Minimal bandwidth for ${\mathbb C}^*$-actions on generalized Grassmannians

Abstract

The bandwidth of a ${\mathbb C}^*$-action of a polarized pair (X, L) is a natural measure of its complexity. In this paper, we study ${\mathbb C}^*$-actions on rational homogeneous spaces, determining which provide minimal bandwidth. We prove that the minimal bandwidth is linked to the smallest coefficient of the fundamental weight, in a base of simple roots, which describes the variety as a marked Dynkin diagram. As a direct application of the results we study the Chow groups of the Cayley plane $\mathrm{E}_6(6)$.

1 Introduction

GKM-theory was introduced in [9] to study the topology of a manifold M, using the skeleton structure of fixed points and 1-dimensional orbits provided by the action of a torus $S^1 \times \ldots \times S^1$. On the algebraic side, an analogue theory for complex projective varieties has been developed in [4], using algebraic tori ${\mathbb C}^* \times \ldots \times {\mathbb C}^*$.

1.1 Contents of the paper

Rational homogeneous varieties (RH varieties for short) appear to be the most natural examples to test the tools of the theory. Given a semisimple group G, we may consider the restriction to a maximal torus $H \subset G$ of the natural G-action on the RH variety G/P. The skeleton structure for the H-action on G/P consists of a polytope $\Delta (G/P) \subset {\mathbb R}^n$ of isolated fixed points and information about the decomposition of the (co)tangent space at these points. Then, chosen a 1-dimensional torus ${\mathbb C}^* \subset H$, the action will be encoded on the projection of the polytope to a particular 1-dimensional lattice (see Construction 2.7). The extremal values of the projection of $\Delta (G/P)$ are denoted by $\mu _{\text {max}}$ and $\mu _{\text {min}}$. The bandwidth of the action is defined to be $|\mu |=\mu _{\text {max}}-\mu _{\text {min}}$ (see [26] for the original paper).

It seems natural that the projection which provides the minimal bandwidth must be linked to the symmetries of the polytope $\Delta (G/P)$, which is given by the Coxeter polytope associated to the marked Dynkin diagram (see [19] for an introduction to the topic). The symmetries of $\Delta (G/P)$ are given by some hyperplanes $h_i$ orthogonal to a base of simple roots $\alpha _i$ of the root system $\Phi (G,H)$.

We can restate the problem in terms of convex geometry (Fig. 1). In fact, computing the minimal bandwidth for ${\mathbb C}^*$-actions on X is equivalent to find a projection of minimum length of $\Delta (G/P)$ on a line. Moreover, in the language of [7] [Chapter 13], ${\mathbb C}^*$-actions of minimal bandwidth correspond to slicing the polytope $\Delta (G/P)$ with the minimum number of slices.

Fig. 1

Slicing a cube: vertex-first, edge-first and face-first. The three ways of slicing the cube reflect the downgradings of the action of a maximal torus $H \subset {\text {Sp}}_6({\mathbb C})$ on the variety of 2-dimensional isotropic subspaces of ${\mathbb C}^5$

The main result of this paper is the following.

Theorem 1.1

Let ${\mathbb C}^*$ act on the polarized pair (X, L), where $X={\mathcal D}(j)$ is an RH variety with Picard number one given by a connected Dynkin diagram ${\mathcal D}$ and one of its nodes $\alpha _j$, $L={\mathcal {O}}_X(1)$ is the (very) ample line bundle given by the fundamental weight $\omega _j$. Then the minimal bandwidth and the projections for which the minimal value is attained are given in Table 1, using the notation developed in Sect. 3.

Table 1 Minimal bandwidth for a ${\mathbb C}^*$-action on a generalized Grassmannian ${\mathcal D}(j)$, where ${\mathcal D}$ is the Dynkin diagram marked at the j-th node

1.2 Why should this be interesting?

Combining the statement of Theorem 1.1 with the Białynicki-Birula decomposition (see [1, 2] for the original decomposition or ([6], Theorem 4.2), [4] [Theorem 3.1]),

$$\begin{aligned} H_\bullet (X,{\mathbb Z})= \bigoplus _{i=1}^s H_{\bullet -2\nu _+(i)} (Y_i,{\mathbb Z})=\bigoplus _{i=1}^s H_{\bullet -2\nu _-(i)} (Y_i,{\mathbb Z}), \end{aligned}$$

(1.1)

(see Sect. 2 for the notation) one can study the homology (and, a posteriori, cohomology and Chow) groups of not so well-known RH varieties (for example, the ones arising from exceptional groups), using the simpler homology ring of its fixed-point components. We will show in Theorem 3.6 that the fixed-point components of the fundamental ${\mathbb C}^*$-actions (see Construction 3.1) are varieties with the action of a certain semisimple subgroup $G' \subset G$.

When these $G'$-varieties are RH (that are almost all the cases), our result allows us to write $H_\bullet (G/P,{\mathbb Z})$ as a sum of shifted homology rings on RH $G'$-varieties, with the minimum number of summands. This decomposition may be potentially useful for enumerative-geometric problems related to RH varieties of exceptional type, for which explicit homological descriptions are not so well-known.

1.3 Outline of the paper

Section 2 is devoted to an introduction to the theory of RH varieties and torus actions, in particular ${\mathbb C}^*$-actions. In Sect. 3 we study the properties of fundamental ${\mathbb C}^*$-actions, that provide the projections on the lines orthogonal to the hyperplanes $h_i$. Also the proof of the Theorem 1.1 is contained in this section. Then, in Sect. 4 we describe the ${\mathbb C}^*$-actions on classical cases, combining Representation Theory with techniques of Projective Geometry. Last, in Sect. 5 we apply our results to describe the Chow groups of the Cayley plane $\mathrm{E}_6(6)$, restating, in particular, some results presented on [13].

2 Preliminaries

A projective variety X is an integral, proper, separated scheme of finite type over ${\mathbb C}$ that admits an ample line bundle $L \in {\text {Pic}}(X)$.

2.1 Rational homogeneous spaces

In this section we provide a short introduction to RH varieties. We refer to [17, 18, 20] for details and to [8] for the Representation Theory part.

Let G be a semisimple algebraic group and let $H \subset B \subset P$ be, respectively, a maximal torus, a Borel subgroup and a parabolic subgroup of G. We denote by ${\mathfrak h}\subset {\mathfrak b}\subset {\mathfrak p}\subset {\mathfrak g}$ their corresponding Lie algebras. Moreover, we define $M(H):={\text {Hom}}(H,{\mathbb C}^*)$ to be the group of characters of H. As usual, given $\lambda \in M(H)$ and $t \in H$, we write $t^\lambda :=\lambda (t)$.

The choice of H and B determine a root system $\Phi =\Phi (G,H) \subset M(H)$, a base of simple roots $\{\alpha _1,\ldots ,\alpha _n\}$ and a decomposition $\Phi =\Phi ^+ \cup \Phi ^-$ onto positive and negative roots such that ${\mathfrak g}={\mathfrak h}\oplus \bigoplus _{\alpha \in \Phi } {\mathfrak g}_\alpha$ and ${\mathfrak b}={\mathfrak h}\oplus \bigoplus _{\alpha \in \Phi ^+} {\mathfrak g}_{\alpha }$, where ${\mathfrak g}_\alpha$ denotes the eigenspace associated to the root $\alpha$:

$$\begin{aligned} {\mathfrak g}_\alpha =\left\{ v \in {\mathfrak g}: t \cdot v= t^\alpha v \text { for all }t \in H\right\} . \end{aligned}$$

We denote by $W=W(G,H)$ the Weyl group of G.

