High rank torus actions on contact manifolds

We prove LeBrun--Salamon conjecture in the following situation: if $X$ is a contact Fano manifold of dimension $2n+1$ whose group of automorphisms is reductive of rank $\geq \max(2,(n-3)/2)$ then $X$ is the adjoint variety of a simple group. The rank assumption is fulfilled not only by the three series of classical linear groups but also by almost all the exceptional ones.


Introduction
A fundamental result of Riemannian geometry is the classification of manifolds with a metric according to their holonomy groups, by De Rham and Berger, [2]. One of the classes of Riemannian manifolds in this classification are quaternion-Kähler manifolds which are of (real) dimension 4n, with n ≥ 2, and holonomy group Sp(n) Sp(1) = [Sp(n) × Sp(1)]/Z 2 ⊂ SO(4n, R). Salamon proved that the twistor space of a compact positive quaternion-Kähler manifold is a (complex) contact Fano manifold of (complex) dimension 2n+1 and admits a Kähler-Einsten metric, [23]; a converse statement holds by a result of LeBrun, [18]. A celebrated LeBrun-Salamon conjecture, [19], asserts that every compact positive quaternion-Kähler manifold is symmetric or, equivalently, every contact Fano manifold with Kähler-Einsten metric is a homogeneous space. In fact, quaternion-Kähler symmetric spaces are known as Wolf spaces and, equivalently, homogeneous complex contact manifolds are known to be the closed orbits in the projectivizations of adjoint representations of simple algebraic groups; we call them adjoint varieties.
In the absence of a proof of LeBrun-Salamon conjecture in its full generality, low-dimensional cases have been verified for n ≤ 4. That is, the conjecture was proved for positive quaternion-Kähler manifolds of (real) dimension ≤ 16 and for contact Fano manifolds with a Kähler-Einsten metric of dimension ≤ 9, see [6] and references therein. Also, the conjecture is known for contact Fano manifolds X (even without assuming that they admit a Kähler-Einsten metric) if the first Chern class of the quotient L = T X/F of the contact distribution F → T X does not generate the second cohomology H 2 (X, Z), [16]. In fact, if the class is divisible in 2010 Mathematics Subject Classification. Primary 14L30; Secondary 14M17, 14M25. First and third author supported by PRIN project "Geometria delle varietà algebriche". First, second and third author supported by MIUR project FFABR. Second and fourth author supported by Polish National Science Center project 2016/23/G/ST1/04828. The results in this paper were partially obtained while the second and fourth author visited the University of Trento. H 2 (X, Z) then X is known to be the projective space P 2n+1 , hence the closed orbit in the adjoint representation of Sp(2n + 2, C). On the other hand, if the rank of the second cohomology is ≥ 2, then X is the (naive) projectivization of the cotangent bundle on P n+1 , hence the closed orbit in the adjoint representation of SL(n+2, C).
LeBrun-Salamon conjecture was proved in some cases with additional assumptions on group actions. Bielawski [4] and Fang, [8,9], considered quaternion-Kähler manifolds with an action of a (real) torus. It is known, [20], that the automorphisms group of a Fano manifold with a Kähler-Einstein metric is reductive, which is an equivalent of a real compact group in the complex case. Thus the automorphisms of such a variety contain an algebraic torus; the rank of a maximal subtorus is, by definition, the rank of the reductive group. The results of Fang imply that contact Fano manifolds with Kähler-Einstein metric admitting an action of an algebraic torus of rank r ≥ n/2 + 3 are adjoint varieties of the simple groups Sp(2n + 2, C) and SL(n + 2, C), which are of rank n + 1. We note that the condition r ≥ n/2 + 3 is not fulfilled for the group SO(n + 4, C) which is of rank n/2 + 2.
The main theorem of the present paper, Theorem 6.1, improves previous results so that not only it covers the adjoint varieties of all the classical series of linear groups but also almost all the exceptional ones. That is, if X is a contact Fano manifold of dimension 2n+1 whose group of automorphisms is of rank ≥ max(2, (n− 3)/2) then X is the adjoint variety of one of the classical linear groups SL(n + 2, C), i.e., type A, Sp(2n + 2, C), i.e., type C, SO(n + 4, C), i.e., type B or D, or a simple group of type G 2 , F 4 , E 6 , E 7 . Although motivated by a problem from Riemannian geometry, the present paper depends solely on methods from algebraic geometry. Thus our language and formulation of results is provided in terms specific for algebraic group actions, specially algebraic torus actions, and complex birational geometry. In fact, Theorem 6.1 follows from Theorem 5.1, which is the technical core of the present paper. Theorem 5.1 asserts that a contact Fano manifold is the adjoint variety of one of the simple groups if the group of its automorphisms is reductive of rank ≥ 2 and, in addition, the action of its Cartan torus has only isolated extremal fixed points. The arguments in the proof of this result are about the action of a suitably chosen two dimensional subtorus in the Cartan torus in question. This downgrading of the torus action and suitable results about adjunction theory for varieties with a C * -action, [22], and about the birational geometry of small bandwidth C * -actions, [21], enable to identify the variety as the adjoint variety of a simple group.
The paper is organized as follows. Section 2 contains general results about algebraic groups and their actions, while Section 3 concentrates on homogeneous manifolds. In particular, we deal with adjoint varieties of simple algebraic groups. As the main result of the present paper covers the case of adjoint varieties of most exceptional algebraic groups, a significant part of its contents concerns properties specific for these groups, see e.g. [17,14]. However, in our approach, the classical linear group series SO(n + 4, C), that is the groups of type B and D, are treated on equal terms with exceptional groups. This is because the Picard groups of their adjoint varieties are generated by the class of the quotient L = T X/F , contrary to the other two types of classical linear groups. This allows us to restate geometrical properties of adjoint varieties related to the so-called Freudenthal magic square in a context appropriate for our purposes.
Section 4 summarizes specific results concerning torus actions on contact manifolds, as in [6,Section 5], while Section 5 contains the proof of Theorem 5.1, the main technical result of the paper. For clarity, the proof is divided into five steps; a short guide through the proof is provided after the formulation of the theorem. Finally, Section 6 contains the proof of Theorem 6.1.
Acknowledgements: The authors would like to thank A. Weber and J. Buczyński for their interesting comments and for providing bibliographic references.

