)

		For any v∈ H^0(F,Ω^1_F) there exists u∈ H^0(X,Ω^1_X) such that v=j^*u, hence 
    σ_F^*v=σ_F^*j^*u=j^*σ^*u=j^*u=v.
 So σ_F^* acts trivially on H^0(F,Ω^1_F). According to Lemma <ref>, σ_F^* induces trivial action on H^0(F,ω_F). Then we have an injective group homomorphism 
		
    Aut_ℚ(X)↪Aut_d(F),  σ↦σ_F,

		where Aut_d(F) is the subgroup of automorphisms of F acting trivially on H^0(F,Ω^j_F) for all j=1,2.
		
		Step 2. We claim that |Aut_d(F)|≤ 2.
		
		Note that F is a minimal surface of general type and of maximal Albanese dimension. Using the same method of Lemma <ref>, we can show that the Albanese map a_F F→ A_F factors through the the quotient map F→ F'=:F/Aut_d(F) and χ(ω_F)=χ(ω_F')>0. Let S be the minimal smooth model of F', by G. Xiao's result |Aut_d(F)|K_S^2≤ K_F^2 <cit.>. Combining this with the Severi inequality 4χ(ω_S)≤ K_S^2, the Bogomolov-Miyaoka-Yau inequality K_F^2≤ 9χ(ω_F) and χ(ω_F)=χ(ω_F')=χ(ω_S), we conclude that |Aut_d(F)|≤ 2, and hence |Aut_ℚ(X)|≤ 2.
	
	

 §.§ Automorphisms of X descended from ()
 Throughout the rest of this section, we assume that g(C_i/G)=1 for all 1≤ i≤ 3 and G is an abelian group. Set Lf(X)=(G× G× G)/KΔ_G where K:=K_1× K_2× K_3, K_i=Ker(G→Aut(C_i)) and Δ_G the diagonal subgroup of G× G× G.
	
		With the above notations, we have an injective group homomorphism 
    Lf(X)↪(X).

	
	
	Recall that homomorphism ψ_i G→(C_i) is given by the G-action on C_i for each 1≤ i≤ 3, then we get a homomorphism 
    ψ:=ψ_1×ψ_2×ψ_3 G× G× G→().
 Since K_i=(ψ_i) for 1≤ i≤ 3, we can see that K=(ψ). As X is the quotient of  under the action of ψ(Δ_G), it follows that
		
    Aut(X)≅ N(ψ(Δ_G))/ψ(Δ_G),
 
		where N(ψ(Δ_G)) the normalizer of ψ(Δ_G) in (). Since G is abelian and ψ is a group homomorphism,
		
    ψ(h g)=ψ(g)ψ(h),  for any  g∈Δ_G and h∈ G× G× G,
 hence ψ(G× G× G)⊆ N(ψ(Δ_G)).  Composing ψ with the qoutient map N(ψ(Δ_G))→ N(ψ(Δ_G))/ψ(Δ_G) and the isomorphism (<ref>), we have a group homomorphism 
    G× G× G→(X)
 with kernel KΔ_G, which induces an injective homomorphism 
    (G× G× G)/KΔ_G↪Aut(X).

		
	
	Note that we have an injective homomorphism 
    j  (G× G× G)/K↪()
 and that ((G× G× G)/K)/Δ_G=Lf(X). For any g∈Lf(X), there exists some g̃∈ (G× G× G)/K such that g̃Δ_G=g, we call the image j(g) of g̃ in Aut() a lifting of g. The following proposition shows that numerically trivial automorphism of X can be lifted to ().
	
	 
		Let σ be an automorphism of X satisfies the property: for all 1≤ i≤ 3,  f_i∘σ=f_i, where f_i X→ C_i/G is a fibration induced by the natural projtection, then there is an injective group homomorphism ⟨σ⟩↪Lf(X).
		
		In particular, we have an injective homomorphism 
    Aut_ℚ(X)↪Lf(X).
 
	
	 Set C̅_i=C_i/G. Since ()/G is the minimal realization of X, we have K∩Δ_G={1}, and we can identify Δ_G as its image in the quotient group (G× G× G)/K under the quotient map. Therefore, we have an injective homomorphism 
    Δ_G↪ G/K_1× G/K_2 × G/K_3.
 Consider the action of G/K_1× G/K_2 × G/K_3 on C_1× C_2× C_3, there is a commutative diagram:
		
    C_1× C_2× C_3[rr]^π[rd]^φ       X [ld]^φ'
       C̅_1×C̅_2×C̅_3

		where C̅_1×C̅_2×C̅_3=(C_1× C_2× C_3)/G/K_1× G/K_2 × G/K_3. Hence 
    Gal(φ)≅ G/K_1× G/K_2 × G/K_3,
       Gal(π)≅Δ_G.
It follows that 
    Gal(φ')=Gal(φ)/Gal(π)≅ (G/K_1× G/K_2 × G/K_3)/Δ_G≅Lf(X).

		Set H:=⟨σ⟩ and Y:=X/H. By assumption, f_i∘σ=f_i for all 1≤ i≤ 3. Note that φ'=f_1× f_2× f_3, it follows that φ' factors through the quotient map φ” X→ Y, 
    X [rr]^φ”[rd]^φ'       Y [ld]
       C̅_1×C̅_2×C̅_3

		Theorefore H≅Gal(φ”)◃Gal(φ')≅Lf(X), and thus there is an injective homomorphism H↪Lf(X).
		