Such a root system $\Phi$ can be associated uniquely to a Dynkin diagram ${\mathcal D}$, whose nodes corresponds to a base $\{\alpha _1,\ldots ,\alpha _n\}$ and the labels correspond to the angles between these simple roots (see [11] [Part III]). Moreover, any parabolic subgroup P is determined by its Lie algebra ${\mathfrak p}$ and one can prove (see [8] [p. 394]) that every parabolic Lie algebra is, up to conjugation, of the form

$$\begin{aligned} {\mathfrak p}={\mathfrak b}\oplus \bigoplus _{\alpha \in \Phi ^+ \setminus \Phi ^+(J)} {\mathfrak g}_{-\alpha } \end{aligned}$$

where $\Phi ^+(J)=\{ \alpha \in \Phi ^+ : \exists \, \alpha _j \in J \text { such that } \alpha -\alpha _j \in \Phi ^+ \cup \{0\}\}$ and $J \subset \{\alpha _1,\ldots ,\alpha _n\}$ is a subset of cardinality equals to the Picard number of G/P, hence we write $P=P_J$.

Definition 2.1

An RH variety is a projective variety on which G acts transitively. It is a quotient $G/P_J$ and we can represent it as a marked Dynkin diagram, using the nodes that define J:

$$\begin{aligned} {\mathcal D}(J):=G/P_J, \end{aligned}$$

where ${\mathcal D}$ is the Dynkin diagram for the Lie algebra ${\mathfrak g}$.

We recall that a weight $\omega \in M(H) \otimes {\mathbb R}$ is an element such that $\alpha ^\vee (\omega ):=2(\omega ,\alpha )/(\alpha ,\alpha ) \in {\mathbb Z}$ for all $\alpha \in \Phi$ and we denote the set of weights, called the weight lattice, by ${\mathcal {P}}_\Phi$. In general, it holds ${\mathbb Z}\Phi \subset M(H) \subset {\mathcal {P}}_\Phi$.

Moreover, $\omega$ is dominant if and only if $\alpha ^\vee (\omega ) \ge 0$ for all $\alpha \in \Phi$. If we denote by $\omega _1,\ldots ,\omega _n$ the fundamental weights, i.e. such that $\alpha _i^\vee (\omega _j)=\delta _{ij}$, then a dominant weight can be written as $\omega =\sum _{j=1}^n m_j\omega _j$ with $m_j \ge 0$. Since the Weyl group W acts transitively on the Weyl chambers (see [11] [Theorem 10.3]), up to change the basis using W, we can suppose that a weight $\omega$ is always dominant.

From now on we suppose that G is the simply-connected group of ${\mathfrak g}$, that is the semisimple group such that $M(H)={\mathcal {P}}_\Phi$. In particular $\omega _j \in M(H)$. Because of this choice, we obtain a 1 : 1 correspondence (see [8] [Theorem 23.16]) between line bundles on ${\mathcal D}(J)$ and irreducible G-representations. In fact, given a line bundle L, there exists a unique dominant weight $\omega =\sum _{\alpha _j \in J} m_j \omega _j \in M(H)$ such that

$$\begin{aligned} H^0({\mathcal D}(J), L)=V(\omega ), \end{aligned}$$

where $V(\omega )$ is the irreducible G-representation with highest weight $\omega$, i.e. $\omega - \lambda$ is positive for every $\lambda \in M_\omega$. Moreover, ${\mathcal D}(J)$ is the unique closed G-orbit contained in $\P (V(\omega ))$ (see [8] [Claim 23.52]). A fundamental G-representation is $V=V(\omega _j)$ and the corresponding line bundle L is the generator of the Picard group of ${\mathcal D}(j)$, called a generalized Grassmannian.

2.2 Torus actions: grid data

In this section we recall results that can be found in [4][Section 2] and then in [21, 22, 26].

Let X be a smooth projective variety of dimension d and let $L \in {\text {Pic}}(X)$ be an ample line bundle. We consider a non-trivial action of an algebraic torus $H\simeq ({\mathbb C}^*)^n$ on X and we write $X^H=Y_1 \sqcup \ldots \sqcup Y_s$ for the decomposition of the fixed-point locus into smooth connected components (see [14]). Moreover, we define ${\mathcal Y}:=\{Y_1,\ldots ,Y_s\}$. Now we consider a linearization of the action $\mu _L^H: H \times L \rightarrow L$, that always exists (see [15] [Proposition 2.4] and the rest of the paper for an introduction).

Remark 2.2

We define a weight map (denoted again by) $\mu _L^H: {\mathcal Y}\rightarrow M(H)$ as follow: take a fixed point $p \in Y \subset X^H$, then the H-action on the fiber $L_p$ produces $\mu _L^H(p) \in M(H)$. Moreover, given another $p' \in Y$, we have that $\mu _L^H(p')=\mu _L^H(p)=\mu _L^H(Y)$ (see [10] for the symplectic point of view of the story).

Notation 2.3

An H-action on (X, L) means an H-action on X and a linearization $\mu _L$ of the action on L, which will be intended as the corresponding weight map $\mu _L:{\mathcal Y}\rightarrow M(H)$ defined as above.

Definition 2.4

The convex-hull of $\mu _L^H(Y_1),\ldots ,\mu _L^H(Y_s)$ is called the polytope of fixed points

$$\begin{aligned} \Delta (X)=\Delta (X,L,H,\mu _L) \subset M(H)_{\mathbb R}. \end{aligned}$$

Again, let $p \in X^H$ and consider the induced H-action on the cotangent space $T_p X^\vee$. This action produces a set of characters $\nu _1(p),\ldots ,\nu _d(p) \in M(H)$ and these characters are again constant on the fixed-point component $Y \supset p$, hence we write $\nu _1(Y),\ldots ,\nu _d(Y)$.

Definition 2.5

The set of non-zero characters among $\nu _1(Y),\ldots ,\nu _d(Y)$ is called the compass of the action, denoted by ${\text {Comp}}(Y,X,H)$.

Note that $\dim Y$ is equal to the number of zero characters among $\nu _i(Y)$.

At this point we are interested in polytope of fixed points and compasses of generalized Grassmannians when H is a maximal torus.

Proposition 2.6

( [21], Lemmas 3.1, 3.3) Let (X, L) be a polarized pair, where $X={\mathcal D}(j)$ is a generalized Grassmannian and $L \in {\text {Pic}}X$ is the generator. Consider the action of a maximal torus $H \subset P_j \subset G$:

$$\begin{aligned} H \times X \ni (h, gP_j) \mapsto hgP_j \in X. \end{aligned}$$

Then the fixed points and their compasses can be obtained as follows

Polytope of fixed points:

$X^H=\{wP_j: w \in W\}$ and the character on the fiber over $wP_i$ is $-w(\omega _j)$;

Compass:

${\text {Comp}}(wP_j,X,H)=\{w(\alpha ): \alpha =\sum _i m_i(\alpha )\alpha _i \in \Phi ^+ \text { and } m_j(\alpha )>0\}$.

We stress out that a linearization is always defined up to the choice of a character. Because of this fact, we will use the following linearization:

$$\begin{aligned} \mu _L^H(wP_j):=\omega _j-w(\omega _j). \end{aligned}$$

(2.1)

Thanks to [4] [Corollary 2.18], we obtain $\mu _L^H(wP_j) \in {\mathbb Z}\Phi$.

