| Sbornik: Mathematics |
|
|
|
|
|
This article is cited in 1 scientific paper (total in 1 paper) The Hartogs extension phenomenon in almost homogeneous algebraic varieties S. V. Feklistov Siberian Federal University, Krasnoyarsk, Russia
Bibliographic databases:
     § 1. IntroductionThe classical Hartogs extension theorem states that for every domain $W\subset\mathbb{C}^{n}$, $n>1$, and a compact set $K\subset W$ such that $W\setminus K$ is connected the restriction homomorphism
$$
\begin{equation*}
H^{0}(W,\mathcal{O})\to H^{0}(W\setminus K, \mathcal{O})
\end{equation*}
\notag
$$
is an isomorphism. A natural question arises if this is true for complex analytic spaces. Definition 1. We say that a noncompact connected complex analytic space $X$ admits the Hartogs phenomenon if for any domain $W\subset X$ and a compact set $K\subset W$ such that $W\setminus K$ is connected, the restriction homomorphism is an isomorphism. In this or a similar formulation this phenomenon has extensively been investigated in many situations, including Stein manifolds and spaces, $(n-1)$-complete normal complex spaces and so on; see [2]–[6], [10], [11], [16], [20], [22], [26] and [27]. Our goal is to investigate the Hartogs phenomenon in almost homogeneous algebraic varieties. Let $X$ be a complex analytic variety (that is, a reduced, irreducible complex analytic space) and $G$ be a connected complex Lie group acting holomorphically on $X$. In this situation $X$ is called a complex analytic $G$-variety. A complex analytic $G$-variety $X$ is said to be almost homogeneous if $X$ has an open $G$-orbit $\Omega$; see [1]. For example, toric varieties, horospherical varieties, flag varieties (an more generally, spherical varieties) are almost homogeneous algebraic varieties. In the context of toric varieties the Hartogs phenomenon was considered in [12], [18] and [19]. We will follow an approach going back to Serre [23]. First we prove a result about cohomology vanishing for some class of complex analytic varieties. Theorem 1. Let $X$ be a noncompact complex analytic variety possessing the following properties: Then $X$ admits the Hartogs phenomenon if and only if $H^{1}_{c}(X,\mathcal{O})=0$. We apply Theorem 1 to normal noncompact complex analytic varieties that possess some properties related to their compactifications. First we give the following definition. Definition 2. A noncompact complex analytic variety $X$ is called $(b,\sigma)$-compactifiable if it admits a compactification $X'$ with the following properties: If $X$ is normal (algebraic), then also we require $X'$ to be normal (algebraic, respectively). If $X$ is a $G$-variety, then we require $X'$ to be a $G$-variety and the compactification map to be $G$-equivariant. Note that the number $\sigma$ is called the irregularity of $X'$ (see [15] for this notion in the context of projective surfaces). The number $b$ is related to the number $e(X)$ of topological ends of $X$ (see [21] for topological ends and [27] for the relation to the Hartogs phenomenon). Namely, we have $b\leqslant e(X)$. In this paper we consider only the case $b=1$, $\sigma=0$. This means that $X'\setminus X$ is connected and $H^{1}(X',\mathcal{O})=0$. For a $(1,0)$-compactifiable complex analytic variety $X$ we have the canonical isomorphism
$$
\begin{equation*}
H^{1}_{c}(X,\mathcal{O})=H^{0}(Z,i^{-1}\mathcal{O})/\mathbb C,
\end{equation*}
\notag
$$
where $Z=X'\setminus X$ and $i\colon Z\hookrightarrow X'$ is the closed embedding. Now let $X$ be a normal $(1,0)$-compactifiable almost homogeneous algebraic $G$-variety, where $G$ is acting algebraically on $X$, and let $X'$ be the corresponding compactification of $X$. Denote the set of $G$-stable prime divisors on $X'$ by $\mathcal{G}(X')$ and define
$$
\begin{equation*}
Y:=X'\setminus \bigcup_{D\in\mathcal{G}(X'),D\subset X} D.
\end{equation*}
\notag
$$
Note that $\{D\in\mathcal{G}(X')\mid D\subset X\}$ can be an empty set. Let $G$ be a connected complex reductive Lie group, $B\subset G$ be a Borel subgroup and $T\subset B$ be a maximal algebraic torus with character lattice $\mathfrak{X}(T)$, lattice of 1-parameter subgroups $\mathfrak{X}^{*}(T):=\operatorname{Hom}(\mathfrak{X}(T),\mathbb{Z})$ and set of dominant characters $\mathfrak{X}_{+}(T)$. In Proposition 2 we prove that the algebra of regular functions $\mathbb{C}[Y]$ is a dense subspace of the topological vector space $H^{0}(Z,i^{-1}\mathcal{O})$ (with the direct limit topology). It follows that $H^{1}_{c}(X,\mathcal{O})=0$ if and only if $\mathbb{C}[Y]=\mathbb{C}$. Since $G$ acts algebraically on $Y$, it follows that $\mathbb{C}[Y]$ is a representation of $G$. Now define the weight monoid of $Y$ by
$$
\begin{equation*}
\Lambda_{+}(Y):=\bigl\{\lambda\in \mathfrak X_{+}(T)\mid \mathbb C[Y]_{\lambda}^{(B)}\neq 0\bigr\},
\end{equation*}
\notag
$$
where $\mathbb C[Y]_{\lambda}^{(B)}:=\bigl\{f\in\mathbb C[Y]\mid \exists\,\lambda\in\mathfrak X(T)\colon b.f=\lambda(b)f\ \forall\, b\in B\bigr\}$. We obtain the following weight criterion. Theorem 2. Let $G$ be a connected complex reductive Lie group, and let $X$ be a normal $(1,0)$-compactifiable almost homogeneous algebraic $G$-variety. Then $X$ admits the Hartogs phenomenon if and only if $\Lambda_{+}(Y)=0$ and $\mathbb{C}[Y]^{B}=\mathbb{C}$. In this paper we consider so-called spherical $G$-varieties. These are almost homogeneous algebraic $G$-varieties, where $G$ is a complex reductive group and a Borel subgroup $B\subset G$ acts on $X$ with open orbit. Let $X$ be a spherical $G$-variety with open $G$-orbit $\Omega$. We define the weight lattice by
$$
\begin{equation*}
M:=\bigl\{\lambda\in \mathfrak X(T)\mid \mathbb C(\Omega)^{(B)}_{\lambda}\neq0\bigr\}.
\end{equation*}
\notag
$$
Note that $M$ is a sublattice of the character lattice $\mathfrak{X}(T)$. Let $M_{\mathbb{R}}:=M\otimes\mathbb{R}$. Denote the set of all $B$-stable prime divisors on $Y$ by $\mathcal{B}(Y)$. Each $B$-stable divisor $D\in \mathcal{B}(Y)$ defines a discrete valuation
$$
\begin{equation*}
v_{D}\colon \mathbb C(\Omega)\setminus\{0\}\to \mathbb{Z}.
\end{equation*}
\notag
$$
Recall that $v_{D}(f)$ is the order of the zero or pole of $f$ at $D$. Also, $v_{D}$ defines a point $a_{D}\in N:=\operatorname{Hom}(M,\mathbb{Z})$ in the dual weight lattice by the formula $\langle a_{D},\lambda\rangle:=v_{D}(f)$ for $f\in \mathbb{C}(\Omega)^{(B)}_{\lambda}$. Consider the following cone in the space $N_{\mathbb{R}}$:
$$
\begin{equation*}
C:=\mathbb{R}_{\geqslant 0}\bigl\langle a_{D}\mid D\in \mathcal{B}(Y)\bigr\rangle.