		By lemma <ref>, take any σ∈Aut_ℚ(X), f_i∘σ=f_i for all 1≤ i≤ 3. According to the argument above, there is an injective homomorphism Aut_ℚ(X)↪Lf(X).
	
	
	 
		For irregular surfaces which is not of maximal Albanese dimension, Cai and Liu find one S surface isogenous to a product of curves with q(S)=1 and Aut_ℚ(S)≅_4 <cit.>, a generator of this group can't lift to an automorphism of the product of curves associated to the minimal realization of  S.   
	
	
	

  §.§.§ Representations of Lf(X)
 Form now to the end of this section, we fix an algebaric data 
    𝒜=(G,K_1,K_2,K_3,_1,_2,_3)
 for some threefold X isogenous to an unmixed product of curves. 
	
		A linear character χ_1×χ_2×χ_3 of group G× G× G is called admissible character for 𝒜 if it satisfies the following conditions:
		
			
  * K_i⊆(χ_i) for all 1≤ i≤ 3;
			
  * if χ_i≠ 1_G, then χ_i(σ)≠ 1 for some σ∈Σ_i;
			
  * χ_1χ_2χ_3=1_G.
		The number of characters χ_i such that χ_i≠ 1_G of an admissible character χ_1×χ_2×χ_3 is called the weight. Denote the set of all admissible characters of weight 3 (resp. weight 2)  by A_3 (resp. A_2), and set A=A_3∪ A_2. 
	
	Let χ_1×χ_2×χ_3 be an admissible characater for 𝒜. Recall that Lf(X)=(G× G× G)/KΔ_G. Since the conditions (1) and (3) in Definition <ref> implies that KΔ_G⊆(χ_1×χ_2×χ_3), the admissible character χ_1×χ_2×χ_3 can be regard as a linear character of the group Lf(X). We next consider the actions of Lf(X) on H^i(X,) for 1≤ i≤ 3 given by
	
    ρ_iLf(X)→(H^i(X,)),   g↦ [ω↦(g^-1)^*ω].
 
	
	
		Under the above notations. We have the following properties:
		
			
  * The representation ρ_1 is trivial;
			
  * and for any ψ∈Lf(X)^* and i=2,3, the character space H^i(X,ℂ)^ψ under the representation ρ_i is non-zero  if and only if ψ is an admissible character. 
		  Moreover, there is a filtration of subgroups of Lf(X)
		
    (ρ_3)⊂(ρ_2)⊂(ρ_1)=Lf(X).

		Therefore, Aut_ℚ(X)=(ρ_3)=(∩_ψ∈ A(ψ))/KΔ_G.
	
	 Restricting the G-action of  on C_i we get representations 
    φ_i G→(H^1(C_i,)).
 By Theoerm <ref> we have 
    H^1(C_i,)=H^1(C_i,)^1_G⊕(⊕_χ∈ I_iH^1(C_i,)^χ)
 where I_i={χ∈ G^*|χ(σ)≠1  for some σ∈Σ_i  and K_i⊂(χ)}. 
		
		By the Künneth theorem of the cohomology of product spaces, we can see that 
		
    H^1(C_1× C_2× C_3,ℂ)   = ⊕_1≤ i<j≤3,
     k≠ i,jH^1(C_i,ℂ)⊗ H^0(C_j,ℂ)⊗ H^0(C_k,ℂ)
    
    			H^2(C_1× C_2× C_3,ℂ)   =W_2⊕( ⊕_1≤ i<j≤3,
     k≠ i,jH^1(C_i,ℂ)⊗ H^1(C_j,ℂ)⊗ H^0(C_k,ℂ))
    
    			H^3(C_1× C_2× C_3,ℂ)   =W_3⊕(H^1(C_1,ℂ)⊗ H^1(C_2,ℂ)⊗ H^1(C_3,ℂ))

		where 
    W_2=⊕_1≤ i<j≤3,
     k≠ i,jH^2(C_i,ℂ)⊗ H^0(C_j,ℂ)⊗ H^0(C_k,ℂ),
       W_3=⊕_1≤ i,j≤3H^2(C_i,ℂ)⊗ H^1(C_j,ℂ)⊗ H^0(C_k,ℂ).

		Since H^i(X,)≅ H^i(,)^G, we obtain following decompositions:
		
    H^1(X,ℂ)   =⊕_1≤ i<j≤3,
     k≠ i,jH^1(C_i,ℂ)^1_G⊗ H^0(C_j,ℂ)⊗ H^0(C_k,ℂ) 
    
    			H^2(X,ℂ)   =W_2⊕(⊕_1≤ i<j≤3,
     k≠ i,j⊕_χ∈ I_i^0∩ I_j^0H^1(C_i,ℂ)^χ⊗ H^1(C_j,ℂ)^χ̅⊗ H^0(C_k,ℂ))
    
    			H^3(X,ℂ)   =W'_3⊕(⊕_χ_i∈ I_i^0,1≤ i≤3,
    χ_1χ_2χ_3=χ_0H^1(C_1,ℂ)^χ_1⊗ H^1(C_2,ℂ)^χ_2⊗ H^1(C_3,ℂ)^χ_3)

		where W'_3=⊕_1≤ i,j≤3H^2(C_i,ℂ)⊗ H^1(C_j,ℂ)^1_G⊗ H^0(C_k,ℂ)
		here I_i^0=I_i∪{1_G} for each 1≤ i≤ 3. From formula (<ref>), Lf(X)=(ρ_1). By formula (<ref>),
		
    H^2(X,)^χ×χ̅× 1_G=H^1(C_1,)^χ H^1(C_2,)^χ̅ H^0(C_3,)
       H^3(X,)^χ×χ̅× 1_G=H^1(C_1,)^χ H^1(C_2,)^χ̅ H^1(C_3,)^1_G
       H^3(X,)^χ_1×χ_2×χ_3=H^1(C_1,)^χ_1 H^1(C_2,)^χ_2 H^1(C_3,)^χ_3

		where χ×χ̅× 1_G∈ A_2  and χ_1×χ_2×χ_3∈ A_3. Therefore, 
    H^i(X,ℂ)^ψ≠ 0 if and only if ψ∈ A.
 Compare formula (<ref>) and formula (<ref>), we see that (ρ_3)⊂(ρ_2). By Proposition <ref>, Aut_ℚ(X)=(ρ_3)=(∩_ψ∈ A(ψ))/KΔ_G.
		
