Automorphisms of projective manifolds

Let $(M,P\nabla_M)$ be a compact projective manifold and $Aut(M,P\nabla_M)$ its group of automorphisms. The purpose of this paper is to study the topological properties of $(M,P\nabla_M)$ if $Aut(M,P\nabla_M))$ is not discrete by applying the results that I have shown in [13] and the Benzekri's functor which associates to a projective manifold a radiant affine manifold. This enables us to show that the orbits of the connected component of $Aut(M,P\nabla_M)$ are immersed projective submanifolds. We also classify $3$-dimensional compact projective manifolds such that $dim(Aut(M,P\nabla_M))\geq 2$.


Introduction
The purpose of this paper is to study the group of automorphisms of projective manifolds.Firstly we recall the definition of (X, G) manifolds, their group of automorphisms and morphisms between (X, G)-structures.We applied the results described in the general framework of (X, G)-manifolds to the category of affine manifolds and projective manifolds.Benzekri has constructed a functor which associates to a projective manifold (M, P ∇ M ) a radiant affine manifold (B(M ), ∇ B(M) ) whose underlying topological space is M × S 1 .It enables us to show that there exists a surjective morphism between the connected component Aut(B(M ), ∇ B(M) ) 0 of the group of affine automorphisms of (B(M ), ∇ B(M) ) and the connected component Aut(M, P ∇ M ) 0 of the group of projective automorphisms of (M, P ∇ M ).
Let (M, ∇ M ) be a compact affine manifold, in [13], I have studied the relations between Aut(M, ∇ M ) and the topology of M .This enables us to show that the orbits of Aut(M, P ∇ M ) 0 are projective immersed submanifolds.In the last section, we study the automorphisms group of 2 and 3 dimensional projective manifolds.We remark that a 2-dimensional projective manifold whose group of automorphisms is not discrete is homeomorphic to the sphere, the 2dimensional projective space or the two dimensional torus.Finally we show that a 3-dimensional projective manifold (M, P ∇ M ) whose developing map is injective and such that dim(Aut(M, P ∇ M ) ≥ 2 is homeomorphic to a spherical manifold, S 2 × S 1 , or a finite cover of M is the total space of a torus bundle.
Remark that (X, G) manifolds play an important role in low dimensional topology: seven of the eight geometry of Thurston are examples of projective geometry (see Cooper and Goldman [8] p. 1220).In [12] p.17, Sullivan and Thurston note that the existence of a (X, G)-structure on every 3-manifold implies the Poincare conjecture.

(X, G)-manifolds.
A (X, G) model is a finite dimensional differentiable manifold X, endowed with an effective and transitive action of a Lie group G which satisfies the unique extension property.This is equivalent to saying that: two elements g, g ′ of G are equal if and only if their respective restriction to a non empty open subset of X are equal.
A (X, G) manifold (M, X, G) is a differentiable manifold M , endowed with an open covering (U i ) i∈I such that for every i ∈ I, there exists a differentiable map f i : U i → X which is a diffeomorphism onto its image and A (X, G) structure defined on M can be lifted to the universal cover M of M .This structure is defined by a local diffeomorphism D M : M → X.This implies that a (X, G) chart of this structure is an open subset U of M such that the restriction of D M to U is a diffeomorphism onto its image.
Let (X, G) and (X ′ , G ′ ) be two models, φ : X → X ′ a differentiable map and Φ : G → G ′ a morphism of groups such that for every g ∈ G, the following diagram is commutative: and Φ(g) • φ to f i (U i ) coincide.We will denote by Aut(X, M, G) the group of (Id G , Id X )-automorphisms of (M, X, G) and by Aut(M, X, G) 0 its connected component.It is a Lie group endowed with the compact open topology.For every element g ∈ Aut( M , X, G), the developing map defines a representation

−→ X
Remark that the group of Deck transformations that we identify to the fundamental group π 1 (M ), of M , is a subgroup of Aut( M , X, G).The restriction h M of H M to the fundamental group π 1 (M ), of M is called the holonomy representation of the (X, G) manifold (M, X, G).
The pullback p M (f ) of an element f of Aut(M, X, G), by the universal covering map, p M : M → M is an element of Aut( M , X, G) which belongs to the normalizer N (π 1 (M )) of π 1 (M ) in Aut( M , X, G).Conversely, every element g of N (π 1 (M )) induces an element A M (g) of Aut(M, X, G) such that the following diagram is commutative: The kernel of the morphism A M : N (π 1 (M )) → Aut(M, X, G) is π 1 (M ) and A M is a local diffeomorphim.We will denote by N (π 1 (M )) 0 the connected component of N (π 1 (M )), it is also the connected component of the commutator of π 1 (M ) in Aut( M , X, G).Since A M is locally invertible, it induces an isomorphism between the Lie algebra n(π 1 (M )) of N (π 1 (M )) and the Lie algebra aut(M, X, G) of Aut(M, X, G).If (M, X, G) is a compact (X, G) manifold, aut(M, X, G) is isomorphic to space of elements of G, the Lie algebra of G, which are invariant by h M (π 1 (M )).