Construction 2.7

Consider a subtorus $H' \subset H$, this inclusion induces a contravariant map:

$$\begin{aligned} \imath ^*: M(H) \rightarrow M(H'). \end{aligned}$$

(2.2)

We denote by ${\mathcal Y}'$ the set of fixed-point components for the action of $H'$ and by $\mu _L^{H'}: {\mathcal Y}' \rightarrow M(H')$ the correspondent character map. This operation is called a downgrading of the action.

From now on, we will be interested in the case $H' \simeq {\mathbb C}^*$, so $M(H')\simeq {\mathbb Z}$.

Definition 2.8

The source and the sink of the ${\mathbb C}^*$-action are the unique fixed-point components $Y_+, Y_- \in {\mathcal Y}$ such that for a generic $x \in X$ we have, respectively,

$$\begin{aligned} \lim _{t \rightarrow 0} t\cdot x \in Y_+ \quad \text {and}\quad \lim _{t \rightarrow 0} t^{-1}\cdot x \in Y_-. \end{aligned}$$

Alternatively, (see [4][Lemma 2.12]) take a vertex $\mu '_L(Y)$ of $\Delta (X,L,{\mathbb C}^*,\mu '_L)$:

• ${\text {Comp}}(Y,X,{\mathbb C}^*) \subset {\mathbb Z}_{>0}$, then Y is the sink.

• ${\text {Comp}}(Y,X,{\mathbb C}^*) \subset {\mathbb Z}_{<0}$, then Y is the source.

Definition 2.9

Consider a ${\mathbb C}^*$-action on (X, L). The bandwidth of the ${\mathbb C}^*$-action is defined by

$$\begin{aligned} |\mu _L^{{\mathbb C}^*}|=\mu _L^{{\mathbb C}^*}(Y_+)-\mu _L^{{\mathbb C}^*}(Y_-). \end{aligned}$$

We consider the induced ${\mathbb C}^*$-action on the tangent space $T X_{|Y}$, which decomposes as

$$\begin{aligned} T X_{|Y}= TY \oplus N(Y|X)= N^+(Y|X) \oplus TY \oplus N^-(Y|X). \end{aligned}$$

where ${\mathbb C}^*$ acts with positive, zero and negative weights and we write

$$\begin{aligned} \nu _\pm (Y):= {\text {rank}}N^\pm (Y|X). \end{aligned}$$

Last, we introduce a special class of ${\mathbb C}^*$-actions ona polarized pair (X, L): the equalized ones. This condition appears in the hypotheses of [22] [Theorem 8.1] and will be central in future papers. In Sect. 4 we show a few examples of non-equalized ${\mathbb C}^*$-actions.

Definition 2.10

A ${\mathbb C}^*$-action on X is equalized at a fixed-point component Y if all the weights on N(Y|X) are $\pm 1$. An action is equalized if it is equalized at every component.

Remark 2.11

( [22], Remark 2.13) Let C be the closure of a 1-dimensional orbit for the ${\mathbb C}^*$-action and let $f: \P ^1 \rightarrow C$ be its normalization. We lift up the ${\mathbb C}^*$-action on $\P ^1$ obtaining two points $y_+,y_-$ as source and sink and we denote by $\delta (C)$ the weight of the lifted action on $T_{y_+}\P ^1$. The ${\mathbb C}^*$-action is equalized if and only if $\delta (C)=1$ for every closure of 1-dimensional orbit.

Lemma 2.12

( [22], Lemma2.12) Let X be a projective smooth variety with $L \in {\text {Pic}}(X)$. Denote by $Y_+$ and $Y_-$ the sink and the source of a ${\mathbb C}^*$-action on (X, L) and by C the closure of a general orbit. Then

$$\begin{aligned} \mu _L^{{\mathbb C}^*}(Y_+)-\mu _L^{{\mathbb C}^*}(Y_-)=\delta (C) \deg f^*L. \end{aligned}$$

(AM vs FM)

3 Downgradings along simple roots and associated ${\mathbb C}^*$-actions

In this section we describe fundamental ${\mathbb C}^*$-actions on generalized Grassmannians, which is the main topic of this paper.

3.1 ${\mathbb Z}$-gradings associated with simple roots

Let $X=G/P_j$ be a generalized Grassmannian and let $H \subset P_j \subset G$ be a maximal torus contained in the maximal parabolic subgroup $P_j$ of the simply-connected group G.

Construction 3.1

Consider a ${\mathbb C}^*$-action on X, this is equivalent to give a map $\beta : {\mathbb C}^* \rightarrow {\text {Aut}}(X)=G_{ad}$, where $G_{ad}$ is the adjoint group for the Lie algebra ${\mathfrak g}$. Thanks to the Jordan-Chevalley decomposition, we can restrict ourselves to consider the restriction of the map to a maximal torus

$$\begin{aligned} \beta : {\mathbb C}^* \rightarrow H_{ad} \quad \Longrightarrow \quad \beta \in M(H_{ad})^\vee . \end{aligned}$$

Since $M(H_{ad})={\mathbb Z}\Phi$, we obtain that $\beta$ can be thought as a map

$$\begin{aligned} \beta : {\mathbb Z}\Phi \rightarrow M({\mathbb C}^*) \simeq {\mathbb Z}. \end{aligned}$$

We will say that $H_i \subset H$ acts on X with a fundamental ${\mathbb C}^*$-action is the corresponding downgrading is as follow:

$$\begin{aligned} \sigma _i: {\mathbb Z}\Phi&\rightarrow {\mathbb Z}\\ \alpha _j&\mapsto \delta _{ij}. \end{aligned}$$

Note that this construction generalize [27] [p. 38]. In particular, given a general ${\mathbb C}^*$-action on X, this will correspond to a linear combination $\sum _i q_i\sigma _i$ of fundamental ${\mathbb C}^*$-actions.

As remarked in Sect. 2, up to change a change of basis, we can always suppose $q_i \ge 0$ (see [11] [p. 67]).

3.2 The transversal group to a ${\mathbb C}^*$-action

Let G be a semisimple group as above and let $H_i \subset H$ giving a fundamental ${\mathbb C}^*$-action on X given by the downgrading along the simple root $\alpha _i$. Consider the Cartan decomposition of ${\mathfrak g}={\mathfrak h}\oplus \bigoplus _{\alpha \in \Phi } {\mathfrak g}_\alpha$, we obtain a ${\mathbb Z}$-grading on ${\mathfrak g}$ by setting

$$\begin{aligned} {\mathfrak g}_0={\mathfrak h}\oplus \bigoplus _{\begin{array}{c} \alpha \in \Phi \\ m_i(\alpha )=0 \end{array}} {\mathfrak g}_\alpha \quad \text {and} \quad {\mathfrak g}_m:=\bigoplus _{\begin{array}{c} \alpha \in \Phi \\ m_i(\alpha )=m \end{array}} {\mathfrak g}_\alpha \text { if }m \ne 0. \end{aligned}$$

Remark 3.2

We have that $\bigoplus _{m\ge 0} {\mathfrak g}_m$ is the Lie algebra of $P_i \subset G$:

$$\begin{aligned} {\mathfrak p}_i= {\mathfrak h}\oplus \bigoplus _{\alpha \in \Phi ^+} {\mathfrak g}_{\alpha } \oplus \bigoplus _{\alpha \in \Phi ^+ \setminus \Phi ^+(i)} {\mathfrak g}_{-\alpha }={\mathfrak h}\oplus \bigoplus _{\alpha \in \Phi ^+} {\mathfrak g}_\alpha \oplus \bigoplus _{\begin{array}{c} \alpha \in \Phi ^+ \\ m_i(\alpha )=0 \end{array}}{\mathfrak g}_{-\alpha }=\bigoplus _{m \ge 0} {\mathfrak g}_m. \end{aligned}$$

where we recall $\Phi ^+(i)=\{ \alpha =\sum _{h=1}^n m_h(\alpha )\alpha _h \in \Phi ^+: m_i(\alpha )>0\}$.