\end{equation*}
\notag
$$
We have the following convex geometric criterion of the Hartogs phenomenon. Corollary 1. Let $X$ be a $(1,0)$-compactifiable spherical variety. Then $X$ admits the Hartogs phenomenon if and only if $C=N_{\mathbb{R}}$. Note that this criterion can also be formulated in terms of coloured fans. Briefly speaking, a spherical variety with open $G$-orbit $\Omega$ is encoded by a coloured fan, a collection of strictly convex cones with common apex in a real vector space that can intersect only along their common faces. For more details, see § 6 in this paper or [13] and [28]. Let $X_{\Sigma}$ be a spherical variety with coloured fan $\Sigma$. We denote the support of $\Sigma$ by $|\Sigma|$ and denote the closure of $N_{\mathbb{R}}\setminus |\Sigma|$ in $N_{\mathbb{R}}$ by $\overline{N_{\mathbb{R}}\setminus |\Sigma|}$. We let $\mathcal{V}(\Omega)$ denote the finitely generated convex rational cone in $N_{\mathbb{Q}}=N\otimes\mathbb{Q}$ consisting of all $G$-invariant valuations (see [13], §§ 4 and 10, or § 6 in this paper), and let $\mathcal{V}_{\mathbb{R}}(\Omega)$ denote the cone generated by the set $\mathcal{V}(\Omega)$ in $N_{\mathbb{R}}$. We denote the set of all prime $B$-stable divisors of $\Omega$ by $\mathcal{B}(\Omega)$. Note that a noncompact spherical variety $X_{\Sigma}$ is $(1,0)$-compactifiable if and only if $\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|$ is a connected set (see Lemma 4). We have the following main result. Theorem 3. Let $X_{\Sigma}$ be a noncompact spherical $G$-variety with open $G$-orbit $\Omega$ and coloured fan $\Sigma$ such that $\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|$ is a connected set. Then $X_{\Sigma}$ admits the Hartogs phenomenon if and only if In particular, let $X_{\Sigma}$ be a noncompact horospherical variety with open $G$-orbit $\Omega$. Let $U$ be the unipotent radical of the Borel subgroup $B$, $S$ be the set of simple roots with respect to $B$ and $S^{\vee}$ be the set of dual simple roots. Let $H$ be the stabilizer of a point $o\in\Omega$ such that $H\supset U^{-}$. Consider a parabolic subgroup $P\supset B$ such that $P^{-}=N_{G}(H)$. Note that the parabolic subgroups containing a fixed Borel subgroup $B$ are parametrized by subsets of simple roots $I\subset S$. Let $I$ be the subset of $S$ that corresponds to the parabolic subgroup $P$. The injective map $\iota\colon M_\mathbb{R}\hookrightarrow \mathfrak{X}(T)\otimes\mathbb{R}$ induces the surjective map
$$
\begin{equation*}
\iota^{*}\colon \mathfrak X^{*}(T)\otimes\mathbb R\twoheadrightarrow N_{\mathbb R}:=N\otimes\mathbb R.
\end{equation*}
\notag
$$
We obtain the following corollary. Corollary 2. Let $X_{\Sigma}$ be a noncompact horospherical $G$-variety with open $G$-orbit $\Omega$ and coloured fan $\Sigma$ such that $N_{\mathbb{R}}\setminus |\Sigma|$ is a connected set. Then $X_{\Sigma}$ admits the Hartogs phenomenon if and only if where $(S\setminus I)^{\vee}=\{\alpha^{\vee}\mid\alpha\in S\setminus I\}$. § 2. Cohomological criterion for the Hartogs phenomenonWe consider only reduced, irreducible complex analytic spaces (that is, complex analytic varieties). Serre established the Hartogs phenomenon for Stein manifolds using the triviality of the cohomology group with compact support $H^{1}_{c}(X,\mathcal{O})$, where $\mathcal{O}$ is the sheaf of holomorphic functions [23]. For more details about sheaf cohomology with compact support, see [6]. We consider a class of complex analytic varieties such that in this class the triviality of $H^{1}_{c}(X,\mathcal{O})$ is a necessary and sufficient condition for the Hartogs phenomenon. Theorem 1. Let $X$ be a noncompact complex analytic variety with the following properties: Then $X$ admits the Hartogs phenomenon if and only if $H^{1}_{c}(X,\mathcal{O})=0$. Proof. First we assume that $H^{1}_{c}(X,\mathcal{O})=0$. Let $W\subset X$ be a domain and $K\subset W$ be a compact set such that $W\setminus K$ is connected. Note that since $X$, $W$ and $W\setminus K$ are connected sets, it follows that $X\setminus K$ is connected set.
We need the following lemma. Lemma 1. Let $K\subset X$ be a compact set such that $X\setminus K$ is connected. If $H^{1}_{c}(X,\mathcal{O})=0$, then the restriction homomorphism $H^{0}(X,\mathcal{O}) \to H^{0}(X\setminus K,\mathcal{O})$ is an isomorphism. Proof.
We consider the following exact sequence of cohomology groups [6]:
Obviously, $H^{0}_{K}(X,\mathcal{O})=0$. So the restriction homomorphism $R_{K}$ is injective. Recall that if $S$ and $T\subset X$ are compact sets and $S\subset T$, then we obtain the canonical homomorphism $\phi_{ST}\colon H^{1}_{S}(X,\mathcal{O})\to H^{1}_{T}(X,\mathcal{O})$. Moreover, we have the canonical isomorphism [6]
$$
\begin{equation*}
\varinjlim_{S} H^{1}_{S}(X,\mathcal{O})\cong H^{1}_{c}(X,\mathcal{O}),
\end{equation*}
\notag
$$
where the inductive limit is taken over all compact subsets $S$ of $X$ (or a cofinal part of them).
Let $f\in H^{0}(X\setminus K,\mathcal{O})$, and consider $F_{K}(f)\in H^{1}_{K}(X,\mathcal{O})$. Since ${H^{1}_{c}(X,\mathcal{O})=0}$, it follows that there exists a compact set $K'\subset X$ such that $K\subset K'$ and $\phi_{KK'}(F_{K}(f))=0$. Note that the following diagram is commutative: It follows that $F_{K'}(f|_{X\setminus K'})=0$. Thus there exists $g\in H^{0}(X,\mathcal{O})$ such that $g|_{X\setminus K'}=f|_{X\setminus K'}$. Recall that $X\setminus K$ is a connected set. Using the uniqueness theorem we obtain $g|_{X\setminus K}=R_{K}(g)=f$. The proof of the lemma is complete. The lemma is proved. Further, we have the following commutative diagram for $K\subset W\subset X\subset X'$: Using Lemma 1 we obtain that the restriction homomorphism $R_2$ is an isomorphism. Since $X\cap (X'\setminus K)=X\setminus K$, it follows that $R_{1}$ is an isomorphism. Since $H^{1}(X',\mathcal{O})=0$, it follows that $H^{1}_{K}(X',\mathcal{O})=0$. Using the excision property [6] we see that $H_{1}$ and $H_{2}$ are canonical isomorphisms. Hence $H^{1}_{K}(W,\mathcal{O})=0$ and the restriction homomorphism $R_{3}$ is an isomorphism. Now we assume that $X$ admits the Hartogs phenomenon. In particular, the restriction homomorphism $H^{0}(X,\mathcal{O})\to H^{0}(X\setminus V_{n},\mathcal{O})$ is an isomorphism, where $\{V_{n}\}$ is a compact exhaustion such that $X\setminus V_n$ is a connected set for any $n$. We have the following commutative diagram for $V_{n}\subset X\subset X'$: Since $R_{2}$ is an isomorphism, so is $R_1$ too. Since $H^{1}(X',\mathcal{O})=0$, it follows that $H^{1}_{V_{n}}(X,\mathcal{O})\cong H^{1}_{V_{n}}(X',\mathcal{O})=0$. By assumption $\{V_{n}\}$ is a compact exhaustion of $X$. Thus $H^{1}_{c}(X,\mathcal{O})\cong \varinjlim_{n} H^{1}_{V_n}(X,\mathcal{O})=0$. The proof of the theorem is complete. Let $X$ be a normal $(1,0)$-compactifiable complex analytic variety, $X'$ be a corresponding compactification of $X$, and let $Z:=X'\setminus X$. Then we have the following corollary. Corollary 3. A normal $(1,0)$-compactifiable complex analytic variety $X$ admits the Hartogs phenomenon if and only if where $i\colon Z\hookrightarrow X'$ is the closed embedding. Proof. First we have the following exact sequence:
$$
\begin{equation*}
0 \to H^{0}_{c}(X,\mathcal{O}) \to H^{0}(X',\mathcal{O}) \to H^{0}(Z,i^{-1}\mathcal{O}) \to H^{1}_{c}(X,\mathcal{O}) \to H^{1}(X',\mathcal{O}) \to \cdots
\end{equation*}
\notag
$$
(see [7], Ch. II, § 10.3).