Affine and projective structures.
Let R n be the n-dimensional real vector space.We denote by Gl(n, R) the group of linear automorphisms of R n and by Af f (n, R) its group of affine transformations.If we fix an origin 0 of R n , for every element f ∈ Af f (n, R), we can write f = (L(f ), a f ) where L(f ) is an element of Gl(R n ) and a f = f (0).The couple (R n , Af f (n, R)) is a model.A (R n , Af f (n, R)) manifold is also called an affine manifold.Equivalently, an (R n , Af f (n, R)) manifold is a ndimensional differentiable manifold M endowed with a connection ∇ M whose curvature and torsion tensors vanish identically.
Remark that the linear part L(h M ) of the holonomy representation h M of an affine manifold (M, ∇ M ) is the holonomy of the connection ∇ M .We say that the n-dimensional affine manifold (M, ∇ M ) is radiant if its holonomy h M fixes an element of R n , this is equivalent to saying that h M and L(h M ) are conjugated by a translation.
The n-dimensional real projective space RP n is the quotient of R n+1 −{0} by the equivalence relation defined by x ≃ y if and only there exists λ ∈ R such that x = λy.If x is an element of R n+1 − {0}, we will denote by [x] RP n its equivalent class.The group Gl(n + 1, R) acts transitively on RP n by the action defined by g.[x] RP n = [g.x]RP n the kernel of this action is the group H n+1 of homothetic maps.We denote by P Gl(n is also called a projective manifold.Equivalently, a projective manifold can be defined by a differentiable manifold M endowed with a projectively flat connection P ∇ M .We will denote it by (M, P ∇ M ).
The n-dimensional sphere S n is the quotient of R n+1 −{0} by the equivalence relation defined by x ≃ y if and only if there exists λ > 0 such that x = λy.Let x be an element of R n+1 − {0}, we will denote by [x] S n its equivalence class for this relation.Remark that if , is an Euclidean metric defined on R n+1 , there exists a bijection between the unit sphere S n , = {x : x ∈ R n+1 , x, x = 1} and S n defined by the restriction of the equivalence relation to S n , .There exists a map D S n : S n → RP n such that for every element The map p n is a covering, thus is the developing map of a projectively flat connection P ∇ S n defined on S n .
A p-dimensional projective submanifold (F, P ∇ F ) of the projective manifold (M, P ∇ M ) is a p-dimensional submanifold F of M endowed with a structure of a projective manifold, such that the canonical embedding i F : (F, P ∇ F ) → (M, P ∇ M ) is a morphism of projective manifolds.
Let F be the universal cover of F , we can lift i F to a projective map îF : Proposition 3.1.The group of automorphisms of the n-dimensional projective manifold S n is isomorphic to Sl(n + 1, R), the group of invertible (n + 1) × (n + 1) matrices such that for every element Proof.Let g be an element of Sl(n + 1, R).For every [x] S n ∈ S n , we write u g (x) = [g(x)] S n .Let [g] be the image of g by the quotient map Sl(n + 1, R) → P Gl(n + 1, R), we have [g] • D S n = D S n • u g .This implies that u g is an element of Aut(S n , P ∇ S n ).Suppose that u g = Id S n , it implies that for every [x] ∈ S n , g(x) = λ(x)x, λ(x) > 0, we deduce that g(x) = λId R n , λ > 0, and λ n+1 = 1 since g ∈ Sl(n + 1, R).This implies that λ = 1.We deduce that u : Sl(n + 1, R) → Aut(S n , P ∇ S n ) defined by u(g) = u g is injective.Let f be an element of Aut(S n , P ∇ S n ), there exists an element and the open embedding i n : R n → RP n defined by ).We deduce that for every affine manifold (M, ∇ M ) whose developing map is D M , there exists a projective structure defined on M whose developing map is i n • D M .Benzecri [4] p.241-242 has defined a functor between the category of projective manifolds of dimension n and the category of radiant affine manifolds of dimension n + 1 which can be described as follows: Firstly, we remark that since the universal cover M of the projective manifold M is simply connected and p n : S n → P R n is a covering map, the theorem 4.1 of Bredon [5] p.143 implies that the development map D M : M → P R n , can be lifted to a local diffeomorphism D ′ M : M → S n which is a projective morphism.Let N (π 1 (M )) be the normalizer of π 1 (M ) in Aut( M , P ∇ M ), for every g ∈ N (π 1 (M )), there exists H ′ M (g) ∈ Aut(S n , P ∇ S n ) such that the following diagram is commutative: .
We will denote by h ′ M the restriction of , which is the developing map of a radiant structure defined on M × S 1 whose holonomy representation h . This radiant affine manifold M × S 1 is the construction of Benzecri, we will often denote this affine structure by (B(M ), ∇ B(M) ) and by p B(M) : M × S 1 → M the projection on the first factor.
Let f : (M, P ∇ M ) → (N, P ∇ N ) be a morphism between n-dimensional projective manifolds; f can lifted to the projective the morphism f : M → Ñ .We deduce the existence of a morphism of affine manifolds Let (N, ∇ N ) be a n-dimensional radiant affine manifold.We suppose that the holonomy of N fixes the origin of R n .The vector field defined on R n by X R n R (x) = x is invariant by the holonomy.Its pullback by the developing map is a vector field X Ñ R of Ñ invariant by π 1 (N ).We deduce that X Ñ R is the pullback of a vector field X N R of N called the radiant vector field of N .
Proposition 3.2.Let (M, P ∇ M ) be a compact projective manifold.There exists a surjective morphism of groups between the connected component of Aut(B(M ), ∇ B(M) ) and the connected component of Aut(M, P ∇ M ).
Proof.Let f be an element of Aut(B(M ), ∇ B(M) ) 0 , the connected component of Aut(B(M ), ∇ B(M) ).Consider an element f of Aut( M × R * + ) 0 over f .For every x ∈ M and t ∈ R * + , we can write f (x, t) = (g(x, t), h(x, t)).The flow of X R n+1 R is in the center of Gl(n + 1, R), we deduce that f commutes with the flow X B(M) R , g(x, t) does not depend of t and h(x, t) = th(x, 1).Let γ be an element of π 1 (M ), since (γ, 2).(x, t) = (γ(x), 2t) is an element of π 1 (B(M )) and f is an element of N (π 1 (B(M ))) 0 , we deduce that (γ, 2) commutes with f and g commute with γ.This implies that there exists an element g of Aut(M, P ∇ M ) whose lifts is g.Remark that since f is an affine transformation, h(x, 1) is a constant.The correspondence P : Aut(B(M ), ∇ B(M) ) 0 → Aut(M, P ∇ M ) 0 defined by P (f ) = g is well defined and is surjective morphism of groups since for every element f ∈ Aut(M, P ∇ M ) 0 , P (b(f )) = f .Let (M, P ∇ M ) be a projective manifold M , the orbits of the radiant flow φ are compact.The images of the elements of φ B(M) t by P are the identity on (M, P ∇ M ).This implies that dim(Aut(M, P ∇ M )) + 1 ≤ dim(Aut(B(M ), ∇ B(M) ))).We deduce that if (M, P ∇ M ) is a projective manifold, such that Aut(M, P ∇ M ) is not discrete, the dimension of Aut(B(M ), ∇ B(M) ) is superior or equal to 2.