The first part of the next statement can be found in [27]. The second part is a consequence of the previous remark, using the usual Levi decomposition of a parabolic subgroup.

Proposition 3.3

( [27], p.35) There exists $G_0 \subset G$ reductive such that ${\mathfrak g}_0={\text {Lie}}(G_0)$, the Lie algebra associated to $G_0$. Moreover, $G_0$ is the Levi part of $P_i$ and the positive roots of the associated root system are $\Phi ^+ \setminus \Phi ^+(i)$.

It is a well-known fact that a reductive Lie algebra can be written as ${\mathfrak {g}}_0={\mathfrak {g}}_0^\text {ss} \oplus {\mathfrak {a}}$ where ${\mathfrak {g}}_0^\text {ss}$ is semisimple and ${\mathfrak {a}}$ is abelian. We know that ${\mathfrak h}$ is generated by the coroots $\alpha _1^\vee , \ldots ,\alpha _n^\vee$. Define ${\mathfrak {h}}^\perp$ as the Cartan algebra generated by all the coroots $\alpha ^\vee _1,\ldots ,\alpha ^\vee _n$ but $\alpha _i^\vee$. Then we can write

$$\begin{aligned} {\mathfrak g}^\perp :={\mathfrak h}^\perp \oplus \bigoplus _{\begin{array}{c} \alpha \in \Phi ^+ \\ m_i(\alpha )=0 \end{array}} ({\mathfrak g}_{-\alpha } \oplus {\mathfrak g}_\alpha ) \quad \text {and} \quad {\mathfrak g}_0={\mathfrak g}^\perp \oplus {\mathfrak {a}} \end{aligned}$$

where ${\mathfrak g}^\perp$ is semisimple by construction and ${\mathfrak {a}}$ is the Lie algebra generated, as a vector space, by the coroot $\alpha _i^\vee$ (is abelian because it is a 1-dimensional Lie algebra).

Definition 3.4

The transversal group $G^\perp \subset G$ is the simply-connected semisimple algebraic group such that ${\text {Lie}}(G^\perp )={\mathfrak g}^\perp$.

Let V be an irreducible G-representation, which gives a natural representation of the Lie algebra ${\mathfrak g}\times V \rightarrow V$. We have again a ${\mathbb Z}$-grading induced by the downgrading along $\alpha _i$:

$$\begin{aligned} V = \bigoplus _{\lambda \in M(H)} V_\lambda =\bigoplus _{m \in {\mathbb Z}} V_m \quad \text {with } V_m=\bigoplus _{m_i(\lambda )= m} V_\lambda . \end{aligned}$$

Remark 3.5

Note that by construction every $V_m$ is ${\mathfrak g}^\perp$-invariant, hence a $G^\perp$-module.

Now let $V=V(k_j\omega _j)$, hence $X\subset \P (V)$. Again, the natural G-action on X descend to an H-action and then to a $H_i$-action. In particular

$$\begin{aligned} X^{H_i}=X \cap \P (V)^{H_i}= \bigsqcup _{m \in {\mathbb Z}} (X \cap \P (V_m)). \end{aligned}$$

Let us note that in general $X \cap \P (V_m)$ is not necessarily connected. Consider the action of $G^\perp$: X is $G'$-invariant and also $\P (V_m)$ is $G^\perp$-invariant, hence $X \cap \P (V_m)$ is $G^\perp$-invariant, so it is union of $G^\perp$-orbits.

Theorem 3.6

It holds that $X \cap \P (V_m)$ is a finite union of disjoint $G^\perp$-orbits.

Proof

It is sufficient to prove that:

$$\begin{aligned} T_x (X \cap \P (V_m))= T_x (G^\perp \cdot x) \quad \text {for every } x \in X \cap \P (V_m). \end{aligned}$$

(3.1)

Let $x \in \tilde{X} \subset X \cap \P (V_m)$ where $\tilde{X}$ is the irreducible component of $X \cap \P (V_m)$ containing x. Note that $G^\perp \cdot x \subset \tilde{X}$. If they have the same dimension, then $G^\perp \cdot x$ is dense in ${\tilde{X}}$.

We know that $G^\perp \cdot x$ is smooth, but ${\tilde{X}}$ might not be in general, hence:

$$\begin{aligned} \dim T_x (G^\perp \cdot x)=\dim (G^\perp \cdot x) \le \dim {\tilde{X}} \le \dim T_x {\tilde{X}} \le \dim T_x (X \cap \P (V_m)). \end{aligned}$$

If $T_x (X \cap \P (V_m)) = T_x (G^\perp \cdot x)$, then $\dim {\tilde{X}}=\dim G^\perp \cdot x$.

Suppose that $G^\perp \cdot x \subset {\tilde{X}}$ is dense. If ${\tilde{X}} \setminus G^\perp \cdot x \ne \emptyset$, $\dim G^\perp \cdot x < \dim {\tilde{X}}$, and it will be $G^\perp$-invariant. Take $x' \in {\tilde{X}} \setminus G^\perp \cdot x$. With the same argument one shows that $G^\perp \cdot x' \subset {\tilde{X}}$ is dense and we get a contradiction. So ${\tilde{X}}=G^\perp \cdot x$ for all $x \in {\tilde{X}}$. It follows that X is disjoint union of $G^\perp$-closed orbits.

Let us prove (3.1): $x \in \P (V)$ is of the form $x=[v]$, then

$$\begin{aligned} T_x (G^\perp \cdot x)&=\frac{T_v (G^\perp \cdot v)}{\langle v \rangle }=\frac{{\mathfrak g}^\perp \cdot v}{\langle v \rangle }\\ T_x (X \cap \P (V_m))&\subset T_x X \cap T_x \P (V_m)=\frac{{\mathfrak g}\cdot v}{\langle v \rangle } \cap \frac{V_m}{\langle v \rangle }=\frac{{\mathfrak g}\cdot v \cap V_m}{\langle v \rangle }. \end{aligned}$$

It remains to prove that ${\mathfrak g}^\perp \cdot v = {\mathfrak g}\cdot v \cap V_m$ for all $v \in V_m$, but this follows from Remark 3.5, hence we have $T_x(X \cap \P (V_m)) \subset T_x(G^\perp \cdot x)$. For the other inclusion, since the irreducible component of $X \cap \P (V_m)$ containing x also contains $G^\perp \cdot x$, then $T_x(G^\perp \cdot x) \subset T_x(X \cap \P (V_m))$. $\square$

3.3 Proof of main theorem

Let (X, L) be a polarized pair with an H-action, where $X={\mathcal D}(j)=G/P_j$ and L is the generator of the Picard group.

This section contains proof of Theorem 1.1, that describes the ${\mathbb C}^*$-actions of minimal bandwidth of a generalized Grassmannian. The proof will be obtained in two steps. In the first one we will show that the minimal bandwidth is achieved for one of the fundamental ${\mathbb C}^*$-actions described in Sect. 3.1. In the second step we will show that the bandwidth of the ${\mathbb C}^*$-action associated to $\sigma _i$ can be computed as the difference of the weights of the action on L at two given points, corresponding to the identity and to the longest element $w_\circ$ of the Weyl group of G.