Since $H^{0}_{c}(X,\mathcal{O})=0$, $H^{0}(X',\mathcal{O})=\mathbb{C}$ and $H^{1}(X',\mathcal{O})=0$, it follows that
$$
\begin{equation*}
H^{1}_{c}(X,\mathcal{O})\cong H^{0}(Z,i^{-1}\mathcal{O})/\mathbb C.
\end{equation*}
\notag
$$
Let $U\subset X'$ be a connected open neighbourhood of $Z$. Since $U$ is also a normal space and $Z$ is a thin set in $U$, $U\setminus Z$ is a connected set by the criterion of connectedness (see [14], Statement 13.8). Thus, $V=X'\setminus U$ is a compact subset of $X$, and $X\setminus V$ is connected. A sequence of nested connected neighbourhoods $\{U_{n}\}_{n=1}^{\infty}$ of $Z$ with properties $\overline{U_{n+1}}\subset U_{n}$ and $\bigcap_{n}U_{n}=Z$ induces a sequence of compact sets $\{V_{n}\}_{n=1}^{\infty}$ in $X$, giving an exhaustion of $X$ such that $X\setminus V_{n}$ is connected. The existence of such a sequence $\{U_{n}\}_{n=1}^{\infty}$ follows from the metrizability of $X'$. Indeed, let $\rho$ be a metric compatible with the topology of $X'$. We define open sets by
$$
\begin{equation*}
U_{n}:=\biggl\{x\in X'\biggm| \inf_{z\in X'\setminus X}\rho(z,x)<\frac{1}{n}\biggr\}, \qquad n=1,2,\dots\,.
\end{equation*}
\notag
$$
The sequence $\{U_{n}\}_{n=1}^{\infty}$ has the properties $\overline{U_{n+1}}\subset U_{n}$ and $\bigcap_{n}U_{n}=Z$. The proof of the corollary is complete. For example, let $X$ be a normal noncompact algebraic $G$-variety and $G$ be a complex linear algebraic group acting algebraically on $X$. By Sumihiro’s results (see [25], Theorem 3) there exists a $G$-equivariant compactification $X'$ of $X$ (that is, there exists a normal compact algebraic variety $X'$ on which an algebraic action of $G$ is defined such that $X$ is embedded in $X'$ as an open subset and the algebraic action of $G$ on $X'$ is an extension of the given algebraic action of $G$ on $X$). We assume additionally that X is (b, 0)-compactifiable with $b \geqslant 1$. If ${b>1}$, then the analytic set $Z=X'\setminus X$ is not connected. However, there exists a $G$-invariant Zariski open subvariety $X''\subset X'$ such that $X\subset X''$ and $X'\setminus X''$ is connected (that is, $X''$ is $(1,0)$-compactifiable). In this case, if $X''$ admits the Hartogs phenomenon, then so does $X$. § 3. Almost homogeneous $G$-varietiesLet $X$ be a complex analytic variety (that is, a reduced, irreducible complex analytic space) and $G$ be a connected complex Lie group acting holomorphically on $X$. In this situation $X$ is called a complex analytic $G$-variety. Definition 3. A complex analytic $G$-variety $X$ is called almost homogeneous if $X$ has an open $G$-orbit $\Omega$. Note that an open $G$-orbit $\Omega$ in an almost homogeneous complex analytic $G$-variety $X$ is unique and connected and $E:=X\setminus \Omega$ is a proper analytic subset (see [1], § 1.7, Proposition 4). The main object under consideration in this paper is normal $(1,0)$-compactifiable almost homogeneous complex algebraic $G$-varieties (see Definition 2 for $(1,0)$-compactifiable varieties), where $G$ is a reductive complex Lie group. Recall that a connected complex Lie group $G$ is called reductive if $G$ has a compact real form $K$ (that is, $K$ is a compact real Lie subgroup of $G$ and ${\operatorname{Lie}(K)\otimes\mathbb{C}=\operatorname{Lie}(G)}$). This definition is closely related to the corresponding definition for connected linear algebraic groups. We recall that a connected linear algebraic group is called reductive if its unipotent radical is trivial. A connected reductive linear algebraic group over $\mathbb{C}$ is reductive in the sense of the first definition. Conversely, each connected reductive complex Lie group $G$ has a unique structure of a reductive linear algebraic group. Thus, one can consider the subalgebra of regular functions $\mathbb{C}[G]$ of the algebra of holomorphic functions $H^{0}(G,\mathcal{O})$. Using Corollary 3 we see that we have to study the space $H^{0}(Z,i^{-1}\mathcal{O})$. In the case when $G$ is a complex reductive Lie group we use the Harish-Chandra theorem (see § 4). We conclude this section by proving the existence of a neighbourhood system of $Z:=X'\setminus X$ with nice properties. Recall that $H^{0}(Z,i^{-1}\mathcal{O})=\varinjlim_{U\supset Z}H^{0}(U,\mathcal{O})$, where the inductive limit can be taken over a cofinal system of neighbourhoods of $Z$. Now let $X$ be an arbitrary complex analytic variety, $K$ be a compact Lie group acting holomorphically on $X$ and $Z$ be a connected compact $K$-invariant analytic set in $X$. Proposition 1. There exists a cofinal system of neighbourhoods $\{U_{n}\}$ of $Z$ such that each $U_{n}$ is a $K$-invariant open neighbourhood. Proof. Note that $X$ is metrizable. Let $\rho$ be a metric compatible with the topology of $X$. Let $\mu$ be a Haar measure on the compact Lie group $K$. Define Clearly, $\rho_K$ is a metric. Since $\mu$ is left-right-invariant, it follows that
Let $U$ be a relatively compact neighbourhood of $Z$. Since $\rho(k.x,k.y)$ is continuous on the compact set $K\times\overline{U}\times\overline{U}$, it follows that $\rho_{K}(x,y)$ is continuous on $U\times U$ with respect to the original topology on $X$. Note that the distance function $\operatorname{dist} (x, Z):=\inf_{z\in Z}\rho_{K}(x,z)$ is a continuous function on $U$. Let $U_{n}:=\bigl\{x\in U\mid \operatorname{dist} (x, Z)<{1}/{n}\bigr\}$. It is an open neighbourhood of $Z$ with respect to the original topology. Note that for all $x\in U_{n}$ and $k\in K$ we have
$$
\begin{equation*}
\inf_{z\in Z}\rho_{K}(k.x,z)=\inf_{z\in Z}\rho_{K}(x,k^{-1}.z)=\inf_{z\in Z}\rho_{K}(x,z)<\frac{1}{n}.