Automorphisms of projective manifolds and automorphisms of radiant affine manifolds.
Let (N, ∇ N ) be an affine manifold.In [13], I have shown that aut(N, ∇ N ), the Lie algebra of Aut(N, ∇ N ) is endowed with an associative product defined by X.Y = ∇ M X Y .We deduce that ∇ B(M) defines on aut(B(M ), ∇ B(M) ) an associative structure which can be pulled back to n(π 1 (B(M )).It results that the Lie algebra H B(M) (n(B(M ), ∇ B(M) )) of the image of N (π 1 (B(M ))) by H B(M) is stable by the canonical product of matrices which is the image of the associative product of n(π ).The theorem 23 of chap.III of [1] implies that we can write: where S M is a semi-simple associative algebra and N M a nilpotent associative algebra.
In [14], by using this associative product, I have shown that the orbits of the canonical action of Af f (N, ∇ N ) 0 on N are immersed affine submanifolds of (N, ∇ N ) and are the leaves of a (singular) foliation.This leads to the following result: Proposition 4.1.Let (M, P ∇ M ) be a projective manifold.The orbits of the action of Aut(M, P ∇ M ) 0 on M are immersed projective submanifolds and are the leaves of a singular foliation.
Proof.The orbits of Aut(B(M ), ∇ B(M) ) 0 are immersed affine submanifolds of B(M ).The proposition 3.2 shows that there exists a surjective map P : ).This implies that the orbits of Aut(M, P ∇ M ) 0 are the images of the orbits of Aut(B(M ), ∇ B(M) ) 0 by the quotient map B(M ) → M .Theorem 4.1 Let (M, P ∇ M ) be a compact oriented projective manifold of dimension superior or equal to 2. Suppose that H M (N (π 1 (M ))) acts transitively on RP n , then (M, P ∇ M ) is isomorphic to a finite quotient of KP m by a subgroup of K where K is the field of real numbers, complex numbers, quaternions or octonions.The action of π 1 (M ) on KP n is induced by its action on K m+1 by homothetic maps.
Proof.The fact that H M (N (π 1 (M ))) acts transitively on RP n implies that H ′ M (N (π 1 (M ))) acts transitively on S n .The theorem of Montgomery Zipplin [11] p.226 implies that a connected compact subgroup K ′ of H ′ M (N (π 1 (M ))) acts transitively on S n .The theorem I p. 456 of Montgomery and Samelson [10] implies that a connected compact simple subgroup C ′ of K ′ acts transitively on S n .The Lie algebra of the connected component C of H ′ −1 M (C ′ ) is isomorphic to the Lie algebra of C ′ since the kernel of H ′ M is discrete.This implies that C is compact.Remark that the orbits of the action of C on M are open.We deduce that C acts transitively on M and M is compact.This implies that D ′ M : M → S n is a covering since it is a local diffeomorphism defined between compact manifolds.This implies that D ′ M is a diffeomorphim since S n and M are simply connected.
We can write The group of automorphisms of the irreducible representation U i is K where K = R, C, H or O.We deduce that R n+1 is a K vector space and the action of induced by its action on K by right multiplication of elements of K.