	
	

 §.§ Translate to a group-theoretic problem

	By Lemma <ref>, the numerically trivial automorphism group Aut_ℚ(X) can be computed from the algebraic data 𝒜=(G,K_1,K_2,K_3,_1,_2,_3). The rest problem is determining the set A of admissible characters for 𝒜. Now we fix some σ_i∈Σ_i for each 1≤ i≤ 3, and consider the 7-tuple (G,K_1,K_2,K_3,σ_1,σ_2,σ_3). In general we can consider the following datum.
	
		Let G be a finite abelian group with the identity 1. Given three cyclic subgroups K_1,K_2,K_3 of G and three nontrivial elements σ_1,σ_2,σ_3 in G. A 7-tuple (G,K_1,K_2,K_3,σ_1,σ_2,σ_3) is called a qausi algebraic data if it satisfies the following conditions:
		
			
  * K_i∩ K_j = {1} for each 1≤ i<j≤3;
			
  * σ_i∉ K_i for all 1≤ i≤ 3;
			
  * ⟨σ_1K_1⟩∩⟨σ_2K_2⟩∩⟨σ_3K_3⟩={1}
		
		The condition (3) will be called the freeness condition.

		We say that a quasi algebraic data (G,K_1,K_2,K_3,σ_1,σ_2,σ_3) is induced from an algebraic data 𝒜=(G,K_1,K_2,K_3,_1,_2,_3) if σ_i∈Σ_i for all 1≤ i≤ 3. 

		We define the admissible set relative to σ_1,σ_2,σ_3 as following 
    A(σ_1,σ_2,σ_3):={χ_1×χ_2×χ_3∈ A| χ_i(σ_i)≠ 1  if χ_i≠ 1_G, 1≤ i≤ 3}
 
	
	Since K_i⊆Ker(χ_i) for all 1≤ i≤ 3, χ_i is a character of the quotient group G/K_i for all 1≤ i≤ 3. Hence the set A(σ_1,σ_2,σ_3) depends only on the cosets σ_1K_1,σ_2K_2,σ_3K_3. For this reason, we define a equivalence between the set of quasi algebraic datum induced from a fixed algebraic data 𝒜: 
	
    (G,K_1,K_2,K_3,σ_1,σ_2,σ_3)≡ (G,K_1,K_2,K_3,σ_1',σ_2',σ_3')σ_iK_i=σ_i'K_i  for all i

	So equivalent quasi algebraic datum have the same admissible set A(σ_1,σ_2,σ_3). 
	
	Set G(σ_1,σ_2,σ_3):=∩_ψ∈ A(σ_1,σ_2,σ_3)Ker(ψ). Since for each χ_1×χ_2×χ_3∈ A(σ_1,σ_2,σ_3), KΔ_G⊆(χ_1×χ_2×χ_3) and thus KΔ_G≤ G(σ_1,σ_2,σ_3). By Proposition <ref>, there is an injective homomorphism 
    Aut_ℚ(X)↪ G(σ_1,σ_2,σ_3)/KΔ_G.
 Theorem <ref>, (3) follows  from the following result.
	
		Under the above notations. Then the group G(σ_1,σ_2,σ_3)/KΔ_G  is a 2-elementary abelian group.
	
	 Given an element (τ_1,τ_2,τ_3)∈ G(σ_1,σ_2,σ_3), let d be the smallest integer such that (τ_1^d,τ_2^d,τ_3^d)∈ KΔ_G. We need to show that d≤ 2.
		
		Set τ_1'=τ_1τ_3^-1,τ_2'=τ_2τ_3^-1, we have (τ_1',τ_2',1)KΔ_G=(τ_1,τ_2,τ_3)KΔ_G. Observe that if (g_1,g_2,1)∈ KΔ_G, then g_1∈ K_1K_3 and g_2∈ K_2K_3.  Let d_i3 be the smallest integer such that τ_i'^d_i3∈ K_iK_3, then it is easy to see that d=[d_13,d_23] for i=1,2. Thus there are three integers d_12,d_13,d_23 such that any two of them have smallest common multiple d, i.e., d_ij is the smallest integer such that (τ_iτ_j^-1)^d_ij∈ K_iK_j for each 1≤ i<j≤ 3.
		
		If d_13≥ 3, then τ_1'∉ K_1K_3. Consider the set of characters 
    I={χ∈ (G/K_1K_3)^*|χ(τ_1')≠ 1}.
 Applying the property P_1,3 of Lemma <ref>, 
    χ_1χ×χ_2×χ_3χ∈ A(σ_1,σ_2,σ_3)

		for some χ∈ I and χ_1×χ_2×χ_3∈ A(σ_1,σ_2,σ_3). However, this shows that χ_1(τ_1')χ_2(τ_2')=χ_1(τ_1')χ(τ_1')χ_2(τ_2')=1, so we get χ(τ_1')=1, which contradicts to χ(τ_1')≠ 1. So we conclude that d_13≤ 2. For the same reason, we can prove that d_13,d_23≤ 2. Thus we get d≤ 2.
		 