\end{equation*}
\notag
$$
It follows that $U_n$ is a $K$-invariant neighbourhood of $Z$. Now, let $V$ be an arbitrary neighbourhood of $Z$. Since there exists $n$ such that ${1}/{n}<\inf_{z\in Z,\,x\in X\setminus V}\rho_{K}(x,z)$, it follows that $U_{n}\subset V$. This completes the proof of Proposition 1. Therefore, $H^{0}(Z,i^{-1}\mathcal{O}) =\varinjlim_{U_n}H^{0}(U_n,\mathcal{O})$, where the inductive limit is taken over a system of neighbourhoods $\{U_n\}$ of $Z$ consisting of $K$-invariant neighbourhoods $U_{n}$. § 4. The Harish-Chandra theorem and its applicationsFirst we recall the necessary elements of the representation theory of compact Lie groups (see [1] and [24]). Let $F$ be a Fréchet space over $\mathbb{C}$, let $K$ be a compact Lie group, and let be a continuous representation (that is, $\rho$ is a homomorphism of $K$ into the group of invertible continuous linear operators on $F$ such that $K\to F$, $k\mapsto \rho(k)f$, is continuous for every vector $f\in F$).We define the following $K$-invariant subspaces of $F$. Let $\widehat{K}$ be the set of equivalence classes of finite-dimensional irreducible linear representations of $K$. For each $\delta\in \widehat{K}$ there is a continuous linear operator
$$
\begin{equation*}
E_{\delta}\colon F\to F, \qquad E_{\delta}(f):=\int_{K}\overline{\chi_{\delta}(k)}\rho(k)f\,d\mu(k),
\end{equation*}
\notag
$$
where $\chi_{\delta}$ is the character of a representation in the class $\delta$. We have the following properties of the operators $E_{\delta}$ (see [1], § 5.1, Theorem 3):
Let $F_{\delta}:=E_{\delta}(F)$. Note that $F_{\delta}$ is a closed $K$-invariant subspace of $F$ for each $\delta\in\widehat{K}$ and $F^{0}=\bigoplus_{\delta\in\widehat{K}}F_{\delta}$. Moreover, $F_{\delta}$ is the isotypic component of type $\delta$ of $F$ (that is, $F_{\delta}$ consists of those vectors in $F$ whose $K$-orbit lies in a finite-dimensional $K$-submodule isomorphic to $mV_{\delta}$; see [1], § 5.1, Theorem 4). We have a series representation for any differentiable vector; this series converges absolutely with respect to any continuous seminorm on $F$. This is the so-called Harish-Chandra theorem (see [1], § 5.1, Theorem 5). Theorem 5. For any vector $f\in F^{\infty}$ where the convergence is absolute with respect to any continuous seminorm on $F$. Now let $X$ be a complex analytic $K$-variety and $K$ be a connected compact Lie group acting holomorphically on $X$. Then $K$ has a linear representation on the Fréchet space $F=H^{0}(X,\mathcal{O})$. In this case $F=F^{\infty}$ (see [1], § 5.2, Proposition). Let $G$ be a connected reductive Lie group with compact real form $K$. Let $\Omega$ be a complex algebraic homogeneous $G$-variety and $\mathbb{C}[\Omega]$ be the algebra of regular functions on $\Omega$. Let $W\subset \Omega$ be a $K$-invariant domain. Proposition 2. For every $f\in H^{0}(W,\mathcal{O})$ we have where the convergence is absolute with respect to any continuous seminorm on $H^{0}(W,\mathcal{O})$ and, moreover, $E_{\delta}f\in \mathbb{C}[\Omega]$. Using Theorem 2 in [1], § 5.3, we complete the proof of the Proposition 2. In other words the algebra $\mathbb{C}[\Omega]$ is dense in $H^{0}(W,\mathcal{O})$. Let $G$ and $K$ be as above, and let $X$ be a normal $(1,0)$-compactifiable almost homogeneous complex algebraic $G$-variety. Let $X'$ be the corresponding $G$-equivariant compactification of $X$, and let $Z:=X'\setminus X$. Denote by $\mathcal{G}(X')$ the set of $G$-stable prime divisors on $X'$, and let
$$
\begin{equation*}
Y:=X'\setminus \bigcup_{D\in\mathcal{G}(X'),\,D\subset X} D.
\end{equation*}
\notag
$$
It is a Zariski open algebraic subvariety of $X'$, which is a normal almost homogeneous complex algebraic $G$-variety. Denote the algebra of regular functions on $Y$ by $\mathbb{C}[Y]$. Note that $\{D\in\mathcal{G}(X'),D\subset X\}$ can be an empty set, in which case $Y=X'$ and $\mathbb{C}[Y]=\mathbb{C}$. The vector space $H^{0}(Z,i^{-1}\mathcal{O})$ has the natural structure of a topological vector space (namely, it carries the direct limit topology). Proof. From Proposition 1 we obtain that there exists a cofinal system of neighbourhoods $\{U_n\}$ of $Z$ such that each $U_{n}$ is $K$-invariant.
Consider the Fréchet space $H^{0}(U_{n},\mathcal{O})$. From Theorem 5 we see that any $f\in H^{0}(U_{n},\mathcal{O})$ has a series representation with uniform convergence on compact subsets of $U_n$. Using Proposition 2 we obtain $E_{\delta}f\in \mathbb{C}[\Omega]$.Since $f\in H^{0}(U_{n},\mathcal{O})$ and $U_{n}$ intersects each prime $G$-stable divisor on $Y$, it follows that the rational function $E_{\delta}(f)$ has no poles at prime $G$-stable divisors on $Y$. Thus, $E_{\delta}(f)\in \mathbb{C}[Y]$. Moreover, $\mathbb{C}[Y]$ is a dense subspace of $H^{0}(U_{n},\mathcal{O})$. Clearly, the canonical homomorphism $\mathbb{C}[Y]\to H^{0}(Z,i^{-1}\mathcal{O})$ is injective. Now let $[f]\in H^{0}(Z,i^{-1}\mathcal{O})$. It is represented by a function $f$ in $H^{0}(U_{n},\mathcal{O})$ for some $n$. Since $\mathbb{C}[Y]$ is a dense subspace of $H^{0}(U_{n},\mathcal{O})$, there exists a sequence $\{f_{n}\}_{n\in\mathbb{N}}\subset \mathbb{C}[Y]$ such that $\lim_{n\to\infty}f_{n}=f$ in $H^{0}(U_{n},\mathcal{O})$. The continuity of the canonical map implies that $\lim_{n\to\infty}[f_{n}]=[f]$. Hence the map $\mathbb{C}[Y]\to H^{0}(Z,i^{-1}\mathcal{O})$ has a dense range. The proof of the lemma is complete.Using Corollary 3 and Lemma 2 we obtain the following proposition. Proposition 3. Let $G$ be a connected complex reductive Lie group, and let $X$ be a normal $(1,0)$-compactifiable almost homogeneous algebraic $G$-variety. Then $X$ admits the Hartogs phenomenon if and only if $\mathbb{C}[Y]=\mathbb{C}$. § 5. Weight criterion for the Hartogs phenomenonLet $X, X'$ and $ Y$ be as in § 4. We have to investigate $\mathbb{C}[Y]$. Since $G$ acts algebraically on $Y$, it follows that $\mathbb{C}[Y]$ is a representation of $G$. We recall some facts of the representation theory of reductive groups (see [9] and [17]). Let $G$ be a connected reductive complex Lie group, $B\subset G$ be a Borel subgroup and $T\subset B$ be a maximal algebraic torus with character lattice $\mathfrak{X}(T)$. For the Lie algebra $\mathfrak{g}=\operatorname{Lie}(G)$ we have the root decomposition
$$
\begin{equation*}
\mathfrak{g}=\mathfrak{t}\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_{\alpha}
\end{equation*}
\notag
$$
with respect to the adjoint representation, where $R\subset \mathcal{X}(T)$ is the root system of $G$. Let $R_{+}\subset R$ be the set of positive roots. This set is uniquely defined by the following condition:
$$
\begin{equation*}
\operatorname{Lie}(B)=\mathfrak{t}\oplus\bigoplus_{\alpha\in R_{+}}\mathfrak{g}_{\alpha}.
\end{equation*}
\notag
$$
Denote the set of simple roots and the set of dual simple roots by $S^{\vee}$ (it is a subset of the lattice of 1-parameter subgroups $\mathfrak{X}^{*}(T)$). Denote the dominant Weyl chamber by $C_{+}$. Recall that
$$
\begin{equation*}
C_{+}:=\bigl\{t\in \mathfrak X(T)\otimes\mathbb{R}\mid \langle t,\alpha^{\vee}\rangle\geqslant 0\ \forall\, \alpha\in S\bigr\}.