The next statement shows that the minimal bandwidth for a ${\mathbb C}^*$-action on (X, L) is obtained by one of the fundamental ${\mathbb C}^*$-actions $H_i$.

Proposition 3.7

Let $H \subset G$ be a maximal torus. Let $H' \subset H$ be a 1-dimensional subtorus acting (non trivially) on (X, L) with weight map $\mu _L^{H'}: {\mathcal Y}' \rightarrow {\mathbb Z}$. Then there exists $i_0$ such that

$$\begin{aligned} |\mu _L^{H'}| \ge |\mu _L^{i_0}|, \end{aligned}$$

where $\mu _L^{i_0}: {\mathcal Y}_{i_0} \rightarrow {\mathbb Z}$ is the weight map induced by the fundamental ${\mathbb C}^*$-action of $H_{i_0}$ on (X, L).

Proof

Because we have chosen the linearization as in (2.1), the weight map $\mu _L^H$ has its image contained in ${\mathbb Z}\Phi$. On the other hand, the $H'$-action on X correspond, thanks to Construction 2.7, to a map $\sum _i p_i \sigma _i: {\mathbb Z}\Phi \rightarrow {\mathbb Z}$. Hence, we can see the weight map of the $H'$-action on X as

$$\begin{aligned} \mu _L^{H'}=\left( \sum _i p_i \sigma _i\right) \circ \mu _L^H: {\mathcal Y}' \rightarrow {\mathbb Z}\Phi \rightarrow {\mathbb Z}. \end{aligned}$$

In particular, given a fixed-point $wP_j$, the map send it to

$$\begin{aligned} wP_j \mapsto \omega _j -w(\omega _j)=\sum _{i} m_i \alpha _i \mapsto \sum _i p_i m_i. \end{aligned}$$

Because the action is non trivial, there exists at least one index $i_0$ such that $p_{i_0} \ge 1$. $\square$

We finish the proof of Theorem 1.1 by looking at the value of the linearization over the fibers of two particular fixed points.

Remark 3.8

In the decomposition of ${\mathfrak g}/{\mathfrak p}_j$ there are only negative roots of $\Phi$. Thanks to Remark 3.2 we have that ${\mathfrak p}_j={\mathfrak b}\oplus \bigoplus _{\alpha \in \Phi ^+ \setminus \Phi ^+(j)} {\mathfrak g}_{-\alpha }$, hence

$$\begin{aligned} {\mathfrak g}/{\mathfrak p}_j=\bigoplus _{\alpha -\alpha _j \in \Phi ^+ \cup \{0\}} {\mathfrak g}_{-\alpha }. \end{aligned}$$

(3.2)

Lemma 3.9

Consider the fundamental ${\mathbb C}^*$-action of $H_i$ on (X, L), then $eP_j \in Y_-$. Moreover, $eP_j=Y_-$ if and only if $i=j$.

Proof

Let us consider $T_{eP_j} X={\mathfrak g}/{\mathfrak p}_j$. Thanks to Remark 3.8, we consider the induced $H_i$-action on $v \in {\mathfrak g}_{-\alpha }$ where $\alpha =\sum _{h=1}^n a_h \alpha _h \in \Phi ^+$ and $a_j \ge 1$:

$$\begin{aligned} s \cdot v= s^{\sigma _i(\alpha )}v=s^{a_i}v. \end{aligned}$$

So we obtain that $H_i$ acts on $T_{eP_j}X$ with all non-negative weights (all positive iff $i=j$ because $a_j \ge 1$). By definition of compass, it follows that all its elements are non-positive (all negative iff $i=j$). $\square$

Remark 3.10

With the choice of the linearization of (2.1), it follows that $\mu _L^i(eP_j)=0$.

From the general theory (see [12] [Section 1.8]), there exists $w_\circ \in W$ such that $w_\circ \Phi ^+=\Phi ^-$. In particular, if W is not of type $\mathrm{A}_n$ with $n>1$, $\mathrm{D}_{2n+1}$ and $\mathrm{E}_6$, then $w_\circ =-{\text {id}}$. Otherwise

$$\begin{aligned} ({\mathrm{A}_n}) \qquad&w_\circ \alpha _i=-\alpha _{n+1-i}; \\ ({\mathrm{D}_{2n+1}}) \qquad&w_\circ \alpha _{2n+1}=-\alpha _{2n}; \\ ({\mathrm{E}_6}) \qquad&w_\circ \{\alpha _1,\alpha _2,\alpha _3,\alpha _4,\alpha _5,\alpha _6\}=\{-\alpha _6,-\alpha _2,-\alpha _5, -\alpha _4,-\alpha _3,-\alpha _1\}. \end{aligned}$$

Lemma 3.11

Consider the fundamental ${\mathbb C}^*$-action of $H_i$ on (X, L), then $w_\circ P_j \in Y_+$.

Proof

Same proof as for Lemma 3.9, using $w_\circ \Phi ^+=\Phi ^-$. $\square$

Combining the previous results of the section , we obtain the following result.

Theorem 3.12

Let $H_i$ acts on (X, L) with weight map $\mu _L^i$, then

$$\begin{aligned} |\mu _L^i|=\sigma _i\left( \omega _j-w_\circ (\omega _j)\right) . \end{aligned}$$

Proof

We recall that $\mu _L^i= \sigma _i \circ \mu _L^H$. Thanks to the two previous lemmas, $\mu _L^i(Y_-)=\mu _L^i(eP_j)=0$ and $\mu _L^i(Y_+)=\mu _L^i(w_\circ P_j)$, hence by definition

$$\begin{aligned} |\mu _L^i|=\mu _L^i(w_\circ P_j)-0=\sigma _i(\mu _L^H(w_\circ P_j))=\sigma _i(\omega _j-w_\circ (\omega _j)). \end{aligned}$$

$\square$

At this point, to obtain the minimal bandwidth of Table 1, it is enough to look at the tables in [3][p.250-275] for the decomposition of the fundamental weights in the basis of simple roots. Moreover, we can generalize the previous result to the context of arbitrary RH variety, thanks to the linearity of $\sigma _i$.

Corollary 3.13

Let $X={\mathcal D}(J)$ be a RH variety of arbitrary Picard number and let L be an arbitrary line bundle corresponding to the dominant weight $\omega =\sum _{\alpha _j \in J} m_j\omega _j$. Suppose that the ${\mathbb C}^*$-action on (X, L) correspond to $\sum _i p_i \sigma _i \in ({\mathbb Z}\Phi )^\vee$, then using linearization as in (2.1)

$$\begin{aligned} |\mu _L^i|=\sum _{\alpha _j \in J} m_j\sigma _i(\omega _j-w_\circ (\omega _j)). \end{aligned}$$

4 Projective description of classic cases

We will use the numbering of nodes of Dynkin diagrams as in [3].