Remark.
Suppose that the dimension of M is even, and H M (N (π 1 (M )) 0 )) acts transitively on RP n .The proof of the previous theorem can be simplified as follows: Every element of H ′ M (π 1 (M )) has a fixed point since every element of Gl(2n + 1, R) has a real eigenvalue.We deduce that H ′ M (π 1 (M )) is the identity and there exists a map f : M → RP n such that D M = f • p M .This implies that f is a covering map and M is homeomorphic to S n or RP n .
Let (M, P ∇ M ) be a projective manifold, suppose that Aut(M, P ∇ M ) 0 is not solvable.This implies that Aut(B(M ), ∇ B(M) ) 0 and the connected component of the normalizer N (π 1 (B(M ))) 0 of π 1 (B(M )) in Aut( B(M ), ∇ B(M) ) are not solvable.We deduce that the image of N (π 1 (B(M )) by H B(M) contains a subgroup H S 1 isomorphic to S 1 .We denote by X ′ B(M) , a vector field which generates the Lie algebra of H S 1 , its pullback by the developing map D B(M) of B(M ) is a vector field XB(M) invariant by the fundamental group of B(M ).We deduce that there exists a vector field X B(M) of B(M ) whose pullback by the universal covering map is XB(M) .Suppose that the developing map is injective, the flow of X B(M) defines an action of S 1 on B(M ) which is transverse and commutes with the radial flow.This implies there exists a vector X M on M which is the image of X B(M) by the map induced by p B(M) : B(M ) → M .The vector field X M induces an action of S 1 on M : We have: Proposition 4.2.Let (M, P ∇ M ) be a compact projective manifold whose developing map is injective, suppose that Aut(M, P ∇ M ) 0 is not solvable, then M is endowed with a non trivial action of S 1 .