\end{equation*}
\notag
$$
Define the set of dominant characters $\mathfrak{X}_{+}(T)$ to be $\mathfrak{X}(T)\cap C_{+}$. We need the following facts. 1. There is a bijective correspondence between the set $\mathfrak{X}_{+}(T)$ of dominant characters and the set $\widehat{G}$ of isomorphism classes of irreducible representations of $G$. 2. For any rational representation $V$ of the reductive group $G$ (that is, a representation such that every $v\in V$ is contained in a finite-dimensional $G$-stable subspace on which $G$ acts algebraically) we have the canonical decomposition
$$
\begin{equation*}
V\cong\bigoplus_{\lambda\in\mathfrak X_{+}(T)}V_{\lambda}^{(B)}\otimes V(\lambda),
\end{equation*}
\notag
$$
where $V_{\lambda}^{(B)}:=\bigl\{v\in V\mid b.v=\lambda(b)v\bigr\}$ is the set of $B$-eigenvectors of weight $\lambda$. Since $\mathbb{C}[Y]$ is a rational representation of $G$ (see [9], Lemma 1.5), it follows that the canonical decomposition for $\mathbb{C}[Y]$ has the form
$$
\begin{equation*}
\mathbb C[Y]\cong\bigoplus_{\lambda\in\mathfrak X_{+}(T)}\mathbb C[Y]_{\lambda}^{(B)}\otimes V(\lambda).
\end{equation*}
\notag
$$
Define the weight monoid of $Y$ by
$$
\begin{equation*}
\Lambda_{+}(Y):=\bigl\{\lambda\in \mathfrak X_{+}(T)\mid \mathbb C[Y]_{\lambda}^{(B)}\neq 0\bigr\}.
\end{equation*}
\notag
$$
Then we obtain the following weight criterion for the Hartogs phenomenon. Theorem 6. Let $G$ be a connected complex reductive Lie group, and let $X$ be a normal $(1,0)$-compactifiable almost homogeneous algebraic $G$-variety. Then $X$ admits the Hartogs phenomenon if and only if $\Lambda_{+}(Y)=0$ and $\mathbb{C}[Y]^{B}=\mathbb{C}$. § 6. Convex geometric criterion for the Hartogs phenomenon in spherical varietiesIn this section we recall some facts from the theory of spherical varieties (see [13] and [28]). Let $G$ be a complex reductive Lie group, $B\subset G$ be a Borel subgroup and $T\subset B$ be a maximal algebraic torus. Definition 5. A normal almost homogeneous complex algebraic $G$-variety with open orbit $\Omega$ is called spherical if $\Omega$ contains an open $B$-orbit $O$. Note that the open $B$-orbit $O$ is an affine variety (see [28], Theorem 3.5). Also, a normal algebraic $G$-variety $X$ is spherical if and only if each $B$-invariant rational function on $X$ is constant (that is, $\mathbb{C}(X)^{B}=\mathbb{C}$; see [13], Theorem 2.8) and if and only if $X$ contains finitely many $B$-orbits (see [13], Theorem 2.11). With the open orbit $\Omega$ we associate the weight lattice
$$
\begin{equation*}
M:=\bigl\{\lambda\in \mathfrak{X}(T)\mid \mathbb{C}(\Omega)^{(B)}_{\lambda}\neq0\bigr\}
\end{equation*}
\notag
$$
and the dual weight lattice $N:=\operatorname{Hom}(M,\mathbb{Z})$. Also, let $M_{\mathbb{R}}:=M\otimes\mathbb{R}$ and $N_{\mathbb{R}}:=N\otimes\mathbb{R}$. Note that $M$ is a sublattice of the character lattice $\mathfrak{X}(T)$ of the torus $T$. Recall that $B=TU$ (here $U$ is the unipotent radical of $B$), and since $U$ has no nontrivial characters, we obtain $\mathfrak{X}(B)=\mathfrak{X}(T)$. We define
$$
\begin{equation*}
\mathbb{C}(\Omega)^{(B)}:=\bigl\{f\in \mathbb{C}(\Omega)\setminus\{0\}\mid \exists\,\lambda\in\mathfrak X(T)\colon b.f=\lambda(b)f\ \forall\, b\in B\bigr\}.
\end{equation*}
\notag
$$
It follows that we have an isomorphism which is given by $\lambda\mapsto [f]$ where $[f]$ is the equivalence class of any function
$$
\begin{equation*}
f\in \mathbb{C}(\Omega)^{(B)}_{\lambda}=\bigl\{f\in \mathbb{C}(\Omega)\mid b.f=\lambda(b)f\ \forall\, b\in B\bigr\}.
\end{equation*}
\notag
$$
Let $Y$ be as in § 4. Denote the set of all prime $B$-stable divisors on $Y$ by $\mathcal{B}(Y)$. Note that $\mathcal{B}(Y)$ is the union of the set of prime $G$-stable divisors on $Y$ and the set of prime $B$-stable but not $G$-stable divisors on $Y$. Each $B$-stable divisor $D\in \mathcal{B}(Y)$ defines a discrete valuation
$$
\begin{equation*}
v_{D}\colon \mathbb{C}(\Omega)\setminus\{0\}\to \mathbb{Z}.
\end{equation*}
\notag
$$
Recall that $v_{D}(f)$ is the order of the zero or pole of $f$ at $D$. Also, the valuation $v_{D}$ defines a point in the dual weight lattice $a_{D}\in N=\operatorname{Hom}(M,\mathbb{Z})$ by setting $\langle a_{D},\lambda\rangle:=v_{D}(f)$ for $f\in \mathbb{C}(\Omega)^{(B)}_{\lambda}$. Set
$$
\begin{equation*}
L:=\bigl\{\lambda\in M_\mathbb{R}\mid \langle a_{D},\lambda\rangle \geqslant 0\ \forall\, D\in\mathcal{B}(Y)\bigr\};
\end{equation*}
\notag
$$
we identify $L$ with its image under the injective map
$$
\begin{equation*}
\iota\colon M_{\mathbb{R}}\hookrightarrow \mathfrak{X}(T)\otimes\mathbb{R}.
\end{equation*}
\notag
$$
We obtain the following description of the set of $B$-eigenvectors of $\mathbb{C}[Y]$. Lemma 3. The set of $B$-semi-invariant vectors in $C[Y]$ has the following form: In particular, this yields a description of the weight monoid of $Y$: Proof. Note that
If $\lambda\in L$, then for any $f\in \mathbb{C}[O]^{(B)}_{\lambda}$ we obtain $\langle a_{D}, \lambda\rangle\geqslant 0$ for all $D\in \mathcal{B}(Y)$. It follows that $\mathbb{C}[Y]^{(B)}_{\lambda}=\mathbb{C}[O]^{(B)}_{\lambda}$. By the definition of $M$ we have $\mathbb{C}(\Omega)^{(B)}_{\lambda}\neq 0$ for all $\lambda\in M$. Since
$$
\begin{equation*}
\mathbb{C}[Y]^{(B)}_{\lambda}=\mathbb{C}(\Omega)^{(B)}_{\lambda}
\end{equation*}
\notag
$$
for $\lambda\in L$, it follows that $\mathbb{C}[Y]^{(B)}_{\lambda}\neq 0$.
Now let $\lambda\notin L$, and assume that there exits $f\in \mathbb{C}[Y]^{(B)}_{\lambda}$ such that $f\neq 0$. Then there exists $D\in\mathcal{B}(Y)$ such that $v_{D}(f)<0$. This is a contradiction; the proof of the lemma is complete. We obtain the following corollary. Corollary 4. Let $X$ be a $(1,0)$-compactifiable spherical $G$-variety. Then $X$ admits the Hartogs phenomenon if and only if $L=0$. Now we reformulate this criterion in terms of the dual weight lattice $N$. Define a cone $C$ in the space $N_{\mathbb{R}}$ by
$$
\begin{equation*}
C:=\mathbb{R}_{\geqslant 0}\bigl\langle a_{D}\mid D\in \mathcal{B}(Y)\bigr\rangle.