Preliminaries
This section contains some background material and language we will use later on. Let us start by introducing some general notation.
Notation and conventions. Throughout this paper all the varieties will be algebraic, projective and defined over the field of complex numbers, mostly smooth (in this case we will also refer to them as projective manifolds). Given a vector bundle E on a variety X, we will denote by P(E) the quotient of the complement of the zero section of E by the action of homotheties. The Lie groups SL(n, C), SO(n, C), . . . will be denoted by SL(n), SO(n), . . . and considered with their structure of algebraic groups. Given a semisimple group G, a rational homogeneous G-variety is a projective variety of the form G/P ; a subgroup P ⊂ G for which G/P is projective is called parabolic. Such a variety is completely determined by combinatorial data: the Dynkin diagram D of G and a subset I of the set D of nodes in D (of cardinality equal to the Picard number of G/P ). More concretely, up to conjugacy, the parabolic subgroup P can always be written as BW (D \ I)B, where B ⊂ G is a Borel subgroup, and W (D \ I) is the subgroup of the Weyl group W of G generated by the reflections associated to the nodes of D \ I. We will then write (see [21,  2.1. Torus actions on smooth projective varieties. We will briefly recall here some general facts on torus actions on smooth complex projective varieties. We refer the interested reader to [6] for a complete account on the concepts and tools that we introduce here. Throughout the section we will denote by (X, L) a polarized pair, consisting of a smooth complex projective variety X and an ample line bundle L on X, and by H (C * ) r an r-dimensional complex torus, acting effectively on X. The lattice of weights of H will be denoted by M(H) := Hom(H, C * ).
The set of points of X fixed by the action of H, denoted by X H , is a union of smooth closed irreducible subvarieties, indexed by a finite set Y: Given a linearization of the action of H on the line bundle L, µ L : H × L → L, the weight of the action of H on the fiber L x on a fixed point x depends only on the fixed point component Y containing x, and we denote it, abusing notation, by µ L (Y ).
The polytope of fixed points (X, L, H, µ L ) (respectively, the polytope of sections Γ(X, L, H, µ L )) is defined as the convex hull in M(H)⊗R of the weights of the action of H on L (respectively, of the weights of the action of H on H 0 (X, L)). We refer to [6, Section 2.1] for details. A fixed point component Y ∈ Y for which µ L (Y ) is a vertex of (X, L, H, µ L ) will be called an extremal fixed point component.
We will sometimes consider the restriction of the action of H on (X, L) to a subtorus H ⊂ H, a process that we call downgrading. We denote by ı : H → H the inclusion, by ı * : M(H) → M(H ) the induced map between the lattices of weights, by Y the set of fixed point components of the induced action of H on X, and by µ L : Y → M(H ) the map associating to every H -fixed point component Y the H -weight of L on every point of Y . The following straightforward lemma describes how the weights of the action behave with respect to the downgrading: Lemma 2.1. With the above notation: Given a fixed point component Y ∈ Y, the action of H on the normal bundle N Y |X of Y in X provides a splitting of this bundle into eigen-subbundles: whose corresponding weights, denoted by −ν i (Y ), are nontrivial. Denoting by r i the rank of N −νi(Y ) (Y ), the set of the elements ν i (Y ), counted r i -times, will be called the compass of the action of H on Y . We will write it as In order to make our exposition self-contained, we will recall from [6] two statements regarding compasses that we will use later on; the first one describes the behavior of the compass with respect to downgrading: Lemma 2.13] Let H be an algebraic torus acting on a smooth projective variety X, and H ⊂ H a subtorus. Let ı * : M(H) → M(H ) be the projection between the corresponding lattices of weights. Take fixed point components Y ⊂ X H and Y ⊂ Y ∩ X H . Then: Another important property of the compass is that its elements at a given fixed point x ∈ X H are vectors pointing from µ L (x) towards the weight µ L (x ) of another fixed point x ∈ X H . More precisely: Lemma 2.3. [6, Corollary 2.14] Let H be an algebraic torus acting on a smooth projective variety X, let L be a line bundle on X, and let µ L be a linearization of the action. Take Y ⊂ X H , and ν ∈ C(Y, X, H). Then there exists Y ⊂ X H and λ ∈ Q >0 such that µ L (Y ) = µ L (Y ) + λν.
The above invariants of H-actions were introduced in [6] with the motivation that, in the case in which H ⊂ G is the maximal torus of a semisimple group, they can be used to characterize G-representations and rational homogeneous Gvarieties. In particular, in this paper we will make use of the following statement, which has been proved in [6,Propositions 2.23,2.24] under the additional assumption that all the fixed point components are isolated points; the proof follows the same line of argumentation.
Proposition 2.4. Let (X, L) and (X , L ) be two polarized pairs. Let H be a complex torus acting on both pairs and denote by Y and Y the set of irreducible fixed components of X H and X H , respectively. Assume that there exists a bijection ψ : Y → Y such that, for every Y ∈ Y: If the actions of H on X and X are restrictions of the actions of a semisimple group G containing H as a maximal torus, and if X is a rational homogeneous G-variety, then (X, L) (X , L ).
Proof. Note that conditions (A 2 ), (A 3 ) are equivalent to require that L |Y is H-equivariantly isomorphic to L |ψ(Y ) and that N Y |X is H-equivariantly isomorphic to N ψ(Y )|X , respectively. This implies an equality of H-equivariant Euler characteristics χ H (X, L) = χ H (X , L ) (see [6,Theorem A.1]); together with the hypothesis on the vanishing of the cohomology, this tells us that H 0 (X, L), H 0 (X , L ) are equal as elements of the representation ring of H.
For the second part, a similar argument provides isomorphisms of H-modules: H 0 (X, L ⊗m ) H 0 (X , L ⊗m ), for m 0. By standard Representation Theory (cf [10, Theorem 14.18]), since these spaces are G-modules, they are isomorphic also as G-modules. Then for m 0, the natural morphism φ : X → P(H 0 (X, L ⊗m ) ∨ ) will be a G-equivariant embedding so that the image of X will be invariant by G. In particular, it will contain the unique closed orbit of the action, which is isomorphic to X . Since condition (A 3 ) implies that dim X = dim X we conclude that (X, L) (X , L ).

2.2.
Actions of C * on smooth projective varieties. The case of actions of the torus C * , that has been extensively studied in the literature (see [7] and references therein), will be particularly useful for our purposes. We introduce here some notation and basic facts we will use when dealing with this type of actions.
We first choose an isomorphism M(C * ) Z. Given a C * -action on a smooth projective variety X, for every fixed point component Y ∈ Y the normal bundle of Y in X splits into two subbundles, on which C * acts with positive and negative weights, respectively: We denote by ν ± (Y ) = rank N ± (Y ) the ranks of these subbundles. Then the following result, due to Bia lynicki-Birula (cf. [3], [7,Theorem 4.4]), allows us to compute the homology groups of X: Theorem 2.5. In the situation described above there is a decomposition: Given a polarized pair (X, L) admitting a C * -action, we identify the weights µ L (Y ) with the corresponding integers. The minimum and maximum value of µ L , denoted by µ min and µ max , are achieved in two unique fixed point components, that we call the sink and the source of the action; the rest of the fixed point components will be called inner the fixed points components which are neither the source nor the sink. The bandwidth of the action of H on (X, L) is defined as the difference |µ| = µ max − µ min . Moreover, we say that the action of H on X is equalized at Y if H acts on N Y |X with weights ±1.
In the cases in which the bandwidth of a C * -action on a pair is small, one expects to have reasonably short lists of examples, under certain assumptions. We refer to [21] for an account on this matter. In this paper we will make use of the following two results, regarding equalized actions of bandwidth two and three. The first tells us that under some conditions an action of bandwidth two is determined by its local behaviour at the sink and the source: Theorem 2.6. [21, Corollary 5.12] Let (X, L) be a polarized pair supporting a C * -action of bandwidth two, equalized at the sink Y −1 and source Y 1 , which are both positive dimensional. Assume moreover that ρ X = 1, that there exists an inner fixed point component, and that the vector bundles N ∨ Y±1/X ⊗ L are semiample. Then X is uniquely determined by (Y ±1 , N Y±1/X ).
We will also make use of the following complete classification of equalized bandwidth 3 actions with isolated extremal fixed points: Theorem 2.7. [22,Theorem 3.5], [21,Theorem 6.8] Let (X, L) be a polarized pair, where X is a projective manifold of dimension n ≥ 3 with a linearized action of C * of bandwidth three, such that its sink and source are isolated points. Assume in addition that the action is equalized, and denote by Y i the union of the inner fixed point components of weight i, i = 1, 2. Then one of the following holds: quadric Q n−3 and a point, and L |Q n−3 O Q n−3 (1). (3) X is one of the following rational homogeneous varieties: L is the ample generator of Pic(X) and the varieties Y i are, respectively