Automorphisms of projective manifolds of dimension 2 and 3.
In dimension 2, we have the following result: Proposition 5.1.Let (M, P ∇ M ) be a 2-dimensional compact connected oriented projective manifold, suppose that Aut(M, P ∇ P ) is not discrete, then M is homeomorphic to the 2-dimensional torus or to the sphere.
Proof.Suppose N M = 0, there exists a non zero element A M ∈ N M such that A 2 M = 0, we deduce that dim(ker(A M )) = 2, dim(Im(A M )) = 1.Remark that Im(A M ) is fixed by the holonomy.
Suppose that N M = 0, we deduce that dim(S M ) ≥ 2, there exists a non zero element distinct of the identity e M ∈ S M such that e 2 M = e M .To see this remark that S M contains either an associative algebra isomorphic to the associative algebra of 2 × 2 real matrices or two idempotents which are linearly independent.The linear map e M is diagonalizable and its eigenvalues are equal to 0 and 1.Since the flow of e M is distinct of the radial flow, we deduce that 0 is an eigenvalue of e M .This implies that either the dimension of the eigenspace associated to 0 is 1, or the the dimension of the eigenspace associated to 1 is 1.We deduce that the holonomy preserves a vector subspace of dimension 1.
We conclude that if Aut(M, P ∇ M )) is not discrete, its holonomy fixed a point of P R 2 .The lemma 2.5 p. 808 in Goldman [9] implies that the Euler number of M is positive.Dimension 3.
In this section, we study the group of automorphisms of a connected 3dimensional compact projective manifold (M, P ∇ M ) whose group of automorphisms is not discrete.
Suppose that Aut(M, P ∇ M ) 0 is not solvable, then Aut(B(M ), ∇ B(M) ) 0 and N (π 1 (B(M ))) 0 are not solvable.We deduce that the connected subgroup of Gl(n + 1, R), H B(M) (N (π 1 (M )) 0 ) contains a subgroup H" isomorphic to S 1 .We denote by X" B(M) a vector field which generates the Lie algebra of H".The pullback Suppose that the set of fixed points of H" is not empty, we can write R 4 = U ⊕ V where U is a 2-dimensional vector subspace corresponding to the non trivial irreducible submodule of H" and V the set of fixed points.Remark that h B(M) (π 1 (B(M ))) preserves U and V since it commutes with H".This implies that there exists a foliation F U (resp.F V ) on B(M ) whose pullback by the universal covering map is the pullback by D B(M) of the foliation of R 4 whose leaves are 2-dimensional affine spaces parallel to U (resp.parallel to V ).Proposition 5.2.Suppose that V ∩ D B(M) ( B(M )) is empty.Then a finite cover of M is a total space of a fibre bundle over S 1 whose fibre is T 2 .
Proof.The vector field defined by Y "(u, v) = u; u ∈ U, v ∈ V is invariant by the holonomy of B(M ).To show this, remark that the restriction of H" to U defines on it a complex structure and since h B(M) (π 1 (B(M ))) commutes with H", its restriction to U are morphisms of that complex structure.The pullback of Y " by D B(M) is the pullback of a vector field Y B(M) of B(M ) by the universal covering map.The image Y M of Y B(M) by p B(M) and X M commute and generate a locally free action of R 2 on M since V ∩ D B(M) ( B(M )) is empty.Chatelet, Rosenberg and Weil [6] implies that M is the total space of a fibre bundle over S 1 whose fibre is T 2 .
If F V has compact leaf, we have the following result: Proposition 5.3.Let (M, P ∇ M ) be a 3-dimensional compact projective manifold whose developing map is injective.Suppose that Aut(M, P ∇ M ) 0 is not solvable and V ∩ D B(M) ( B(M )) is not empty.Then the holonomy of (M, P ∇ M ) is solvable.
Proof.Let F0 be a connected component of V ∩ D B(M) ( B(M )) its image by the universal covering map is a compact leaf F 0 compact leaf of F V which is a 2-dimension compact affine manifold, we deduce that its fundamental group is solvable.Let r be the restriction of h(π 1 (B(M )) to V , since h B(M) (π 1 (B(M )) preserves V , we have an exact sequence: The groups Ker(r) is solvable since it restriction to U commutes with a non trivial linear action of S 1 .The group Im(r) is also solvable since it is contained in h F0 (π 1 (F 0 )), we deduce that h B(M) (π 1 (B(M )) is solvable.
In this section we study 3-dimensional projective manifolds whose group of automorphisms is solvable.We can decompose the associative algebra n(π 1 (B(M )) by writing: n(π 1 (B(M )) = S M ⊕ N M , where S M is a semi-simple associative algebra and N M a nilpotent associative algebra.We deduce that S M is the direct product of associative algebras isomorphic to either R or C and is commutative.It results that the fact that Aut(M, P ∇ M ) 0 is not commutative implies that N M is not commutative.
Aut(M, P ∇ M ) 0 is solvable and is not commutative.Theorem 5.1.Suppose that N M is not commutative, then h B(M) (π 1 (B(M ))), the image of the holonomy of B(M ) is solvable.

Proof.
First step: Suppose that the square of every element of N M is zero.Let A, B ∈ N M such that AB = BA.Suppose that dim(ker(A)) = 3.It implies that dim(Im(A)) = 1.Since (A+B) 2 = 0, we deduce that AB +BA = 0 The morphism f ′ is equivariant with respect to the action of π 1 (B(M )) on M ×R * + and π 1 (B(N )) on Ñ ×R * + , and covers a morphism b(f ) : B(M ) → B(N ).