	
	
		If X belongs to class III, then Aut_ℚ(X)≅ (ℤ_2)^k for k=0,1,2.
	
	
		By Remark <ref>, the order of Aut_ℚ(X) is at most 4, and by Theorem <ref>, Aut_ℚ(X)≅ (ℤ_2)^k for k=0,1,2.
		
	
	

  §.§.§ Configurations of qausi algebraic datum
Given a quasi algebaric data 𝒜. To construt admissible characters in A(σ_1,σ_2,σ_3), we need to inverstigate the poset structure of subgroups K_i,K_jK_k, H:=K_1K_2K_3 of G and the incidence relation between σ_i and these subgroups.
	
		Let 𝒫(𝒜):={{1},K_i,K_jK_k,H|1≤ i≤3,1≤ j<k≤ 3} be a set of subgroups of G, the partial oder on 𝒫(𝒜) is given by U≤ V iff U⊆ V for U,V∈𝒫(𝒜). Set 𝒫(𝒜)^*:=𝒫(𝒜)∪{G} with the partial order U≤ G for all U∈𝒫(𝒜). We say that 𝒜 is of general type if all the subgroups K_i,K_jK_k,H are distinct, otherwise it is called special type. The configuration associated to the algebraic data 𝒜 is a map:
		
    C^𝒜{1,2,3}→𝒫(𝒜)^*

		defined by C^𝒜(i) being the smallest element U∈𝒫(𝒜)^* containing σ_i.
		
		If 𝒜 is of general type, we call 𝒜 reduced if  
    C^𝒜(i)∈{K_j,K_k,K_jK_k,G}
 for {i,j,k}={1,2,3}.
		
		If 𝒜 is of special type with H=K_iK_j, we call 𝒜 reduced if 
    C^𝒜(i)∈{K_j,G},C^𝒜(j)∈{K_i,G} and C^𝒜(k)∈{K_i,K_j,H,G}

		for {i,j,k}={1,2,3}.
	
	
	We can visualize a poset as a graph whose vertexes are subgroups, two vertexes adjoint one edge iff the corresponding subgroups have strict order relation, and we draw this graph from top to bottum with vertexes corresponding to subgroups from H to {1}. The following poset 𝒫(𝒜) is the bigest one when 𝒜 is of general type, i.e., all subgroups K_i,K_jK_k, H are distinct.
	
	

		
			(0,0) circle (2pt) node[right]K_1 – (-0.25,1)circle (2pt) node[left]K_1K_2– (1,2)circle (2pt) node[above]H
			(-0.25,1) –(1,0) circle (2pt) node[right]K_2–(2.25,1) node[right]K_2K_3(0,0)–(1,1)circle  (2pt) node[right]K_1K_3–(2,0) (1,1)–(1,2)
			(1,2)–(2.25,1) circle (2pt)–(2,0)circle (2pt) node[right]K_3
			(0,0)–(1,-0.5) circle(2pt) node[below]{1} (1,0)–(1,-0.5) (2,0)–(1,-0.5);
		
	
  
	
	If 𝒜 is of special type with H=K_1K_2, then H is isomorphic to K_1× K_2, and K_3 is a diagonal subgroup of H, i.e., K_1∩ K_3={1}, K_2∩ K_3={1}. There are three cases of 𝒜, they are:
		
  * K_1K_3,K_2K_3 are both proper subgroups of H;
		
  * H=K_2K_3 and K_1K_3 is a proper subgroup of H;
		
  * H=K_1K_3=K_2K_3.
	 
	We draw the corresponding graphes as bellow respectively.
	

		
			(0,0) circle (2pt) node[right]K_1–(0.5,0.5)circle (2pt)node[left]K_1K_3–(1,1)circle (2pt)node[above]H
			(0.5,0.5)–(1,0)circle (2pt)node[right]K_3–(1.5,0.5)circle (2pt)node[right]K_2K_3–(2,0)circle (2pt)node[right]K_2
			(1,1)–(1,0) (1,1)–(1.5,0.5) 
			(0,0)–(1,-0.5) circle(2pt) node[below]{1} (1,0)–(1,-0.5) (2,0)–(1,-0.5);
		
		
			(0,0) circle (2pt) node[right]K_1–(0.5,0.5)circle (2pt)node[left]K_1K_3–(1,1)circle (2pt)node[above]H
			(0.5,0.5)–(1,0)circle (2pt)node[right]K_3
			(1,1)–(1,0) (1,1)–(2,0) circle (2pt)node[right]K_2
			(0,0)–(1,-0.5) circle(2pt) node[below]{1} (1,0)–(1,-0.5) (2,0)–(1,-0.5);
		
		
			(0,0) circle (2pt) node[right]K_1–(1,1)circle (2pt)node[above]H (1,1)–(1,0)circle (2pt)node[right]K_3
			(1,1)–(2,0) circle (2pt)node[right]K_2
			(0,0)–(1,-0.5) circle(2pt) node[below]{1} (1,0)–(1,-0.5) (2,0)–(1,-0.5);
		
	
 
	For each class of quasi algebraic datum, we want to choose a simple representative which is a reduced quasi algebraic data, we have the following proposition.
	
		Each algebraic data 𝒜 is equivalent to a reduced algebraic data 𝒜'.
	