\end{equation*}
\notag
$$
Then we obtain the following criterion. Corollary 5. Let $X$ be a $(1,0)$-compactifiable spherical $G$-variety. Then $X$ admits the Hartogs phenomenon if and only if $C=N_{\mathbb{R}}$. Proof. Since $C^{\vee}=L$, where $C^{\vee}$ is the dual cone of $C$, it follows that $C=N_{\mathbb{R}}$ if and only if $L=0$. The proof of the corollary is complete. This criterion can also be formulated in terms of coloured fans. First we recall the definition of the valuation cone (for more details, see [13], §§ 4 and 10, or [28], Ch. 4 and Appendix B). Let $v\colon\mathbb{C}(\Omega)\setminus\{0\}\to \mathbb{Q}$ be a discrete $\mathbb{Q}$-valued valuation. This valuation is called $G$-invariant if $v(g.f)=v(f)$ for all $f\in\mathbb{C}(\Omega)$ and $g\in G$. The set of $G$-invariant valuations of $\mathbb{C}(\Omega)$ is denoted by $\mathcal{V}(\Omega)$. Note that a $G$-invariant valuation $v$ defines a point $a_{v}\in N_{\mathbb{Q}}=N\otimes\mathbb{Q}$ by for some $f\in \mathbb{C}(\Omega)^{(B)}_{\lambda}$. By Corollary 4.9 in [13] we can regard $\mathcal{V}(\Omega)$ as a subset of $N_{\mathbb{Q}}$. Note that it is a finitely generated convex rational cone of maximal dimension (see [13], Corollary 10.6), and it is called the valuation cone of $\Omega$. We denote the cone in $N_{\mathbb{R}}$ generated by $\mathcal{V}(\Omega)$ by $\mathcal{V}_{\mathbb{R}}(\Omega)$.Now we denote the set of all prime $B$-stable divisors of $\Omega$ by $\mathcal{B}(\Omega)$. For each $D\in\mathcal{B}(\Omega)$ we denote by $\overline{D}$ the closure of $D$ in $Y$, where $Y$ is as in § 4. Since $v_{D}=v_{\overline{D}}$ (after the identification $\mathbb{C}(\Omega)=\mathbb{C}(Y)$), we can identify $\mathcal{B}(\Omega)$ with the subset $\{\overline{D}\mid D\in \mathcal{B}(\Omega)\}$ of $\mathcal{B}(Y)$. We recall the definitions of a coloured cone and a coloured fan. Definition 6. A coloured cone is a pair $(\sigma,\mathcal{F})$, where $\sigma\subset N_{\mathbb{R}}$ and $\mathcal{F}\subset \mathcal{B}(\Omega)$, with the following properties: A coloured face of a coloured cone $(\sigma,\mathcal{F})$ is a pair $(\sigma',\mathcal{F}')$ such that $\sigma'$ is a face of $\sigma$, the relative interior of $\sigma'$ intersects $\mathcal{V}(\Omega)$ nontrivially and $\mathcal{F}'=\bigl\{D\in\mathcal{F}\mid a_{D}\in \sigma'\bigr\}$. A coloured fan is a finite set $\Sigma$ of coloured cones with the following properties:
The support of a coloured fan $\Sigma$ is the set $|\Sigma|:=\bigcup_{(\sigma,\mathcal{F})\in\Sigma}\sigma$. Let $X$ be a spherical $G$-variety with open orbit $\Omega$. For any $G$-orbit $Y$ of $X$ denote the $G$-stable subset $\{x\in X\mid \overline{G.x}\supset Y\}$ by $X_Y$ (it is a $G$-stable open subvariety of $X$ containing the unique closed $G$-orbit $Y$). Let $D_{1},\dots, D_{m}$ be the $G$-stable divisors of $X_Y$. Let $\mathcal{F}$ be the set of $B$-stable but not $G$-stable divisors $D$ in $X$ that contain the closed orbit lying in $X_Y$. Let $\sigma$ be the cone of $N_{\mathbb{R}}$ generated by the points $a_{v}$, $v\in\mathcal{F}$, and the points $a_{D_i}\in N$, $i\in\{1,\dots, m\}$. Thus we obtain a coloured cone $(\sigma,\mathcal{F})$. Moreover, the set of coloured cones constructed in this way forms a coloured fan $\Sigma_{X}$. We have the following Luna-Vust theorem on the classification of $\Omega$-embeddings. Theorem 7. The map $X\to\Sigma_{X}$ is a bijection between the set of isomorphism classes of spherical $G$-varieties (with the same open orbit $\Omega$) and the set of coloured fans. Let us now state a few properties of spherical varieties in terms of coloured fans. Remark 1. Let $X_{\Sigma}$ be a spherical variety with open orbit $\Omega$ and coloured fan $\Sigma$. Then the following hold. 1. $X_{\Sigma}$ is compact if and only if $\Sigma$ is complete (i.e. $|\Sigma|\supset\mathcal{V}_{\mathbb{R}}(\Omega)$); 2. There is a bijective correspondence between the $G$-orbits in $X_{\Sigma}$ and the coloured cones in $\Sigma$. 3. If $W_{1}$ and $W_{2}$ are $G$-orbits with coloured cones $(\sigma_{1},\mathcal{F}_1)$ and $(\sigma_{2},\mathcal{F}_2)$, respectively, then $W_{1}\subset \overline{W_{2}}$ if and only if $(\sigma_{2},\mathcal{F}_2)$ is a coloured face of the coloured cone $(\sigma_{1},\mathcal{F}_1)$. Now we prove the following lemma. Lemma 4. Let $X_{\Sigma}$ be a noncompact spherical variety with coloured fan $\Sigma$. Then $X_{\Sigma}$ is $(1,0)$-compactifiable if and only if the open set $\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|$ is connected. Proof. Let $X_{\Sigma'}$ be a compactification of $X_{\Sigma}$. In this case we have $\Sigma\subset \Sigma'$.
First, using Corollaire 1 in [8] and the GAGA principle we have $H^{1}(X_{\Sigma'},\mathcal{O})=0$ for a compact spherical variety $X_{\Sigma'}$. So we have to prove that $Z:=X_{\Sigma'}\setminus X_{\Sigma}$ is connected if and only if $\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|$ is connected. Now by $O(\sigma,\mathcal{F})$ we denote the $G$-orbit corresponding to the coloured cone $(\sigma,\mathcal{F})$. Since the set $Z=X_{\Sigma'}\setminus X_{\Sigma}$ is $G$-stable, it is a union of $G$-orbits. Namely, we have
$$
\begin{equation*}
Z=\bigcup_{(\sigma,\mathcal{F})\in \Sigma'\setminus\Sigma}O(\sigma,\mathcal{F}).
\end{equation*}
\notag
$$
Further, $Z$ is connected if and only if for any two $G$-orbits $V=O(\sigma,\mathcal{F})$ and $ V'=O(\sigma',\mathcal{F}')$ such that $(\sigma,\mathcal{F}), (\sigma',\mathcal{F}')\in\Sigma'\setminus\Sigma$ there exists a sequence of $G$-orbits $\{V_{i}=O(\sigma_i,\mathcal{F}_i)\}_{i=0}^{n}$ with the following properties:
But this is equivalent to the fact that for any two coloured cones
$$
\begin{equation*}
(\sigma,\mathcal{F}),(\sigma',\mathcal{F}') \in \Sigma'\setminus\Sigma
\end{equation*}
\notag
$$
there exists a sequence of coloured cones $\{C_{i}=(\sigma_i,\mathcal{F}_i)\}_{i=0}^{n}$ with the following properties:
This is equivalent to $\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|$ being a connected set. The proof of the lemma is complete. We recall that
$$
\begin{equation*}
Y=X_{\Sigma'}\setminus \bigcup_{D\in\mathcal{G}(X_{\Sigma'}),\,D\subset X_{\Sigma}} D,
\end{equation*}
\notag
$$
where $\mathcal{G}(X_{\Sigma'})$ is the set of all prime $G$-stable divisors on $X_{\Sigma'}$. Let $\mathcal{OG}_{k}(X_{\Sigma'})$ be the set of all $k$-codimensional $G$-orbits on $X_{\Sigma'}$, and let
$$
\begin{equation*}
\mathcal{OG}(X_{\Sigma'}):=\bigcup_{k=1}^{\dim X_{\Sigma'}}\mathcal{OG}_{k}(X_{\Sigma'}).