Torus actions on adjoint varieties
In this section we will describe torus actions on rational homogeneous varieties, paying special attention to the case of adjoint varieties, whose characterization is the goal of this paper. We will start by briefly recalling the action of the maximal tori of their defining semisimple groups, then we will focus on some particular downgradings of those actions, that we will use later in the proof of Theorem 5.1. The following well known statement (see [7,Section 3.4]) describes the set of H-fixed point of every rational homogeneous variety G/P , with P ⊃ B; we include its proof for lack of references: Now, given P ⊃ B, we note first that the natural projection π P : G/B → G/P is H-equivariant, hence π P ((G/B) H ) ⊂ (G/P ) H ; the converse follows from the fact that π P is projective, therefore the inverse image of a fixed point of G/P , which is H-invariant, contains a fixed point. We describe now the compasses of the H-action at fixed points wP , w ∈ W . Lemma 3.3. If P = BW (D \ I)B, denoting by Φ + (D \ I) the set of positive roots of Φ that are linear combinations of elements α j , j ∈ D \ I, then: Proof. As above, we choose for every w ∈ W a preimage in N G (H), and denote it by w. Being g and p respectively the Lie algebra of G and P , we may write isomorphisms of H-modules: Then we use the Cartan decomposition to split g/p as a direct sum of H-eigenspaces and conclude by noting that Ad w (g α ) = g w(α) for all α ∈ Φ.
Let us finally consider the case in which g is simple (abusing notation, we will say in this case that the semisimple group G is simple), and let X G be the corresponding adjoint variety of G, that is the closed orbit of the action of G in the projectivization of the adjoint representation of G, X G → P(g). The following result, in which β denotes the highest weight of the adjoint representation, and v ∈ g β is a nonzero eigenvector, is a consequence of Lemma 3.1.
Corollary 3.4. The set of fixed points of X G by the action of H ⊂ G is: G is bijective to the set of long roots of G.
Proof. By [10, Claim 23.52] we know that X G is the G-orbit of the class of a highest weight vector [v] of the adjoint representation; that is v is a nonzero element of g β , with β the longest positive root of G with respect to a base of simple roots. Then Lemma 3.1 gives us the description of X H G . The second part follows from the fact that Ad w (v) ∈ g w(β) for every w ∈ W . We conclude by noting that W acts transitively on the set of long roots of G (cf. [13, Lemma C]).
3.2. Some special downgradings of ranks one and two. In this section we keep the same notations introduced above, and assume that G is simple. Let us denote by (G) ⊂ M(H) ⊗ R the root polytope of G, that is the polytope generated by the roots of G. We will now define some special downgradings of the action of H on X G , that we will use later on.
Proof. In the case G 2 it is known, by Borel-de Siebenthal theory [5], that the set of long roots of G 2 is a closed root subsystem of the root system of G 2 determining a subgroup S 2 ⊂ G 2 isomorphic to SL (3). Note that S 2 and G 2 have the same maximal torus H and the same root polytopes.
In all the remaining cases we may always choose the long roots α, α forming an angle of 2π/3 among the elements of the base D = {α 1 , . . . , α r } of simple roots of G (see Remark 3.6 below); let us denote them by α i , α j .
We consider the subgroup W ⊂ W of the Weyl group W of G generated by the reflections r i , r j (corresponding to α i and α j ), and the parabolic subgroup P = BW B ⊂ G. We then consider a Levi decomposition of P , P = U L, where U is the unipotent radical of P , and L is reductive, and the commutator S 2 = [L, L] ⊂ L ⊂ P ⊂ G, which is semisimple. The maximal torus of S 2 is H 2 := H ∩ S 2 . By construction, denoting by h, s 2 , h 2 the Lie algebras of H, S 2 and H 2 , by ı * : M(H) ⊗ Z R → M(H 2 ) ⊗ Z R the linear map induced by the inclusion H 2 → H, and by g β ⊂ g the eigenspace associated to a root β of G, we may write: Note that ı * is an orthogonal projection and sends the lattice M(H) to M(H 2 ). Moreover, since all the roots in Φ ∩ (Zα i + Zα j ) are long, then ı * sends any root of G which is not in Φ ∩ (Zα i + Zα j ) to the interior of (S 2 ). This shows that (G) → (S 2 ) has one point fibers over the vertices of the hexagon (S 2 ). Finally we note that in all the cases there exists a root of G not in Φ∩(Zα i +Zα j ) and not orthogonal to the subspace generated by α i , α j , hence its projection to M(H 2 ) ⊗ Z R is a nonzero lattice point in the interior of (S 2 ). This shows that M(H 2 ) contains properly the root lattice of S 2 , so necessarily S 2 SL(3).
Remark 3.6. In the setting of Lemma 3.5, for the cases different from G 2 we may use the following particular choice of the pairs (i, j) defining S 2 ⊂ G: Here we are following the standard reference [13, p. 58] for the numbering of nodes of the corresponding Dynkin diagrams.
Notation 3.7. Let us now describe the downgradings of the action of H ⊂ G on the adjoint variety X G that we are going to consider in this paper.
(S 2 ) With the notation of Lemma 3.5, we will consider a subgroup S 2 ⊂ G, isomorphic to SL(3), a maximal torus H 2 ⊂ S 2 (obtained by intersecting the maximal torus H ⊂ G with S 2 ) given by the choice of two long roots α, α as in Remark 3.6, for the cases different from G 2 . In Figure 2 we represented the points of M(H 2 ) ∩ (S 2 ), which are the possible images of the roots of G via ı * .
where we recall that Φ ⊂ M is the root system of G. By our choice of S 1 ⊂ S 2 ⊂ G we have that, for i = 1, 2, dim(g i m ) = 1 whenever m is a root of S i : these are precisely the vertices of the root polytope (S i ). The rest of values of m ∈ M i for which g i m = 0 correspond to inner points of (S i ). Furthermore, the vector spaces g i m are representations of the subalgebra g i 0 ⊂ g of elements of M i -degree 0 and of its semisimple part g i,ss 0 , i = 1, 2. With the choice of the projection ı : M → M 2 presented in Remark 3.6, one obtains the following description of those subalgebras: 3.3. Freudenthal varieties. We will now consider the adjoint variety X G and compute the fixed point components of the actions of the tori H 2 and H 1 introduced above (see Notation 3.7). In the cases in which G is exceptional, the varieties that we will obtain are essentially those obtained in the Freudenthal magic square studied by Landsberg and Manivel in [17], see Table 1. In particular we will see that certain fixed point components for the torus H 1 coincide with the varieties of type (2) and (3) obtained in the classification of C * -actions of equalized bandwidth three actions with isolated sink and source (cf. Theorem 2.7). These varieties appear in the literature as Freudenthal varieties (cf. [14,17]); in our setting they will be used to recognize adjoint varieties among contact manifolds by means of equivariant K-theory, as we will see in Section 5.
We start by noting that where: By construction, Y m is an isolated point when m is a vertex of (S 2 ), and the same holds for the vertices of (S 1 ). We will now study the fixed point components corresponding to inner lattices points.
Let us first describe, for i = 1, 2, a subgroup G i,ss 0 ⊂ G, i = 1, 2, whose Lie algebra is g i,ss 0 . We will consider only the case i = 2, being i = 1 analogous; we follow the lines of argumentation of [24, Section 2.3.1]. We consider a Z-basis {µ 1 , µ 2 } of M ∨ 2 and, for each j = 1, 2, the derivation D j : g → g defined by D j (x) = µ j (m)x for every x ∈ g 2 m , and for every m ∈ M 2 . Since g is semisimple, each D j is an inner derivation and there exists ξ j ∈ g 2 0 such that D j = ad ξj . Then g 2 0 can be described as {x ∈ g| [x, ξ j ] = 0 for j = 1, 2}, or as the Lie algebra of the subgroup: Proof. This is an adaptation of [24, Theorem 2.6] to our setting. We will show that Y m is a finite union of closed G 2 0 -orbits, from which the statement follows; the case of Z m is analogous. By construction, the varieties Y m are G 2 0 -invariant, so they are unions of G 2 0 -orbits, and we only need to check that all these orbits are closed. To this end, we note first that for every x ∈ Y m we have: This follows by quotienting by x the equalities: Now if an orbit O were not closed, its boundary would contain an orbit O of smaller dimension, and by the equality (2) one has that T X G ∩ T P(gm) would have fibers of smaller dimension on the points of O , contradicting semicontinuity.
Given a fixed point component Λ of X Hi G , i = 1, 2, of weight m, we may consider the action of the torus H/H i on it. By Lemma 2.1 this action has only isolated fixed points whose weights are the long roots of G of M(H i )-degree equal to m. In particular, this shows that Λ is the G i,ss 0 -orbit of the class of an element x ∈ g α , where g α ⊂ g i m and α is a long root of G. For instance, if G is of type F 4 such a root does not exist in g 2 0 , so Y 0 is empty in this case. Moreover, this implies that the fixed point components of X Hi G are the minimal orbits of the irreducible representations of G i,ss 0 with highest weights equal to one of the long roots of G. This allows us to study case by case the fixed locus of X G under the action of H i . Table 1 contains the description of the varieties Y m , Z m , for m = 0 and for nonzero inner points m ∈ (S i ) ∩ M i : In the cases B and D the index r is, respectively, bigger than or equal to 3 and 4. The symbol v n indicates the n-th Veronese embedding.