	 Suppose 𝒜 is of general type and not reduced, then by Definition <ref>, there is some i such that C(i)^𝒜=K_iK_j or C(i)^𝒜=H, it follows that σ_i∈ K_iK_j or σ_i∈ H. If σ_i∈ K_iK_j, we can write σ_i=k_iσ_i' for some k_i∈ K_i and σ_i'∈ K_j, hence σ_iK_i=σ_i'K_i, so we can replace σ_i in 𝒜 by σ_i' and get a new algebraic data 𝒜' such that C^𝒜'(i)=K_j. Similarly for the case C^𝒜(i)=H, we will get a new algebraic data 𝒜' with C^𝒜'(i)=K_jK_k, where {i,j,k}={1,2,3}. Continoue this procedure, we will get a reduced algebraic data 𝒜' which is equivalent to the original one. The same argument works for 𝒜 being of special type.
		
	
	

  §.§.§ Construction of admissible characters

	
		Let 𝒜 be a quasi algebraic data, then the admissible set A(σ_1,σ_2,σ_3) is nonempty.
		
		Moreover, the set A(σ_1,σ_2,σ_3) satisfies the property P_i,j: for {i,j,k}={1,2,3} and g∉ K_iK_j with o(gK_iK_j)≥ 3, 
    χ_iχ×χ_jχ×χ_k∈ A(σ_1,σ_2,σ_3)

		for some χ_i×χ_j×χ_k∈ A(σ_1,σ_2,σ_3) and χ∈(G/K_iK_j)^* such that χ(g)≠ 1.
	
	
	
		By Proposition <ref>, we can assume 𝒜 is reduced. If 𝒜 is of general type. Recall that C^𝒜(i) takes values in the set {K_j,K_k,K_jK_k,G}. The proof will be divided into three steps according to the number ν of C^𝒜(i)=G.
		

  
General type ν = 2. Suppose C^𝒜(1)=C^𝒜(3)=G, then σ_1∉ K_1K_3 and  σ_3∉ K_1K_3. There is a character  ϕ∈ (G/K_1K_3)^* such that 
    ϕ(σ_1)≠ 1 and ϕ(σ_3)≠ 1.
 Thus ϕ× 1_G×ϕ∈ A(σ_1,σ_2,σ_3), which is of weight 2.

		Moreover, if C^𝒜(i)≠ G for some 1≤ i≤ 3. Suppose that 
    C^𝒜(1)=K_3 and  C^𝒜(2)=C^𝒜(3)=G.
 We have
		
    β_3(σ_1)≠1 for some β_3∈(H/K_1K_2)^*
       ϕ(σ_2)≠1 and ϕ(σ_3)≠ 1 for some ϕ∈(G/K_2K_3)^*

		Note that β_3^G(σ_2)=0 provided that σ_2∉ H, by Proposition <ref>, there is a consituent ψ_3 of β_3^G such that ϕ(σ_2)ψ_3(σ_2)≠1. It follows that 
    ψ_3×ϕψ_3×ϕ∈ A(σ_1,σ_2,σ_3),
 which is of weight 3.
		

  
General type ν=1.Suppose C^𝒜(3)=G. Since 
    C^𝒜(1)∈{K_2,K_3,K_2K_3} and  C^𝒜(2)∈{K_1,K_3,K_1K_3},
 there are nine possible choices of C^𝒜(1) and C^𝒜(2). In each choice, the constructions are similar, we just illustrate one case. Suppose 
    C^𝒜(1)=K_3,C^𝒜(2)=K_1 and C^𝒜(3)=G.
 Since σ_3∉ H, there is a character ϕ∈(G/H)^* such that ϕ(σ_3)≠1. We first take a character β_1∈(H/K_2K_3)^* with β_1(σ_2)≠ 1, as σ_3∉ H, we have β_1^G(σ_3)=0, by Proposition <ref>, there is a constituent ψ_1 of β_1^G such that 
		
    ϕ(σ_3)ψ_1(σ_3)≠1.

		Take a character ψ_3∈(G/K_1K_2)^* such that ψ_3(σ_1)≠ 1, then we obtain
		
    ϕ(σ_2)ψ_1(σ_2)ψ_3(σ_2)=β_1(σ_2)≠1,

		as σ_2∈ K_1 and K_1⊆(ϕ),K_1⊆(ψ_3). Combine (<ref>) and (<ref>), we deduce that ψ_3×ϕψ_1ψ_3×ϕψ_1∈ A(σ_1,σ_2,σ_3).
		

  
General type ν=0.We divide three cases.
		

  
Case (a): Suppose that 
    C^𝒜(1)∈{K_3,K_2K_3},C^𝒜(2)∈{K_1,K_1K_3} and C^𝒜(3)∈{K_2,K_1K_2}.
 We can take characters ψ_1∈(G/K_2K_3)^*,ψ_2∈(G/K_1K_3)^*,ψ_3∈(G/K_1K_2)^* such that ψ_i(σ_i)≠1 for all i. It is easy to verify that 
		
    ψ_2ψ_3×ψ_1ψ_3×ψ_1ψ_2∈ A(σ_1,σ_2,σ_3)

		

  
Case (b): Suppose that 
    C^𝒜(1)∈{K_3,K_2K_3},C^𝒜(2)∈{K_3,K_1K_3} and C^𝒜(3)∈{K_1,K_1K_2}.
 We can take characters ψ_1∈(G/K_2K_3)^*,ψ_3∈(G/K_1K_2)^* such that 
    ψ_1(σ_3)≠1,ψ_3(σ_1)≠1 and ψ_3(σ_2)≠1.
 Therefore 
		
    ψ_3×ψ_1ψ_3×ψ_1∈ A(σ_1,σ_2,σ_3).