\end{equation*}
\notag
$$
We consider the spherical variety
$$
\begin{equation*}
\widehat{Y}:=X_{\Sigma'}\setminus\bigcup_{O\in\mathcal{OG}(X_{\Sigma'}),\,\overline{O}\subset X_{\Sigma}}O,
\end{equation*}
\notag
$$
where $\overline{O}$ is the closure of $O$ in $X_{\Sigma'}$. Since $\mathcal{G}(X_{\Sigma'})=\bigl\{\overline{O}\mid O\in \mathcal{OG}_{1}(X_{\Sigma'})\bigr\}$, it follows that $\widehat{Y}\subset Y$ and $\operatorname{codim}(Y\setminus\widehat{Y})>1$. Therefore, $\mathbb{C}[Y]=\mathbb{C}[\widehat{Y}]$, $\mathcal{B}(Y)=\mathcal{B}(\widehat{Y})$. The coloured fan $\Sigma_{\widehat{Y}}$ of $\widehat{Y}$ is obtained as follows. Proof. Consider a coloured cone $(\tau,\mathcal{F}')\in\Sigma'$ and let $O=O(\tau,\mathcal{F}')$ be the corresponding $G$-orbit. Recall that the set $Z=X_{\Sigma'}\setminus X_{\Sigma}$ is a union of $G$-orbits, namely,
The following statements are obvious: $(\tau,\mathcal{F}')\in \Sigma_{\widehat{Y}}$ if and only if $\overline{O}\cap Z\neq\varnothing $, if and only if there exists a coloured cone $(\sigma,\mathcal{F})\in\Sigma'\setminus\Sigma$ such that $(\tau,\mathcal{F}')$ is a coloured face of $(\sigma,\mathcal{F})$. The proof of the lemma is complete. Lemma 6. The equality $|\Sigma_{\widehat{Y}}|\cap\mathcal{V}_{\mathbb{R}}(\Omega)=\overline{\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|}$ holds. Proof. By Lemma 5
A point $p\in\mathcal{V}_{\mathbb{R}}(\Omega)$ belongs to $\overline{\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|}$ if and only if there exists a sequence $\{p_n\}$ of points $p_n\in \mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|$ such that $p_{n}\to p$ as $n\to\infty$. We can assume that every $p_n$ belongs to $\sigma\cap\mathcal{V}_{\mathbb{R}}(\Omega)$ for some coloured cone $(\sigma,\mathcal{F})\in \Sigma'\setminus\Sigma$. Then $p_{n}\in |\Sigma_{\widehat{Y}}|\cap\mathcal{V}_{\mathbb{R}}(\Omega)$. Since $|\Sigma_{\widehat{Y}}|\cap\mathcal{V}_{\mathbb{R}}(\Omega)$ is a closed set, it follows that $p\in |\Sigma_{\widehat{Y}}|\cap\mathcal{V}_{\mathbb{R}}(\Omega)$. The proof of the lemma is complete. We recall that
$$
\begin{equation*}
C=\mathbb{R}_{\geqslant 0}\bigl\langle a_{D}\mid D\in \mathcal{B}(Y)\bigr\rangle\subset N_{\mathbb R}.
\end{equation*}
\notag
$$
Now we prove the following lemma. Lemma 7. The equality $C=\mathbb{R}_{\geqslant 0}\bigl\langle\overline{\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|}\cup \{a_{D}\mid D\in \mathcal{B}(\Omega)\}\bigr\rangle$ holds. Proof. Note that $\mathcal{B}(\widehat{Y})$ is the union of the set $\mathcal{G}(\widehat{Y})$ of prime $G$-stable divisors in $\widehat{Y}$ and the set $\{\overline{D}\mid D\in\mathcal{B}(\Omega)\}$, where $\overline{D}$ is the closure in $\widehat{Y}$ of a prime $B$-stable divisor $D\subset \Omega$.
By $\mathcal{F}(\widehat{Y})$ we denote the union of all subsets $\mathcal{F}\subset\mathcal{B}(\Omega)$ such that $(\sigma,\mathcal{F})\in \Sigma_{\widehat{Y}}$. Thus we have the following equalities:
$$
\begin{equation*}
\begin{aligned} \, C &=\mathbb{R}_{\geqslant 0}\bigl\langle a_{D}\mid D\in \mathcal{B}(Y)\bigr\rangle=\mathbb{R}_{\geqslant 0}\bigl\langle a_{D}\mid D\in \mathcal{B}(\widehat{Y})\bigr\rangle \\ &=\mathbb{R}_{\geqslant 0}\bigl\langle \bigl\{a_{D}\mid D\in \mathcal{G}(\widehat{Y})\cup\mathcal{F}(\widehat{Y})\bigr\}\cup\bigl\{a_{D}\mid D\in\mathcal{B}(\Omega)\bigr\}\bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
Further, we have the decomposition
$$
\begin{equation*}
|\Sigma_{\widehat{Y}}|=(|\Sigma_{\widehat{Y}}|\cap\mathcal{V}_{\mathbb{R}}(\Omega))\cup \overline{|\Sigma_{\widehat{Y}}|\setminus\mathcal{V}_{\mathbb{R}}(\Omega)} =\overline{\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|}\cup \overline{|\Sigma_{\widehat{Y}}|\setminus\mathcal{V}_{\mathbb{R}}(\Omega)}.
\end{equation*}
\notag
$$
Since the set of generators of all coloured cones in $\Sigma_{\widehat{Y}}$ is the set
$$
\begin{equation*}
\bigl\{a_{D}\mid D\in \mathcal{G}(\widehat{Y})\cup\mathcal{F}(\widehat{Y})\bigr\},
\end{equation*}
\notag
$$
it follows that
$$
\begin{equation*}
\mathbb{R}_{\geqslant 0}\bigl\langle a_{D}\mid D\in \mathcal{G}(\widehat{Y})\cup\mathcal{F}(\widehat{Y})\bigr\rangle=\mathbb{R}_{\geqslant 0}\bigl\langle|\Sigma_{\widehat{Y}}|\bigr\rangle=\mathbb{R}_{\geqslant 0}\bigl\langle\overline{\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|}\cup \overline{|\Sigma_{\widehat{Y}}|\setminus\mathcal{V}_{\mathbb{R}}(\Omega)}\bigr\rangle.