Contact manifolds with a compatible torus action
In this section we collect some basic background on contact manifolds and torus actions on them (cf. [16], [6, §4.1]).

Contact manifolds.
A contact manifold is a smooth projective variety X of dimension 2n + 1 together with a line bundle L (for short, we will say that the pair (X, L) is a contact manifold) fitting in an exact sequence of vector bundles: such that the corresponding O'Neill tensor induces a skew-symmetric isomorphism The distribution F ⊂ T X , the line bundle L and the morphism σ are called, respectively, the contact distribution, the contact line bundle and the contact form of (X, L). Note that, combining (3) and (4) we may write Remark 4.1. If a contact manifold X is a Fano manifold, it is known (cf. [16,Proposition 2.13]) that either Pic(X) Z, or X P(Ω P n+1 ). Note also that Equation (5) tells us that X is Fano if and only if L is ample. In the case in which X P(Ω P n+1 ), we have that L = O P(Ω P n+1 ) (1); if Pic(X) Z, then L is the ample generator of Pic(X), except for the case X P 2n+1 , where L O P 2n+1 (2).
The following Lemma allows us to identify H 0 (X, L) with the adjoint representation of the group of automorphisms of a Fano contact manifold X different from P 2n+1 and P(Ω P n+1 ); we denote by CAut(X) ⊂ Aut(X) the group of automorphisms of a contact manifold X preserving the contact structure.
Lemma 4.2. Let (X, L) be a contact Fano manifold with dim X = 2n + 1, and Pic X = ZL. Then CAut(X) = Aut(X). In particular Aut(X) extends to an action on L so that the contact map σ is Aut(X)-equivariant, and we have Aut(X)-equivariant isomorphisms Lie(Aut(X)) H 0 (X, T X ) H 0 (X, L).
Proof. Since X P 2n+1 by Remark 4.1, applying [15, Sect. 2.3], we know that X is covered by a complete unsplit family of rational curves of degree 1 with respect to L, called contact lines. By [15,Theorem 4.4,Corollary 4.5], the VMRT of this family spans P(F ), and so the contact structure F is unique. In particular every automorphism of X preserves F , hence Aut(X) = CAut(X). The second part of the statement follows by [1, Proposition 1.1].
Let us recall that a smooth subvariety Y of a contact manifold X as above is called isotropic if T Y ⊂ F |Y . By definition, an isotropic subvariety of X has dimension at most n; an isotropic subvariety of dimension exactly n is called a Legendrian subvariety of X. For a Legendrian subvariety Y ⊂ X we have a commutative diagram with short exact rows and columns: The lower row of the diagram can be interpreted as follows: Proof. The argument presented here belongs to an unpublished manuscript of the fourth author with J. Buczyński, based on [16, Section 2]. Let us denote by L and and L Y the Atiyah extensions corresponding to the line bundles L and L |Y on X and Y , respectively; they fit into the following commutative diagram of vector bundles on Y with exact rows and columns: We use now the fact that L ∨ admits a nondegenerate skew-symmetric form L ∨ ⊗ L ∨ → L induced by the contact form in X, (cf. [16, Section 2.2]). Moreover, the fact that Y ⊂ X is Legendrian (Corollary 4.10) implies that the subbundle N ∨ Y |X ⊂ L |Y is Lagrangian with respect to it. Hence the inner vertical exact sequence in the above diagram twisted with L |Y± and its dual fit in the following commutative diagram with exact rows: In particular, we get the isomorphism N Y |X L ⊗ L |Y . Twisting with L |Y the second row of the above diagram we conclude.