4.1 The case $\mathrm{A}_n$: smooth drums

We start by $\mathrm{A}_n(1)=\P ^n$. We consider $H \subset {\text {Sl}}_{n+1}({\mathbb C})$, the maximal torus given by the diagonal matrices. At this point we have to compute the polytope of fixed points $\Delta (\P ^n):=\Delta (\P ^n, {\mathcal {O}}(1),H,\mu _{{\mathcal {O}}(1)})$. By [4] [Corollary 2.18] we know that

$$\begin{aligned} \mu _{{\mathcal {O}}(1)}(p')-\mu _{{\mathcal {O}}(1)}(p)=\lambda \nu \end{aligned}$$

where $p,p'$ are fixed points (since H is maximal, every fixed-point component is a fixed point by Proposition 2.6) and $\nu \in {\text {Comp}}(p,\P ^n,H)$. In particular, thanks to Proposition 2.6, we have

$$\begin{aligned} {\text {Comp}}(eP_1,\P ^n,H)=\left\{ \alpha \in \Phi ^+: m_1(\alpha )>0\right\} =\left\{ \alpha _1,\alpha _1+\alpha _2,\ldots ,\alpha _1+\alpha _2+\ldots +\alpha _n\right\} . \end{aligned}$$

Using [19], we know that the polytope of fixed points is an n-simplex. Moreover, because $\mu _{{\mathcal {O}}(1)}(eP_1)=0$, one can reconstruct all the other weights and they corresponds to the elements in the compass!

The H-actions on $\P ^n$ is given by

$$\begin{aligned} \left( t,[x_0:\ldots :x_n]\right) \mapsto \left[ x_0: t^{\alpha _1}x_1:\ldots : t^{\alpha _1+\ldots +\alpha _n}x_n \right] , \end{aligned}$$

where $t^{\sum _i m_i \alpha _i}:=\prod _i t_i^{m_i \alpha _i}$ with $t_i \in {\mathbb C}^*$. Finally, we choose the downgrading along $\alpha _1$:

$$\begin{aligned} (t_1,[x_0:\ldots :x_n]) \mapsto \left[ x_0: t_1^{1}x_1:\ldots :t_1^{1}x_n \right] . \end{aligned}$$

This ${\mathbb C}^*$-action has two fixed-point components:

• (1) the point $[e_0]:=[1:0:\ldots :0]$ with weight 0,

• (2) the hyperplane $\{x_0 = 0\}$ with weight 1.

Hence we recover that the $H_1$-action on $(\P ^n, {\mathcal {O}}(1))$ is 1.

We can repeat all this construction for the Grassmannians $\mathrm{A}_n(k)$, obtaining again minimal bandwidth one. That is because these are examples of RH varieties which are smooth drums in the sense of [22][Section 4]. In fact, [22] [Theorem 4.6] characterizes the varieties X which have a ${\mathbb C}^*$-action of bandwidth one. In the case X is rational homogeneous, it has to be one of the smooth projective horospherical varieties of Picard number one described in [23]. In particular, the description of Grassmannians as horospherical varieties can be found in [23][Proposition 1.9].

4.2 The case $\mathrm{D}_n$

We start with an even dimensional quadric $\mathrm{D}_n(1)$ given by ${\mathcal {Q}}^{2n-2}=\left\{ x_0x_1+\ldots +x_{2n-2}x_{2n-1}=0\right\} \subset \P ^{2n-1}$. Arguing as in the case $\mathrm{A}_n$, the ${\mathbb C}^*$-action obtained by the downgrading along $\alpha _n$:

$$\begin{aligned} \left( t_n,[x_0:\ldots :x_{2n-1}]\right) \mapsto \left[ x_0:t_n^{1}x_1:\ldots :x_{2n-2}:t_n^{1}x_{2n-1} \right] . \end{aligned}$$

The fixed-point components are two $\P ^{n-1}=\mathrm{A}_{n-1}(1)$, as Theorem 3.3 and Proposition 3.6 predict. This is a bandwidth one ${\mathbb C}^*$-action (note that this is [23] [Proposition 1.8]).

If we compute the ${\mathbb C}^*$-action using the downgrading along $\alpha _1$, reduced to the action on $(\mathrm{D}_n(1), {\mathcal {O}}(1))$, the result is

$$\begin{aligned} (t_1,[x_0:\ldots :x_{2n-1}]) \mapsto \left[ t_1^{-1}x_0: t_1x_1 : x_2 : \ldots : x_{2n-1} \right] . \end{aligned}$$

(4.1)

We have three fixed-point components this time:

• (1) the sink $Y_-=[e_0]$ with weight $-1$,

• (2) a smaller quadric, $\mathrm{D}_{n-1}(1)={\mathcal {Q}}^{2n-4}={\mathcal {Q}}^{2n-2} \cap \{x_0=x_1=0\}$, with weight 0,

• (3) the source $Y_+=[e_1]$ with weight 1.

In order to produce a bandwidth two action on $\mathrm{D}_n(k)$ for $k <n-1$, which has to be minimal by Theorem 1.1, we consider the action induced by (4.1) on the isotropic Grassmannians. Because the line spanned by $Y_+$ and $Y_+$ is not contained in the quadric, the invariant lines are:

• (1) any line spanned by $Y_+$ and a point of ${\mathcal {Q}}^{2n-4}$, forming an even dimensional quadric $\mathrm{D}_{n-1}(1)$ with weight 1;

• (2) any line spanned by $Y_-$ and a point of ${\mathcal {Q}}^{2n-4}$, which is $\mathrm{D}_{n-1}(1)$ with weight $-1$;

• (3) any line contained in $\mathrm{D}_{n-1}(1)$; the variety of these lines is $\mathrm{D}_{n-1}(2)$ on which we have weight 0.

We have summarized the situation in Fig. 2.

Fig. 2

The induced action from the downgrading along $\alpha _1$ on $\mathrm{D}_n(2)$

This argument can be generalized for $\mathrm {D}_n(k)$ with $k<n-1$.

Last, we look at the induced action on $\mathrm{D}_n(n)$ (the case of $\mathrm{D}_n(n-1)$ is analogous). It is a well-known fact that there are two families of $\P ^{n-1}$’s contained in ${\mathcal {Q}}^{2n-2}$, that we denote by $\P ^{n-1}_a$’s and $\P ^{n-1}_b$’s. We suppose that $\mathrm{D}_n(n)$ parametrize the family of $\P ^{n-1}_a$’s. At this point, a fixed-point $\P ^{n-1}_a$ in $\mathrm{D}_n(n)$ can be only if two types:

• (1) spanned by $Y_-$ and a $\P ^{n-2}_a$ contained in ${\mathcal {Q}}^{2n-4}$, denoted by $Y_{-1}$,

• (2) spanned by $Y_+$ and a $\P ^{n-2}_a$ contained in ${\mathcal {Q}}^{2n-4}$, $Y_1$.

We expect that this can be reduced to a bandwidth one action, since

$$\begin{aligned} \omega _n=\frac{1}{2}\left( \alpha _1+2\alpha _2+\ldots +(n-2)\alpha _{n-2} +\frac{1}{2}(n-1) \alpha _{n-1}+\frac{1}{2}n\alpha _n\right) . \end{aligned}$$

In fact, this is exactly [23] [Proposition 1.10].

4.3 The case $\mathrm{B}_n$

This case is very similar to the previous one, so we describe only the differences. The odd quadric $\mathrm{B}_n(1)={\mathcal {Q}}^{2n-1}=\left\{ x_0x_1+\ldots +x_{2n-2}x_{2n-1} + x_{2n}^2=0\right\}$ is the closed orbit into the projectivization of the representation given by

$$\begin{aligned} \omega _1=\alpha _1+\ldots +\alpha _n. \end{aligned}$$

The main difference with the $\mathrm{D}_n$-case is that, although we have the same polytope of fixed points (an n-orthoplex), the action given by the downgrading along $\alpha _n$ is given by

$$\begin{aligned} \left( t_n,[x_0:\ldots :x_{2n}]\right) \mapsto \left[ t_n^{-1}x_0:t_nx_1:\ldots :t_n^{-1}x_{2n-2}:t_nx_{2n-1}: x_{2n} \right] . \end{aligned}$$

(4.2)

Hence the $x_{2n}$-coordinate does not allow us to construct the bandwidth one action, contrary to the case of the even quadric ${\mathcal {Q}}^{2n-2}={\mathcal {Q}}^{2n-1} \cap \{ x_{2n}=0\}$.