		

  
Case (c): Suppose that 
    C^𝒜(1)=K_2,C^𝒜(2)=K_1 and C^𝒜(3)=K_1K_2.

		Then we can write σ_3=k_1k_2 for some k_i∈ K_i for i=1,2. Set m=o(σ_1),n=o(σ_2),m'=o(k_1) and n'=o(k_2).
 
		Since K_2K_3 and K_1K_3 are proper subgroups of H, there are characters β_1∈ (H/K_2K_3)^* and β_2∈(H/K_1K_3)^* whose restriciton on K_1 and K_2 are primitive characters, respectively. If max{m,n,m',n'}≥ 3, since K_2⊆(β_1) and K_1⊆(β_2), we can apply Lemma <ref> to the group K_1K_2 with g_1=σ_1,g_2=σ_2,h=σ_3,α_1=(β_2)_K_1K_2,α_2=(β_1)_K_1K_2, then we have 
		
    β_2^s(σ_1)≠ 1 and β_1(σ_3)β_2^s(σ_3)≠ 1     if m≥3 or m'≥3,
    β_1^s(σ_2)≠ 1 and β_1^s(σ_3)β_2(σ_3)≠ 1     if n≥3 or n'≥3,
 for some s=1,2,3,5. It follows that
		
    ψ_2^s×ψ_1×ψ_1ψ_2^s∈ A(σ_1,σ_2,σ_3)     if m≥3 or m'≥3,
    ψ_2×ψ_1^s×ψ_1^sψ_2∈ A(σ_1,σ_2,σ_3)     if n≥3 or n'≥3,
 for some s=1,2,3,5, where ψ_1,ψ_3 are characters of G whose restricition on H are equal to β_1,β_3, respectively. Now we can assume that m=n=m'=n'=2, then we can write σ_3 = σ_1σ_2. We can verify that σ_3∈⟨σ_1 K_1⟩∩⟨σ_2 K_2⟩∩⟨σ_3 K_3⟩. For this case the corresponding quasi algebraic data violates the freeness condition of Definition <ref>.
		

  
Special type.Now we assume that 𝒜 is of special type H=K_1K_2. Recall that 
    C^𝒜(1)∈{K_2,G},C^𝒜(2)∈{K_1,G} and C^𝒜(3)∈{K_1,K_2,H,G}.
 If ν=1,2, we can construct admissible character as for the general type ν=1,2. For the case C^𝒜(1)=K_2,C^𝒜(2)=K_1,C^𝒜(3)∈{K_1,K_2}, we can construct admissible character as the case (b) of 𝒜 being of general type with ν=0. So we can assume that 
    C^𝒜(1)=K_2,C^𝒜(2)=K_1 and C^𝒜(3)=H.
 Let m_1=o(σ_1),m_2=o(σ_2).  Sinces H≅ K_1× K_2, and K_3 is a subgroup such that K_1∩ K_3={1} and K_2∩ K_3={1}. We can apply Lemma <ref>, there are characters α_i∈ H^* for i=1,2 such that 
    K_1⊆(α_2),K_2⊆(α_1) and K_3⊆(α_1α_2),
 and the restriction of α_i on K_i is primitive for each i=1,2. Since σ_3∈ H, we can write σ_3=k_1k_2 for k_i∈ K_i, i=1,2. Similary as the case (c) of 𝒜 being of general type with μ=0, if max{m_1,m_2,o(k_1),o(k_2)}≥ 3, then we can apply Lemma <ref> to the group H with g_1=σ_2,g_2=σ_1,h=σ_3 and α_1,α_2, we have that either ψ_2×ψ^s_1×ψ^s_1ψ_2∈ A(σ_1,σ_2,σ_3) or ψ^s_2×ψ_1×ψ_1ψ^s_2∈ A(σ_1,σ_2,σ_3) for some integer s, where ψ_1,ψ_2 are any characters of G whose restriciton on H is equal to α_1,α_2, respectively. In the case m_1=m_2=o(k_1)=o(k_2), the corresponding quasi algebraic data violates the freeness condition of Definition <ref>.

		

  
Proof of property P_1,3. If σ_1∉ K_1K_3 and σ_3∉ K_1K_3, then we can find a character ϕ×1_G×ϕ∈ A(σ_1,σ_2,σ_3) such that
		
    ϕ∈(G/K_1K_3)^*, ϕ(σ_1)≠ 1  and ϕ(σ_3)≠ 1.
 Since o(gK_1K_3)≥ 3, we can apply lemma <ref> to the quotient group G/K_1K_3 and its three nontrivial elements σ_1K_1K_3,σ_3K_1K_3, gK_1K_3, there are characters ϕ,χ∈(G/K_1K_3)^* such that 
    ϕ(σ_1)≠1,ϕ(σ_3)≠ 1,χ(g)≠ 1,ϕ(σ_1)χ(σ_1)≠ 1 and ϕ(σ_3)χ(σ_3)≠ 1.
 It follows that ϕ×1_G×ϕ∈ A(σ_1,σ_2,σ_3),χ∈ I and ϕχ×1_G×ϕχ∈ A(σ_1,σ_2,σ_3).