4.2.
Compatible torus actions on contact manifolds. We will now consider contact manifolds (X, L) for which there exists an action of a torus H such that the contact form σ is equivariant; we will say that such an action is compatible with the contact structure. In particular we will have an H-action on F , and the isomorphism (4) will be H-equivariant.
The largest compatible torus action on a contact manifold (X, L) is the one in which H is a maximal torus of the identity component CAut(X) • of the group of contact automorphisms of X. In the case in which this group is reductive (for instance if X is the twistor space of a compact quaternion-Kähler manifold), we have a precise description of the polytope of sections of this action, as a direct application of Lemma 4.2:  Since d = 0 and T Y is at every point the eigenspace of T X |Y associated to the weight zero, it follows that T Y is contained in the kernel of σ |Y : T X |Y → L |Y , that is F |Y . On the other hand, applying Lemma 4.5 to the weights of T Y ⊂ F |Y , we conclude that F |Y contains an H-invariant vector subbundle F of rank equal to dim Y , whose weights are all equal to d, so that F ⊕T Y is a subbundle of T X |Y . The proof is finished observing that F is mapped isomorphically onto Ω Y ⊗ L via the composition of the isomorphism (4) with the induced projection F ∨ ⊗ L → Ω Y ⊗ L. For the second part note that, from the above arguments, the summand of weight d in T X |Y is an extension of F Ω Y ⊗ L and L |Y .
We now focus on the case H = C * . The next statement shows that the central components of a compatible C * -action on (X, L) inherit its contact structure: Lemma 4.7. Let (X, L) be a contact manifold supporting a nontrivial compatible C * -action. Then, for any irreducible component Y 0 such that µ L (Y 0 ) = 0, the eigensubbundle of weight zero (F |Y0 ) 0 ⊂ F |Y0 defines a contact form on Y 0 . Moreover, the ranks ν + (Y 0 ), ν − (Y 0 ) of the positive and negative parts of N Y0|X are equal and, Proof. Since T Y0 is the part of weight zero of T X|Y0 , we have a weight decomposition (see Equation (1) in Section 2.2): and the induced map σ |T Y 0 : T Y0 → L |Y0 is surjective. Thus we have a decomposition of the contact distribution along Y 0 : Furthermore, Equation (5) immediately provides the following: Corollary 4.8. Let (X, L) be a Fano contact manifold supporting a nontrivial C * -action compatible with its contact structure. If Y 0 is an irreducible component such that µ L (Y 0 ) = 0, then (Y 0 , L |Y0 ) is a Fano contact manifold.
We finish this Section with two statements on contact varieties admitting a nontrivial compatible C * -action of bandwidth 2, that will be used in Section 5.
Lemma 4.9. Let (X, L) be a contact manifold supporting a nontrivial compatible C * -action of bandwidth two. Then the weights on the sink and the source of the action of C * on L are equal respectively to −1 and 1.
Proof. Let us denote by Y − the sink of the C * -action on L, and by Y + the source. Denote by d ± the weights of L on Y ± . At every point of Y − , the weights of the action on (T X ) |Y− are all non positive, therefore, by (5), d − < 0. In a similar way we can prove that d + > 0. By the assumption on the bandwidth we have d + = d − + 2, and the result follows.
Combining Lemma 4.6 and Lemma 4.9, we get the following: Corollary 4.10. Let (X, L) be a contact manifold of dimension 2n + 1, supporting a nontrivial compatible C * -action of bandwidth two. Then the extremal fixed components Y ± are Legendrian subvarieties of X, F |Y± is isomorphic to T Y± ⊕ (Ω Y± ⊗ L) and, in particular, dim Y ± = n.
Proof. Let us deal with Y − , being the case of Y + completely analogous. Since the weights of the action on F |Y− are all non positive and, by Lemma 4.5, symmetric with respect to −1/2, it follows that the only weights are 0 and −1, appearing both with multiplicity n. We then conclude arguing as in Lemma 4.6.

Actions on contact manifolds with isolated extremal fixed points
Throughout this section, given a contact manifold (X, L), we will denote by CAut(X) • the identity component of the group of contact automorphisms of X. For a simple algebraic group G we denote by X G the closed orbit in the adjoint representation of G and by L G the pull-back of O(1) via the embedding X G → P(g) where g is the Lie algebra of G. The goal of this section is to prove the following: Theorem 5.1. Let (X, L) be a contact Fano manifold with dim X = 2n + 1, and Pic X = ZL. Suppose that G = CAut(X) • is reductive of rank r ≥ 2, and that the action of the maximal torus H ⊂ G on X has isolated points as extremal fixed point components. Then G is simple of one of the following types: and (X, L) = (X G , L G ).
Remark 5.2. Note that the statement is known for n ≤ 4 without any assumptions on the rank of G and on X H (see [6, Section 1.1] and the references therein). In the proof of the Theorem we will then assume that n ≥ 5. Note also that for the cases n = 5, 6, the statement has been proved in [22,Theorem 5.3] without the assumption on the extremal fixed points.
Outline and methods of the proof of Theorem 5.1. We achieve the proof of Theorem 5.1 in five steps. In Step I we prove the simplicity of G. To this end, in Lemma 5.3 we first observe that the polytope of sections and of fixed points coincide and that they are also equal to (G). Then in Corollary 5.4 we deduce the simplicity of G, so that we know all the possibilities for (X, L, H), and we eliminate the cases in which G is of type C or of type A r with r ≥ 3 (see Lemma 5.6).
Step II and III contain the key point which allows us to extend the previous results of [6,22] to any dimension of the contact variety. In these steps we will make use of classification results of bandwith three and two varieties which have been recently obtained in [21]. These varieties arise as submanifolds of X (see Propositions 5.16 and 5.18) and play a central role to determine the combinatorial data which will be needed. In Step II we use Lemma 3.5 to reduce the action of G on (X, L) to the action of a subgroup S 2 ⊂ G isomorphic to SL(3). Then we focus on the induced action of a rank two torus H 2 ⊂ S 2 on (X, L), and in Step III we describe all the fixed point components and compasses with respect to the action of this smaller torus. As a consequence, in Step IV, we prove that such combinatorial data coincide with the fixed point components and compasses of the action of H 2 on the corresponding adjoint variety (X G , L G ) (cf. Corollaries 5.21 and 5.23). All the information collected in these steps about the H 2 -action allows us to conclude in Step V that the fixed point components and the compasses of the action of the maximal torus H on (X, L) and on (X G , L G ) are the same (see Lemmas 5.24 and 5.25). Applying Proposition 2.4 we obtain that (X, L) (X G , L G ) as stated.

5.1.
Step I: Simplicity and type of the automorphism group. The following statement, whose hypotheses are obviously fulfilled under the assumptions of Theorem 5.1, provides the equality of the polytopes of fixed points and of sections of the action of the maximal torus H ⊂ G = CAut(X) • on (X, L). Applying the machinery developed in [6] we achieve the goal of this step:

5.2.
Step II: Downgradings to subtori of rank two and one. In this subsection we consider the restrictions of the action of G on (X, L) to a subgroup S 2 ⊂ G isomorphic to SL(3) satisfying the requirements of Lemma 3.5, and then to certain subgroups of S 2 isomorphic to SL(2) as described in Section 3.2 (see Notation 3.7). We will consider one of these subgroups for each of the roots of S 2 , so we will start by introducing some notation (see Figure 3).  In Step III we will describe completely the possible isomorphism classes of the fixed point components of the action of H 2 . Here we will only introduce some notation about them, and their most basic properties. We note first that the weights of the induced action of H 2 and H i 1 on L are elements of Hex and of the set {−2, −1, 0, 1, 2}, respectively. In fact, by Lemma 5.3 and Lemma 2.1: Lemma 5.8. The polytope of fixed points (X, L, H 2 ) is equal to (S 2 ). Remark 5.9. Using Lemma 3.5 the inverse image in (G) of every vertex of (S 2 ) (resp. (S 1 )) is unique. Then the basic properties of downgradings (see Lemma 2.1 (iii,v)) tell us that the extremal fixed components of the action of H 2 (resp. H 1 ) on X are isolated points.
Notation 5.10. In the sequel, the extremal H 2 -fixed points associated to the weights α i will be denoted by y i , the fixed point components associated to weights β i will be denoted by Y βi,k , and the fixed point components associated to the weight zero will be denoted by Y 0,k (for k in a finite set of indices). Moreover we set Y βi := k Y βi,k , for i = 0, . . . , 5, Y 0 := k Y 0,k .
Notation 5.11. For every i = 0, . . . , 5, the irreducible fixed point components of the H i 1 -action will be denoted by where the first subindex indicates the H i 1 -weight of L, and k belongs to a finite set of indices for every weight. The next statement shows that the components of weights ±1 are unique, so we will simply write Z i ±1 := Z i ±1,k . In particular Z i 1 will contain all the irreducible H 2 -fixed components of weights α i , β i−1 , β i−2 , α i−2 . Moreover for every component Y βi,k there exists a unique H i 1 -fixed component of weight zero containing it; however, it is not true a priori that given a H i 1 -fixed component of weight zero Z i 0,k there exists an H 2 -fixed component of weight β i contained in it. In any case, we may adjust the indices k so that we may write Y βi,k ⊂ Z i 0,k . Proof. We will do the proof in the case i = 1. We note first that, arguing as in Remark 5.9, y 0 is the only isolated H 1 1 -fixed point associated to the weight 2. Then applying Theorem 2.5 (iii), the dimension of H 2 (X, Z) equals the number of irreducible H 1 1 -fixed point components associated to the weight 1. Since Pic(X) Z, we conclude that there is a unique irreducible component Z 1 1 of this kind. We end this step by studying the induced action of H 2 /H i 1 on Z i 1 and Z i 0,k : with bandwidth equal to three and two, respectively.
Proof. As in Notation 5.11, the extremal fixed points of the H 2 /H i 1 -action have weights α i and α i−2 , thus the bandwidth of the action is three. In the case of Z i 0,k we note first that Y βi,k ⊂ Z i 0,k is, by construction, an extremal fixed point of the H 2 /H i 1 -action on Z i 0,k . Applying Lemma 4.6 it follows that −β i appears with multiplicity dim Y βi,k + 1 ≥ 1 in C(Y βi,k , X, H 2 ) ∩ ker(π * i ) = C(Y βi,k , Z i 0,k , H 2 /H i 1 ) (see Lemma 2.2), so Y βi,k Z i 0,k . Then the other extremal component of the H 2 /H i 1 -action on Z i 0,k must be an H 2 -fixed component of weight either zero or −β i = β i+3 . The first case is not possible by Lemma 4.7, so we conclude that the other extremal component is associated to the weight β i+3 , and that the bandwidth of the action is two.

5.3.
Step III: Computing the fixed components and compasses of the action of H 2 . We start with a statement collecting properties of compasses at extremal fixed point components. Keeping in mind that (X, H 2 , L) = (S 2 ) (cf. Lemma 5.8), the result follows by applying [6,Corollary 5.6] to our case.
Proof. It is enough to note that the action of S 2 on (X, L) satisfies [6, Assumptions 5.1], so that the proof follows from [6, Corollary 5.6]; in fact the only nontrivial assumptions to be checked in that list are (4) and (5), and they hold by Lemma 5.8 and Remark 5.9, respectively.
The following lemma, that describes the compass of the H 2 -action on X at an extremal fixed point, is a generalization of [6,Lemma 5.15], where the statement has been proved for n = 3, 4.
Proposition 5.15. Let us consider the action of H 2 on (X, L). The compasses at the fixed point components which are associated to weights different from zero are: Figure 4. Compasses at fixed components y 0 and Y β2 .
Proof. By Lemma 5.14 (1,2) the elements α i+1 − α i , α i−1 − α i , −α i belong to the compass C(y i , X, H 2 ) with multiplicity one. Using item (3) of the same Lemma, β i − α i , β i−1 − α i are the only other possible elements in the compass, and by Lemma 4.5 they occur with equal multiplicity. The extremal fixed components are points (Remark 5.9), hence the number of elements in C(y i , X, H 2 ) is equal to dim X. The multiplicity of β i − α i , β i−1 − α i in the compass must then be equal to n − 1 and the first claim follows.
We now describe the compass at a component Y βi,k . To this end, consider the one dimensional subtorus H i 1 . Taking the component Z i+1 1 containing it (see Lemma 5.12), and applying Lemma 2.2 we get: where the last equality is gotten using the first part of this statement. By Lemma 2.3, the only element of C(Y βi,k , X, H 2 ) that can be projected to 1 is α i − β i , then this element has multiplicity one in C(Y βi,k , X, H 2 ). The same method with a different choice of the one dimensional subtorus shows that also α i+1 − β i occurs with multiplicity one. Using the symmetry given by Lemma 4.5 we obtain that β i+2 − β i , β i−2 − β i have multiplicity one as well. Moreover, applying Lemma 4.6 we know that the element −β i occurs with multiplicity dim Y βi,k + 1. Finally, since the number of elements of C(Y βi,k , X, H 2 ), counted with multiplicity, must be equal to dim X − dim Y βi,k , and using again Lemma 4.5, it follows that the elements β i−1 − β i and β i+1 − β i have both multiplicity equal to n − 2 − dim Y βi,k . Now we determine the fixed point components of the H 2 action on (X, L), by analyzing first in detail the bandwidth three varieties Z i 1 (see Lemma 5.12).
Proposition 5.16. Let Y βi−1 ⊂ Z i 1 be as in Notation 5.11 and 5.10. Then dim Z i 1 = n − 1, the pair (Z i 1 , Y βi−1 ) is one of the following: (D 6 (6), A 5 (4)), (6) (E 7 (7), E 6 (6)), and the restriction of L to every positive dimensional component Proof. As observed in Lemma 5.13, we have a bandwidth three action of we claim first that this action is equalized. In fact, given any by Lemma 2.2. Using the description of C(Y, X, H 2 ) given in Proposition 5.15 for every fixed component Y ⊂ Z i 1 , this shows that the only possible elements of C(Y, Z i 1 , H 2 /H i 1 ) are ±1. Note also that Z i 1 has isolated extremal fixed points y i , y i−2 , hence dim Z i 1 is equal to the number of elements of C(y i , X, H 2 )∩ker π * i , which is n − 1 by the same Proposition.
Since for the proof of Theorem 5.1 we have assumed that n ≥ 5 (see Remark 5.2), we may apply Theorem 2.7 to (Z i 1 , L |Z i 1 ) to obtain the description of the fixed point components Y βi−1,k and of L |Y β i−1 ,k . The fact that Z i 1 cannot be of type (1) in Theorem 2.7 when dim Z i 1 ≥ 3 has been proved in [22,Corollary 5.9].
Proof. Note that, as observed in Lemma 5.13 the action of H 2 /H i 1 C * on (Z i 0,k , L |Z i 0,k ) has bandwidth two. By Proposition 5.15 and Lemma 2.2 (ii) this action is equalized at the sink and the source, hence it is equalized by [21,Lemma 5.8]. Moreover, by Corollary 4.8, Z i 0,k and subsequently every positive dimensional irreducible component of Z i 0,k ∩ Y 0 is a contact manifold whose contact line bundle is the restriction of L.
By Remark 5.17 and Corollary 4.10 the extremal components Y βi,k , Y βi+3,k of the H 2 /H i 1 action on Z i 0,k are isomorphic to each other, and isomorphic to one of the connected components Y of the varieties Y βi appearing in Proposition 5.16. Moreover, again by Corollary 4.10, we get dim Z i 0,k = 2 dim Y + 1. When Y is a point one has dim Z i 0,k = 1, and we obviously have Z i we have dim Z i 0,k = 5, and L |A2(2) O P 2 (2) by Proposition 5.16; using [21, Lemma 2.9 (i)], L is then the second power of the ample generator of Pic(Z i 0,k ), therefore the index of Z i 0,k is six and so Z i 0,k P 5 . Note that a C * -action on P 5 with two fixed disjoint P 2 's does not have other fixed point components. If Y A 2 (2)×A 2 (1) then dim Z i 0,k = 9, and the Picard number of Z i 0,k must be larger than one (see [21,Lemma 2.9(1)]). Then we conclude that Z i 0,k A 5 (1, 5) (see Remark 4.1). In this case the bandwidth two C * -action has been described in [21,Example 5.18], where it is shown that the central component is A 2 (1, 2) A 2 (1,2).
In all the other cases, since the normal bundles N Y β i ,k |Z i 0,k , N Y β i+3 ,k |Z i 0,k are uniquely determined by Y and L, and their duals twisted with L are globally generated (Proposition 4.3), we can use Theorem 2.6 to conclude that the variety Z i 0,k is also determined by these data. In particular all the Z i 0,k are isomorphic to the corresponding H 2 -fixed components obtained in the case in which X is an adjoint variety (see Table 1), so we obtain the list of triples in the statement.  Proof. Applying [6, Corollary 2.14] we know that each element in the compass C(Y 0,k , X, H 2 ) is proportional either to a root α i or β i . We first show that no element of C(Y 0,k , X, H 2 ) can be proportional to a root α i . By contradiction, assume for instance that λα 0 ∈ C(Y 0,k , X, H 2 ), λ ∈ Q >0 . Consider a subtorus H 1 ⊂ H 2 corresponding to the projection π * : M (H 2 ) → M (H 1 ), sending α 0 to 0, and a variety Z ⊂ X H 1 which contains Y 0,k (see Lemma 2.1). Since we have assumed that λα 0 ∈ C(Y 0,k , X, H 2 ), using Lemma 2.2 (ii) we also have λα 0 ∈ C(Y 0,k , Z , H 2 /H 1 ). It follows that Z Y 0,k , and that Z contains an H 2 -fixed point component of weight α 0 . By Lemma 4.5, it contains also a fixed point component of weight α 3 , and using Lemma 2.1 (iii) these two components are the isolated points y 0 and y 3 . Applying Corollary 4.8, Z is a contact manifold, on which the extremal H 2 /H 1fixed components y 0 , y 3 are Legendrian (Corollary 4.10), therefore Z ∼ = P 1 and Y 0,k = ∅, a contradiction.
At this point, by [6,Corollary 2.14], the only possible elements in C(Y 0,k , X, H 2 ) are proportional to the weights β i 's and, since Y 0,k = X, we have at least one element, say β 1 = λ 1 β 1 , λ 1 ∈ Q >0 . Using Lemma 4.5 we deduce that also β 4 := −β 1 ∈ C(Y 0,k , X, H 2 ), with the same multiplicity as β 1 . We take now the subtorus H 1 1 ⊂ H 2 corresponding to the the orthogonal projection π * 1 : M (H 2 ) → M (H 1 ) sending β 1 to zero (see Figure 3). By Lemma 2.2 (i), the existence of β i in the compass of Y 0,k tells us that there exists a component Z 1 0,k containing it and, as in the previous paragraph, one may show that Z 1 0,k contains two H 2 -fixed point components corresponding to the weights β 1 and β 4 ; note that since Z 1 0,k contains Y 0,k these components cannot be isolated points. The possibilities for these extremal fixed point components are listed in Proposition 5.18; each of them determines Z 1 0,k and, subsequently the multiplicities mult(β i ) of the elements β 1 , β 4 as elements of the compass C(Y 0,k , Z 1 0,k , H 2 /H 1 ) = C(Y 0,k , X, H 2 ) ∩ ker(π * 1 ) (cf. Lemma 2.2 (ii)). By Proposition 5.18 this leaves us with the following possibilities: Note that, each possible Z 1 0,k determines the isomorphism class of the fixed components Z i 1 , by Propositions 5.18 and 5.16; in particular this determines the value of n−1 = dim Z i 1 and consequently the dimension on X, that we have written in the fourth column of the table.
By repeating the same argument we can show that every element β i = λ i β i (λ i ∈ Q >0 ) in the compass has the same multiplicity of β 1 and β 4 . Since the number of element in the compass, counted with multiplicity, equals dim X − dim Y 0,k and this number in all the cases is six times the multiplicity of β 1 we can conclude that all the β i 's appear in the compass, with multiplicity (dim X − dim Y 0,r )/6.
Our arguments above also show that Y 0,k is contained in three fixed point components Z i 0,k , i = 1, 2, 3. Finally using the description of the compass at components of type Y βi,k ⊂ Z i 0,k (Proposition 5.15) and Lemma 2.2, we may write which suffices to show that λ i = 1 so that β i = β i , for all i.