Remark 4.1

The ${\mathbb C}^*$-action (4.2) on $\mathrm{B}_n(1)$ is not equalized. Thanks to Lemma 2.12 we have that

$$\begin{aligned} 2=\delta (C) \deg f^*{\mathcal {O}}(1) \end{aligned}$$

where C is the closure of a general orbit. Suppose that the closure of such an orbit is a line. Then it is a line passing by the points $p_1:=[x_0:0:x_2:0:\ldots :x_{2n-2}:0:0]$ and $p_2:=[0:x_1:0:x_3:\ldots :0:x_{2n-1}:0]$ which has the form

$$\begin{aligned} \overline{\left\{ [x_0:sx_1:\ldots :x_{2n-2}:sx_{2n-1}:0]: s \in {\mathbb C}^*\right\} }. \end{aligned}$$

But, for a general choice of $p_1$ and $p_2$, the points in the line do not satisfy the equation of ${\mathcal {Q}}^{2n-1}$. Hence the general orbit is at least a conic, hence $\deg f^*{\mathcal {O}}(1)\ge 2$ and $\delta (C) \le 1$. But $\delta (C)$ is an integer strictly bigger than zero, so $\delta (C)=1$ and $\deg f^*{\mathcal {O}}(1)=2$. Now consider $q_1:=[1:0:\ldots :0]$ and $q_2:=[0:\ldots :0:1:0]$, the line passing by them is

$$\begin{aligned} {\tilde{C}}=\overline{\left\{ [1:0:\ldots :0:s:0] : s \in {\mathbb C}^*\right\} } \subset {\mathcal {Q}}^{2n-1}. \end{aligned}$$

So $\delta ({\tilde{C}})=2$ for this line and the action is not equalized by Remark 2.11.

On the other hand, the downgrading along $\alpha _1$ produces the same action of the $\mathrm{D}_n$-case, so we skip the computations. One can also compute the minimal bandwidth of $\mathrm{B}_n(n)$ using the isomorphism $\mathrm{B}_n(n) \simeq \mathrm{D}_{n+1}(n+1)$.

4.4 The case $\mathrm{C}_n$

The isomorphism $\mathrm{C}_n(1) \simeq \mathrm{A}_{2n+1}(1)= \P ^{2n-1}$ gives us a bandwidth one action. This is given by the downgrading along $\alpha _n$, in fact

$$\begin{aligned} \omega _1=\alpha _1+\ldots +\alpha _{n-1}+\frac{1}{2}\alpha _n. \end{aligned}$$

If one consider the ${\mathbb C}^*$-action given by $\alpha _1$, this is exactly the same action as in the case of the quadrics:

$$\begin{aligned} \left( t_1,[x_0:\ldots :x_{2n}]\right) \mapsto \left[ t_1^{-1}x_0:t_1x_1:x_2:\ldots :x_{2n-1} \right] . \end{aligned}$$

Hence the induced action on $\mathrm{C}_n(k)$ gives us a bandwidth two action, where the fixed-point components are:

• (1) a $\mathrm{C}_{n-1}(k-1)$ spanned by $Y_-$ and a $\P ^{k-2} \subset \mathrm{C}_{n-1}(1)=\P ^{2n-3}$, with weight $-1$,

• (2) the isotropic Grassmannian $\mathrm{C}_{n-1}(k)$ with weight 0,

• (3) again a $\mathrm{C}_{n-1}(k-1)$ spanned by $Y_+$ and a $\P ^{k-2} \subset \mathrm{C}_{n-1}(1)$ with weight 1.

There are no other fixed-point components because, if $\Omega$ is the symplectic form defining ${\text {Sp}}(2n)$, then an isotropic line is spanned by [a] and [b] such that $\Omega (a,b)=0$ and $\Omega (Y_-,Y_+)\ne 0$. In particular, there is no isotropic linear space of any dimension passing through $Y_+$ and $Y_-$.

We conclude with a Remark on the ${\mathbb C}^*$-action given by $\alpha _1$ along $\mathrm{C}_n(n)$.

Remark 4.2

The ${\mathbb C}^*$-action on $\mathrm{C}_n(n)$ given by the downgrading along $\alpha _1$ is not equalized. Fix an isotropic $\P ^{n-2} \in \mathrm{C}_{n-1}(1)$. Then consider the $\P ^{n}$ generated by the fixed-point $\P ^{n-2}$, by $Y_+$ and $Y_-$. Inside this $\P ^{n}$ there is a pencil of isotropic $\P ^{n-1}$’s, i.e. a line between the fixed-point components of $\mathrm{C}_n(n)$: consider the line between $Y_+$ and $Y_-$ given by

$$\begin{aligned} C=\overline{\{[\underbrace{1:0:\ldots :0}_n:\underbrace{s:0:\ldots :0}_{n}] \in \P ^{2n-1} : s \in {\mathbb C}^*\}}, \end{aligned}$$

then the $\P ^{n-1}$’s are generated by the fixed-point $\P ^{n-2}$’s and by a point in C.

Then, by Lemma 2.12 it must have $2=\delta (C) \deg f^*{\mathcal {O}}(1)$ and $\deg f^*{\mathcal {O}}(1)=1$ by the previous argument. So we have found the closure of a 1-dimensional orbit for which $\delta (C)=2$, hence the action is not equalized.

In the case $n=2$, this coincide with Remark 4.1.

5 Chow groups of the complex Cayley plane

The Cayley plane X can be described in various ways: it is the fourth Severi variety of dimension 16, hence the results of [16] tell us that $X=\P ^2({\mathbb O})$, where ${\mathbb O}={\mathbf {O}} \otimes _{\mathbb R}{\mathbb C}$ (${\mathbf {O}}$ is the algebra of octonions). We prefer to describe X as an RH variety, hence $X:=\mathrm{E}_6(6)$.

Remark 5.1

Thanks to Theorem 1.1, the minimal bandwidth of X is 2 and a fundamental ${\mathbb C}^*$-action that attains this minimum is given by the downgrading along $\alpha _6$ (using the numbering of [3]). Moreover, using Theorem 3.3, the fixed-point components $Y_0,Y_1,Y_2$ are of type $\mathrm{D}_5$.

Proposition 5.2

Consider the fundamental ${\mathbb C}^*$-action of $H_6$ on $X \subset \P (V(\omega _6))$. Then

$$\begin{aligned} X^{H_6}=\{*\} \sqcup \mathrm{D}_5(5) \sqcup \mathrm{D}_5(1). \end{aligned}$$

Moreover, the links between fixed-point components are given by

Proof

The fundamental representation $V(\omega _6)$ has dimension 27, hence $X \subset \P ^{26}$ and the action of the maximal torus H on X has at most 27 fixed points.

Using Lemma 3.9, we conclude that $Y_0$ is a point. Moreover, the compass for H at $Y_2$ is given by

$$\begin{aligned} {\text {Comp}}(Y_2,X,H)=\{\alpha \in \Phi ^+(\mathrm{E}_6,H) : \alpha -\alpha _6 \ge 0\} \end{aligned}$$

thanks to Proposition 2.6.