		Since σ_1∉ K_1K_3 is equivalent to C^𝒜(1)≠ K_3, next step we consider the case C^𝒜(1)= K_3. By the constructions as above, there exists a character χ_1×χ_2
		×χ_3 of weight 3, i.e., χ_i≠ 1_G for all 1≤ i≤ 3. Since σ_1∈ K_3, for any χ∈ I, we have χ(σ_1)=1, hence χ_1(σ_1)χ(σ_1)=χ_1(σ_1)≠ 1. Since o(gK_1K_3)≥ 3, we can take χ∈ I such that χ(g) is a primitive o(gK_1K_3)-th root of unit, thus we have χ^2∈ I. Observe that if χ_3(σ_3)χ(σ_3)=χ_3(σ_3)χ^2(σ_3)=1, then χ(σ_3)=χ_3(σ_3)=1, which contradict to χ_3(σ_3)≠ 1. So we can choose χ∈ I such that χ_3(σ_3)χ^s(σ_3)≠ 1 for some s=1,2. It follows that χ_1χ^s×χ_2
		×χ_3χ^s∈ A(σ_1,σ_2,σ_3) for some s=1,2.
		
	
	

§ EXAMPLES

	
	Let n_1,n_2,n_3 be three positive integers. In Examples <ref>, <ref>  and <ref>, let G=⟨ e_1,e_2,e_3⟩≅_2n_1×_2n_2×_2n_3 where e_1,e_2,e_3 are generators of G of orders 2n_1,2n_2,2n_3 respectively. Let ϕ_i be the characters on G such that ϕ_i(e_j)=e^2πδ_ij√(-1)/2n_i, where δ_ij is the Kronecker symbol. For each 1≤ i≤ 3, let K_i=⟨ e_i⟩. We take the following generating vector of G/K_i
	
    _1   =(e_2K_1, e_3K_1;σ_1K_1,…,σ_1K_1_m_1),
    _2   =( e_1K_2, e_3K_2;σ_2K_2,…,σ_2K_2_m_2),
    _3   =( e_1K_3, e_2K_3;σ_3K_3,…,σ_3K_3_m_3).

	Where m_i is an integer which is divisible by o(σ_iK_i) for each 1≤ i≤ 3. So the type of _i is 
    [1;o(σ_iK_i),…,o(σ_iK_i)_m_i].
 By Riemann's existence theorem, there is an algebraic curve C_i with a G-action ψ_i G→(C_i) such that K_i=(ψ_i) whose quotient C_i/G is an elliptic curve for each 1≤ i≤3. Since the set of nontrivial stabilizers of the G-action on the C_i is Σ_i=σ_iK_i.
	
	[Aut_ℚ(X)≅_2×_2]  

		Take σ_1=e_3^n_3,σ_2=e_1^n_1 and σ_3=e_2^n_2. Thus ∩_iΣ_i={1_G}, and the 7-tuple (G,K_1,K_2,K_3,_1,_2,_3) forms an algebraic datum.  By Remark <ref>, it  determines a threefold X=()/G isogenous to an unmixed product of curves with q(X)=3. It is easy to see that the corresponding admissible sets are
		
    A_2=∅ and  A_3={ϕ_2^k_2ϕ_3^k_3×ϕ_1^k_1ϕ_3^-k_3×ϕ_1^-k_1ϕ_2^-k_2|k_1,k_2,k_3 are odd integers}.

		Let τ=(e^x_1_2e^x_2_3,e_1^x_3e_3^x_4,1)∈ G× G× G be a representative of an element of Aut_ℚ(X) where x_i∈. Then τ KΔ_G∈Aut_ℚ(X) if and only if the following equations hold:
		
    ϕ_1(e_1)^k_1x_3ϕ_2(e_2)^k_2x_1ϕ_3(e_3)^k_3(x_2-x_4)=1

		for all odd integers k_i. In particular from the two equations 
		
    ϕ_1(e_1)^x_3ϕ_2(e_2)^x_1ϕ_3(e_3)^x_2-x_4=ϕ_1(e_1)^3x_3ϕ_2(e_2)^x_1ϕ_3(e_3)^x_2-x_4=1
 
		we get ϕ_1(e_1)^2x_3=1. Since ϕ_1(e_1) is a primitive 2n_1-th root of unit, 2n_1|2x_3, thus the possible value of x_3 is n_1 or 0. Applying this argument again we have x_1=n_2,0 and x_2-x_4=n_3, 0. By Lemma <ref> we have 
    Aut_ℚ(X)=⟨(e_2^n_2,e_1^n_1,1)KΔ_G,(e_2^n_2,e_3^n_3,1)KΔ_G⟩≅_2×_2.

	
	
	[Aut_ℚ(X)≅_2]  

		Take σ_1=e_3^n_3,σ_2=e_1^n_1 and σ_3=e_1^n_1e_2^n_2,
		for the same reason as Example <ref>, the 7-tuple (G,K_1,K_2,K_3,_1,_2,_3) forms an algebraic datum. Let X be the corresponding threefold. We can see  that the corresponding admissible sets are
		
    A_2=∅ and 
       A_3={ϕ_2^k_2ϕ_3^k_3×ϕ_1^k_1ϕ_3^-k_3×ϕ_1^-k_1ϕ_2^-k_2|k_1,k_2 are odd, k_3 is even}.
 
		By Lemma <ref> we obtain 
    Aut_ℚ(X)=⟨(e_2^n_2,1,1)KΔ_G⟩≅_2.


		Take σ_1=e_3^n_3,σ_2=e_1^2 and σ_3=e_2^n_2. Similarly,  we have 
    A_2=∅ and 
       A_3={ϕ_2^k_2ϕ_3^k_3×ϕ_1^k_1ϕ_3^-k_3×ϕ_1^-k_1ϕ_2^-k_2|k_1 is arbitrary, k_2,k_3 are odd}.
 By Lemma <ref>, it follows that 
    Aut_ℚ(X)=⟨(e_2^n_2,e_3^n_3,1)KΔ_G⟩≅_2.
 