5.4.
Step IV: The actions of H 2 on X and X G have the same combinatorial data. The results in the previous Section allow to state the following: Corollary 5.21. Under the assumptions of Theorem 5.1, there exists a simple group G and a bijection ψ : X H2 → X H2 G such that for every Y ∈ X H2 : Proof. Given a pair (X, L) as in ). This shows that there exists a bijection ψ among the set of components of X H2 and the set of fixed point components of X H2 G for a certain simple group G (cf. Table 1), and a torus of dimension two in G that we identify with H 2 (chosen as in Step II). Now we note that, by Proposition 5.20, the above data determine the compass at every component Y 0,k , so that C(Y 0,k , X, H 2 ) = C(ψ(Y 0,k ), X G , H 2 ). In particular 2n+1 = dim X = dim X G and, applying Proposition 5.15 to (X, L) and to (X G , L G ), the compasses C(Y, X, H 2 ), We will show now that X G is the adjoint variety of the group G. We will make use of the following technical lemma: Note that given two positive integers n, m, 2n 2 + n = 2m 2 − m implies that n = (2m − 1)/2, a contradiction. Moreover, one may easily check that the only case in which b n or d m have the same dimension of another Lie algebra of the list is dim(b 6 ) = dim(e 6 ) = 78. In this case, we may study explicitly the projections ı * , described in Lemma 3.5, to conclude that dim(b 6 ) 0 = 24, dim(e 6 ) 0 = 18.
Corollary 5.23. The Lie algebras g and g of the groups G and G are isomorphic and, in particular, X G X G .
Proof. We will show that the assumptions of Proposition 2.4 are fulfilled for the actions of H 2 on (X, L) and (X G , L G ), so that H 0 (X, L) and H 0 (X G , L G ) are isomorphic as H 2 -moduli. Since H 0 (X, L) is isomorphic to g as an H-module, and the same holds for X G , it follows that g g as H 2 -modules. In this way we conclude that they are also isomorphic as Lie algebras by Lemma 5.22.
By Corollary 5.21 we are left to check that L |Y and N −ν (Y ) are isomorphic to their counterparts in X G , for every component Y ⊂ X H2 and every ν ∈ C(Y, X, H).
Since every Y is contained in an H i 1 -fixed point component Z (by Lemma 2.1), and the restriction of L on these components is determined by the isomorphism class of Z (see Propositions 5.16,5.18 and 5.20), the first assumption is fulfilled. Now we check the equality of the graded parts of the normal bundles N −ν (Y ). By Propositions 5.15, 5.20 and Lemma 2.2, the bundles N −ν (Y ) are M(H 2 /H i 1 )graded parts of the normal bundle of Y on the H i 1 -fixed point component containing it (hence they coincide with their counterparts in X G ), for every ν ∈ C(Y, X, H 2 ) different from the following cases (see Figures 4,5): In the first two cases, since y αi is an isolated point, there is nothing to prove. In the latter we simply note that the contact isomorphism restricts to an isomorphism . This completes the proof.