Let us note that there are exactly 16 elements in this compass, one for every fixed point for H which is sent to $Y_1$ by the downgrading along $\alpha _6$. Hence $Y_1 \subset \P ^{15}$ is an RH variety of type $\mathrm{D}_5$. The only variety with these properties is the 10-dimensional spinor variety $\mathrm{D}_5(5)$ (cf. [13] [Proposition 4.2]). Last, there are 6 elements in the compass for the action of H at $Y_1$. Every point of $Y_1$ is linked to the fixed point $Y_2$ by a single direction in the compass, hence

$$\begin{aligned} \nu _+(Y_1):=|\{ \nu _i(Y_1) >0\}|=1 \quad \text {and} \quad \nu _-(Y_1):=|\{\nu _i(Y_1) <0\}|=5. \end{aligned}$$

This information allow us to recover, respectively, the $\P ^0$ and the $\P ^4$ on the diagram.

Finally, We know that $Y_1$ has to parametrize a family of $\P ^4$ of $Y_2$, which is RH of type $\mathrm{D}_5$. It follows that $Y_0=\mathrm{D}_5(1)$. Last, there are 8 (negative) elements in the compass at $Y_2$, and the incidence diagram tells us that every point in $Y_2$ is linked to a 6-dimensional quadric $\mathrm{D}_4(1) \subset \P ^7$, as we expect. $\square$

We summarize the information about positive and negative elements in the compass at the fixed-point components in the following table.

$$\begin{aligned} \begin{array}{c|c|c|c} &{} Y_0 &{} Y_1 &{} Y_2 \\ \hline \nu _+(Y_i) &{} 16 &{} 5 &{} 0 \\ \hline \nu _-(Y_i) &{} 0 &{} 1 &{} 8 \end{array} \end{aligned}$$

Using this table we can rewrite the positive decomposition (1.1) as

$$\begin{aligned} H_\bullet (X,{\mathbb Z})=H_{\bullet }(\mathrm{D}_5(1),{\mathbb Z}) \oplus H_{\bullet -10}(\mathrm{D}_5(5),{\mathbb Z}) \oplus H_{\bullet -32}(\{*\},{\mathbb Z}). \end{aligned}$$

(5.1)

We are in the hypothesis of the Poincaré duality, hence we have

$$\begin{aligned} H_m(Y_i,{\mathbb Z}) \simeq H^{2\dim (Y_i)-m}(Y_i,{\mathbb Z}), \end{aligned}$$

and we obtain the additive decomposition of the cohomology ring as

$$\begin{aligned} H^\bullet (X,{\mathbb Z})=H^{\bullet -16}(\mathrm{D}_5(1),{\mathbb Z}) \oplus H^{\bullet -2}(\mathrm{D}_5(5),{\mathbb Z}) \oplus H^{\bullet }(\{*\},{\mathbb Z}). \end{aligned}$$

(5.2)

Finally we can decompose additively the Chow groups of X as

$$\begin{aligned} A^\bullet (X)=A^{\bullet -8}(\mathrm{D}_5(1)) \oplus A^{\bullet -1}(\mathrm{D}_5(5)) \oplus A^\bullet (\{*\}). \end{aligned}$$

(5.3)

Remark 5.3

As pointed out in [13] [Section 3], one can study the Chow ring of X using the corresponding Hasse diagram. In particular, Iliev and Manivel describe a Schubert subvariety of X isomorphic to $\mathrm{D}_5(5)$, finding a Hasse sub-diagram that corresponds to the one associated to $\mathrm{D}_5(5)$.

Our description in terms of ${\mathbb C}^*$-action on X allows us to interpret the complement of the Hasse sub-diagram of $\mathrm{D}_5(5)$ in X as the union of two sub-diagrams, corresponding to the Chow groups of the sink (a point) and the source ($\mathrm{D}_5(1)$) of the ${\mathbb C}^*$-action.

By the way, we cannot recover the ring structure of $A^\bullet (X)$, because our methods are based essentially on the Białynicki-Birula theorem. This requires a case-by-case analysis, as, for example, Iliev and Manivel did in [13]. For sure, the knowledge of such fixed-point decomposition can help the analysis (Fig. 3).

Fig. 3

Hasse diagram of the Chow ring of X

Fig. 4

Shifted Hasse sub-diagrams corresponding to the fixed-point components $\mathrm{D}_5(5)$ and $\mathrm{D}_5(1)$

Note that these arguments can be redone for an arbitrary generalized Grassmannian of exceptional type $\mathrm{E}_7, \mathrm{E}_8, \mathrm{F}_4, \mathrm{G}_2$. The recipe is as follows.

• (1) Choose the generalized Grassmannian ${\mathcal D}(j)$ such that $H_j$ provides the ${\mathbb C}^*$-action of minimal bandwidth. By Lemma 3.9, the sink is an isolated point.

• (2) Because in the Weyl group of G we have $w_\circ =-{\text {id}}$, also the source is an isolated fixed point.

• (3) Now, use the compass to understand which are the fixed-point components next to the sink and the source. They are the VMRT of ${\mathcal D}(j)$.

• (4) At this point, because the minimal bandwidth of ${\mathcal D}(j)$ is at most 4, it may remain to understand the middle fixed-point component. Arguing as in the case of $\mathrm{E}_6(6)$, one can understand what is the missing fixed-point component.

• (5) If we are interested in another RH variety ${\mathcal D}(k)$, then ${\mathcal D}(k)$ parametrizes a family of certain subvarieties of ${\mathcal D}(j)$ and one can induce the fixed-point components of the action on ${\mathcal D}(k)$ using the ones of ${\mathcal D}(j)$ as we have done in the section of classic cases.

Remark 5.4

The same picture of Fig. 4 appears in [24]. In the language of Pech, the Hasse diagram describes the $P_1$-orbits in $\mathrm{E}_6(6)$, which are nothing but vector bundles over the fixed-point components $Y_i$. Our picture describes, in the language of Pech, the $P_6$-orbits in $\mathrm{E}_6(6)$. In fact, let us note that by Remark 3.2 we have $G_0 \subset P_6$ as the reductive part. As pointed out by Perrin in [25] [Proposition 5] the $P_6$-orbits are of the form $P_6 wP_6/P_6$ for $w \in W(\mathrm{E}_6)$ and there is an affine fibration

$$\begin{aligned} f: P_6 wP_6/P_6 \rightarrow G_0/(G_0 \cap {\text {conj}}_w(P_6))\simeq G^\perp /(G^\perp \cap {\text {conj}}_w(P_6)). \end{aligned}$$

Thanks to Theorem 3.6, $Y_i$ are of the form $G^\perp /(G^\perp \cap {\text {conj}}_w(P_6))$.

On the other hand we have the Theorem of Białynicki-Birula (see the statement in [26]): define for a fixed-point component $Y \in {\mathcal Y}$ the cells

$$\begin{aligned} X^\pm (Y):= \left\{ x \in X : \lim _{t \rightarrow 0} t^{\pm 1} \cdot x \in Y\right\} , \end{aligned}$$

then there is ${\mathbb C}^*$-isomorphism $X^\pm (Y) \simeq N^\pm (Y|X)$ and the natural maps $X^\pm (Y) \rightarrow Y$ are ${\mathbb C}^{\nu _\pm (Y)}$-fibrations. Now, thanks to the first formula of [5], we have

$$\begin{aligned} H_{\bullet -2\nu _\pm (Y)}(X^\pm (Y), {\mathbb Z}) \simeq H_{\bullet -2\nu _\pm (Y)}(Y,{\mathbb Z}) \end{aligned}$$

and we recover our description in terms of Hasse diagrams via the fixed-point components.