	
	
	[Product quotient with terminal singularities]  

		Let n=n_1 and m=n_2=n_3. Take σ_1=e_2^me_3^m,σ_2=e_3^m and σ_3=e_2^m. Since σ_1 K_2= σ_2 K_2, σ_1 K_3= σ_3 K_3 and σ_iK_i has fixed point of G/K_i-action on C_i for all i=1,2,3, hence (σ_1,σ_1,σ_1) has fixed points on . Therefore, the corresponding G-action on  is not free.  So the quotient X_m,n=()/G has singularities of type 1/2(1,1,1) which is not Gorenstein. Even though, cohohomolgies H^*(X_m,n,) of X_m,n can be identified with H^*(,)^G. The same arguments in Section <ref> apply to X_m,n, we can see that Lemma <ref> still holds for X_m,n. The  corresponding admissible sets are
		
    A_2={ϕ_2^k_2×1_G×ϕ_2^-k_2,ϕ_3^k_3×ϕ_3^-k_3×1_G|k_2,k_3  are odd} and A_3=∅.

		It follows that 
    Aut_ℚ(X_m,n)=⟨ (1,e_1,1)KΔ_G⟩≅_2n.

		For σ∈ G and 1≤ i≤3, we denote the set of points on C_i whose stabilizer is exactly ⟨σ⟩ K_i by Fix_C_i(σ). By <cit.>, we have 
		
    |Fix_C_1(e_2^me_3^m)|=4m^2,|Fix_C_2(e_3^m)|=|Fix_C_3(e_2^m)|=4mn;
       |Fix_(σ)|=4^3m^4n^2.

		Therefore, the basket of singularities of X_m,n is 
    Sing(X_m,n)={4^3m^4n^2/4m^2n·1/2(1,1,1)}={16m^2n·1/2(1,1,1)}.
 It is easy to see that K_X_m,n.c_2(X_m,n)=3·2^3∏_i=1^3(g(C_i)-1)/|G|= 24m^2n. By Riemann-Roch formula for singular varieties <cit.>, we have 
		
    χ(ω_X_m,n)=1/24K_X_m,n. c_2(X_m,n)-16m^2n·1/16= 0.
 According to the classification of threefolds with vanishing holomorphic Euler characteristic by J. A. Chen, O. Debarre and Z. Jiang <cit.>, the variety X_m,n belongs to the examples constructed by R. Lararsfeld and L. Ein <cit.>.
	
	
	[Aut_ℚ(X)≅ℤ_2 and q(X)>3]
		Let G=⟨ e_1,e_2,e_3⟩≅ℤ_2×ℤ_2×ℤ_2, K_1=K_2={1} and K_3=⟨ e_1,e_2⟩, ϕ_i is the dual character of e_i. We can take generating vectors for G as the following:
		
    _1=(e_2,e_3;e_1,e_1)
       _2=(e_1,e_3;e_2, e_2).

		Take  _3=(4· e_3K_3;e_3K_3,e_3K_3) a generating vector for G/K_3. It is easy to see that the 7-tuple (G,K_1,K_2,K_3,_1,_2,_3) forms an algebraic datum. Then  the corresponding admissible sets are
		
    A_2={ϕ_1ϕ_2×ϕ_1ϕ_2× 1_G}
       A_3={ϕ_1ϕ_2ϕ_3×ϕ_1ϕ_2×ϕ_3,ϕ_1ϕ_2×ϕ_1ϕ_2ϕ_3×ϕ_3}

		By Lemma <ref> and Corollary <ref> we know that Aut_ℚ(X)≅⟨ e_1,e_2,1⟩ KΔ_G.
		
	
	
	[Some K_i is not cyclic and Aut_ℚ(X)≅_2]  

		Let G=^4_2 with generators e_1,e_2,e_3,e_4, and let 
    K_1=⟨ e_4⟩, K_2=⟨ e_2⟩, K_3=⟨ e_1,e_3⟩.
 We take generating vectors as following 
		
    _1=(e_2K_1,e_3K_1;e_1K_1, e_1K_1),
       _2=(e_1K_2,e_4K_2;e_3K_2, e_3K_2),
       _3=(e_2K_3, e_3K_3;e_2K_3,e_2K_3).

		It is easy to see that the 7-tuple (G,K_1,K_2,K_3,_1,_2,_3) forms an algebraic datum.
		The  corresponding admissible sets are 
    A_2={ϕ_1ϕ_3×ϕ_1ϕ_3×1_G} and 
       A_3={ϕ_1ϕ_2ϕ_3×ϕ_1ϕ_3×ϕ_2,ϕ_1ϕ_2ϕ_3×ϕ_1ϕ_3ϕ_4×ϕ_2ϕ_4}
 
		It follows that Aut_ℚ(X)=⟨ (e_3,e_1,1)KΔ_G⟩≅_2.
	
	
	Let X be a threefold isogenous to a product of curves, not necessary unmixed type, with maximal Albanese dimension. Does Aut_ℚ(X)≅_2^k for some k=0,1,2?
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

§ ACKNOWLEDGEMENTS

		
		
		The author would like to thank his adivisor Jinxing Cai for constant support and encouragement, and he also would like to thank Wenfei Liu, Lei Zhang, Zhan Li and Yifan Chen for useful suggestions and discussions during the conference at Xiamen University in June, 2018. He would like to thank Xiamen University for its hospitality.

	
	
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