5.5.
Step V: Conclusion. In this final step we will conclude the proof of Theorem 5.1 by applying Proposition 2.4, part (C 2 ). In order to do so we need to study the fixed point components of the action on (X, L) of the maximal torus H of G = CAut • (X), the normal bundles of these components, the restrictions of L, and compare these data with those obtained for (X G , L G ) (see Section 3).
We start by comparing X H and X H G , and the weights of L and L G .
Lemma 5.24. Under the assumptions of Theorem 5.1, dim X = dim X G . Moreover, the H-action on X has only isolated fixed points, X H is in 1-to-1 correspondence with X H G , and the corresponding weights of the action on L are the vertices of the root polytope (G).
Proof. By Corollary 5.4 we know that G is a simple group. Using Corollaries 5.21 and 5.23, we have that X H2 and X H2 G are in 1-to-1 correspondence, and that the compasses at two corresponding components are the same. By Remark 5.9, the extremal fixed points of X and X G are isolated, hence the cardinalities of the compasses at these points equal the dimensions of the varieties in question, and we get dim X = dim X G .
We now consider the downgrading of the actions of H 2 on X and X G to a general one dimensional subtorus H 1 ⊂ H 2 . Applying [7, Lemma 4.1] we obtain that X H1 = X H2 and X H1 G = X H2 G , and we get a bijection among X H1 and X H1 G . By the properties of downgrading (Lemma 2.2 (i)), the compasses of the action of H 1 on X and X G on corresponding components are equal, and so, by Theorem 2.5, the singular homology groups of X and X G are equal as well. In particular, X has no odd degree homology, and the Euler characteristics of X and X G are equal.
On the other hand, again by Theorem 2.5, the Euler characteristic of X G is equal to the cardinality of X H G , which is bijective via the weight map µ L G to the set of long roots of G (Corollary 3.4), that is to the set of vertices of the root polytope (G). Since by Lemma 5.3 one has (G) = (X, L, H) and, by hypothesis, the vertices of (X, L, H) are the weights of a set of isolated fixed points of X, it follows that the action of H on X cannot have other fixed point components (because they would contribute to the Euler characteristic of X). This concludes the proof.
Since by the above lemma X H consists of isolated points, in order to apply Proposition 2.4 we need only to prove that the compasses of the action of H on X are the same as the ones of the corresponding adjoint variety X G . By abuse, we will denote corresponding fixed points in X and X G by the same letter.
Lemma 5.25. Under the assumptions of Theorem 5.1, for every fixed point y ∈ X H the compass C(y, X, H) is equal to C(y, X G , H).
Proof. We will make use of our description of the compasses of the H 2action on X (Propositions 5.15 and 5.20), and of the H i 1 -fixed point components Z i 0,k , Z i 1 , being H i 1 the subtorus of H 2 corresponding to the orthogonal projection π * i : M(H 2 ) → M(H i 1 ) sending β i to 0 (see Figures 3, 4, 5, and Notation 5.11). As we already observed (see the proof of Corollary 5.21, and Corollary 5.23), for every i and every index k the subvarieties Z i 0,k , Z i 1 are the same for X and X G . Given an H-fixed point y ∈ X, we will distinguish three cases, according to the value of ı * (µ L (y)), being ı * : M(H) → M(H 2 ) the projection.
Finally, in the case of a fixed point y such that ı * (µ L (y)) = 0, the argument is analogous: the compass C(y, X, H) will be the union of the compasses of the varieties Z i 0,k containing y, which are the same for X and for X G .

High rank torus actions on contact manifolds
As explained in the Introduction, we use Theorem 5.1 to improve Fang's theorems ( [8,9]). In the language of projective geometry those results can be read as characterizations of (some) adjoint varieties as contact Fano manifolds whose groups of automorphisms have rank bigger than a certain bound. We reduce that bound so that it can be used to characterize most adjoint varieties (see Figure 1): Theorem 6.1. Let (X, L) be a contact Fano manifold of dimension 2n + 1 with Pic X = ZL. Suppose that the identity component G of the group of contact automorphisms of X is reductive of rank r ≥ max(2, (n − 3)/2). Then (X, L) = (X G , L G ), and G is simple of one of the following types: B r (r ≥ 3), D r (r ≥ 4), E 6 , E 7 , F 4 , G 2 .
Remark 6.2. Note that the adjoint variety of type E 8 does not satisfy the assumptions, as in this case dim X G = 57, n = 28, r = 8. By Remark 5.2 we may assume n ≥ 5. On the other hand the hypothesis r ≥ 2 is needed only for the cases n = 5, 6, in which the statement has been proved in [22,Theorem 5.3]. Without loss of generality, we may then assume that n ≥ 7.
The Theorem will follow from Theorem 5.1 by showing that the extremal fixed points of the action of a maximal torus H ⊂ G are isolated. In order to do that, we will use some preliminary results. Proof. As in the proof of [6, Proposition 3.9] we consider a full flag of faces of (X, L, H) containing the vertex determined by the weight of L| Y . Quotienting M(H) by the sublattices generated by the lattice points contained in these faces, we obtain a sequence of subtori: where H i is of dimension r − i, and a sequence of smooth irreducible subvarieties Y =: Y 0 Y 1 · · · Y r−1 satisfying that Y i is a fixed point component for the action of H i ; note that this sequence is strictly increasing by Lemma 2.1. Since each Y i is invariant by H j , for j ≤ i, then we have an action of H i−1 /H i C * on Y i , with Y i−1 as a fixed point component.
In particular Y r−1 is associated to a facet of (X, L, H), so it is an extremal fixed point component for the action of H r−1 C * ; and the weight of the action of H r−1 on L at the fixed component Y r−1 is different from zero. Applying Lemma 4.6, we conclude that Y r−1 is an isotropic submanifold of X; therefore dim Y r−1 ≤ n.
Assume first that, for some i, we have dim Y i −dim Y i−1 = 1. Then, by Lemma 6.3, one has Y i P m for some m. So Y 0 is a positive dimensional subvariety of P m and clearly dim H 0 (Y 0 , L| Y 0 ) > 1.
For the second part of the statement, we note that H 0 (Y, L |Y ) = 0 holds trivially for every zero dimensional extremal fixed point component, hence the conclusion follows by Lemma 5.3.
With all the above materials at hand we may finally prove our main statement.
Proof of Theorem 6.1. In view of Remark 6.2, we assume that n ≥ 7, so that the rank of G is bigger than or equal to (n − 3)/2. By Lemma 6.4 one has Γ(X, L, H) = (X, L, H) = (G). Moreover, because of Lemma 4.2 we deduce that the multiplicities of the weights at the vertices of Γ(X, L, H) are the same as for the adjoint representation, thus equal to 1, that is H 0 (Y, L |Y ) = 1 for every extremal fixed point component. Hence by Lemma 6.4 the extremal fixed components must be isolated points, and we may conclude applying Theorem 5.1.