\end{equation*}
\notag
$$
Now, if a ray $\mathbb{R}_{\geqslant0}\langle a_{D}\rangle$ intersects $N_{\mathbb{R}}\setminus\mathcal{V}(\Omega)$, then $D$ is not $G$-stable and thus $D\in\mathcal{F}(\widehat{Y})\subset\mathcal{B}(\Omega)$. We obtain the following equalities:
$$
\begin{equation*}
\begin{aligned} \, C &=\mathbb{R}_{\geqslant 0}\bigl\langle \bigl\{a_{D}\mid D\in \mathcal{G}(\widehat{Y})\cup\mathcal{F}(\widehat{Y})\bigr\}\cup\{a_{D}\mid D\in\mathcal{B}(\Omega)\}\bigr\rangle \\ &=\mathbb{R}_{\geqslant 0}\bigl\langle\overline{\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|}\cup \overline{|\Sigma_{\widehat{Y}}|\setminus\mathcal{V}_{\mathbb{R}}(\Omega)}\cup\{a_{D}\mid D\in\mathcal{B}(\Omega)\}\bigr\rangle \\ &=\mathbb{R}_{\geqslant 0}\bigl\langle\overline{\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|}\cup \{a_{D}\mid D\in \mathcal{B}(\Omega)\}\bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
The proof of the lemma is complete. Thus we obtain the following convex-geometric criterion. Theorem 8. Let $X_{\Sigma}$ be a noncompact spherical $G$-variety with open $G$-orbit $\Omega$ and coloured fan $\Sigma$ such that $\mathcal{V}_{\mathbb{R}}(\Omega)\setminus |\Sigma|$ is a connected set. Then $X_{\Sigma}$ admits the Hartogs phenomenon if and only if Remark 2. Let $\Omega$ be a noncompact spherical homogeneous $G$-variety. The coloured fan of $\Omega$ is $\Sigma=\{(0,\varnothing)\}$. The formula 6.1. The case of horospherical varietiesFirst we recall some notions and notation. Let $G$ be a complex reductive Lie group, $B\subset G$ be a Borel subgroup, $T\subset B$ be a maximal algebraic torus and $U$ be the unipotent radical of $B$ (it is a maximal unipotent subgroup of $G$). Let $W$ be the Weyl group of $G$ with respect to $T$, let $R$ be the root system of $(G,T)$ and $S$ be the set of simple roots with respect to $B$. The group $W$ contains the root reflections $r_{\alpha}$ ($\alpha\in R$) acting on $\mathfrak{X}(T)$ by $r_{\alpha}(\lambda)=\lambda-\langle \lambda, \alpha^{\vee}\rangle\alpha$, and it is generated by the reflections corresponding to simple roots. Note that parabolic subgroups containing a given Borel subgroup $B$ are parametrized by subsets of simple roots $I\subset S$. The Lie algebra of the corresponding parabolic subgroup $P=P_{I}$ is
$$
\begin{equation*}
\operatorname{Lie}(P)=\mathfrak{t}\oplus\bigoplus_{\alpha\in R_{+}\cup R_I}\mathfrak{g}_{\alpha},
\end{equation*}
\notag
$$
where $R_I\subset R$ is the root subsystem generated by $I$. There is a unique Levi decomposition $P=P_{u}\leftthreetimes L$, where $P_{u}$ is the unipotent radical of $P$ and $L=L_{I}\subset P_{I}$ is the unique Levi subgroup containing a fixed maximal torus $T\subset B$. The root system of $(L_{I},T)$ is $R_{I}$. The opposite parabolic subgroup $P^{-}=P^{-}_{I}\supset B^{-}$ associated with $I$ intersects $P_{I}$ in $L_I$ and has the Levi decomposition $P^{-}=P^{-}_{u}\leftthreetimes L$, where $P_{u}^{-}$ is the unipotent radical of $P^{-}$. Now we recall the definition of an horospherical variety. Definition 7. A homogeneous $G$-variety $\Omega$ is called horospherical if the stabilizer of any point in $\Omega$ contains a maximal unipotent subgroup of $G$. A normal almost homogeneous complex algebraic $G$-variety with open orbit $\Omega$ is called horospherical if $\Omega$ is horospherical. Note that an horospherical variety is spherical in the sense of Definition 5. Let $H$ be the stabilizer of any point $o$ in $\Omega$, thus $G/H\cong\Omega$, $g\mapsto g.o$. We may assume that $H\supset U^{-}$. By Lemma 7.4 in [28] we have $H=P^{-}_{u}\leftthreetimes L_{0}$ for some parabolic subgroup $P\supset B$ (namely, $P$ is such that $P^{-}=N_{G}(H)$) with Levi subgroup $L\supset L_{0}\supset L'$ (here $L'$ is the commutator subgroup of $L$) and unipotent radical $P_u$. We may assume that $L\supset T$. Note that the injective map $\iota\colon M_\mathbb{R}\hookrightarrow \mathfrak{X}(T)\otimes\mathbb{R}$ induces the surjective map
$$
\begin{equation*}
\iota^{*}\colon \mathfrak X^{*}(T)\otimes\mathbb R\twoheadrightarrow N_{\mathbb R}:=N\otimes\mathbb R.
\end{equation*}
\notag
$$
Now we recall some facts about horospherical homogeneous $G$-varieties (see [28], § 28.1). Remark 3. The following hold: Thus we obtain the following convex-geometric criterion for horospherical varieties. Corollary 6. Let $X_{\Sigma}$ be a noncompact horospherical $G$-variety with open $G$-orbit $\Omega$ and coloured fan $\Sigma$ such that $N_{\mathbb{R}}\setminus |\Sigma|$ is a connected set. Then $X_{\Sigma}$ admits the Hartogs phenomenon if and only if where $(S\setminus I)^{\vee}=\{\alpha^{\vee}\mid\alpha\in S\setminus I\}$. Remark 4. Using Corollary 6 or Remark 2 we deduce that every horospherical homogeneous $G$-variety $\Omega$ such that $\operatorname{rk}(N)>1$ admits the Hartogs phenomenon. For a horospherical homogeneous $G$-variety $\Omega$ of rank $1$ we have two cases. If $\iota^{*}((S\setminus I)^{\vee})\neq\{0\}$ then there always exists a spherical embedding $X\supset\Omega$ such that $N_{\mathbb{R}}\setminus |\Sigma_{X}|$ is connected and $\mathbb{R}_{\geqslant 0}\langle\overline{N_{\mathbb{R}}\setminus |\Sigma_{X}|}\cup \iota^{*}((S\setminus I)^{\vee})\rangle=N_\mathbb{R}$. In this case, since $X$ admits the Hartogs phenomenon, so also does $\Omega$. If $\iota^{*}((S\setminus I)^{\vee})=\{0\}$, then $\Omega=\mathbb{C}^{*}\times\Omega_0$, where $\Omega_{0}$ is a compact spherical homogeneous $G$-variety (namely, $\Omega_0\cong G/P^{-}$). In this case $\Omega$ does not admit the Hartogs phenomenon. Remark 5. From Remarks 2 and 4 we obtain that the Hartogs phenomenon holds for every noncompact spherical homogeneous $G$-variety, except $\mathbb{C}^{*}\times G/P^{-}$. Further, for $\Omega=G/U^{-}$ we have $I=\varnothing$ and $N=\mathfrak{X}^{*}(T)$; thus we obtain the following. Corollary 7. Let $X_{\Sigma}$ be a horospherical $G$-variety with open $G$-orbit $\Omega=G/U^{-}$ and coloured fan $\Sigma$ such that $(\mathfrak{X}^{*}(T)\otimes\mathbb{R})\setminus |\Sigma|$ is a connected set. Then $X_{\Sigma}$ admits the Hartogs phenomenon if and only if § 7. ExamplesIn this section we consider examples of horospherical varieties with open orbit $(\mathrm{SL}(2)\times\mathbb{C}^{*})/U^{-}$, where $U^{-}$ is the maximal unipotent subgroup of $\mathrm{SL}(2)\times\mathbb{C}^{*}$ consisting of the lower triangular matrices with ones on the diagonal. 7.1. Some calculations for $(\mathrm{SL}(2)\times {\mathbb{C}}^{*})/U^{-}$Consider
$$
\begin{equation*}
G=\mathrm{SL}(2)\times\mathbb{C}^{*}=\left\{ \begin{pmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0& 0 & a_{33} \end{pmatrix} \Biggm| a_{11}a_{22}-a_{12}a_{21}=1 ,\,a_{33}\in\mathbb{C}^{*}\right\},
\end{equation*}
\notag
$$
and let
$$
\begin{equation*}
B=\left\{ \begin{pmatrix} a_{11} & a_{12} & 0 \\ 0 & a_{22} & 0 \\ 0& 0 & a_{33} \end{pmatrix} \Biggm| a_{11}a_{22}=1 ,\,a_{33}\in\mathbb{C}^{*}\right\}.
\end{equation*}
\notag
$$
Consider the following subgroups of $G$:
Note that $\mathfrak{X}(T)=\mathbb{Z}^{2}$. Denoting $t=\begin{pmatrix} t_{1} & 0 & 0\\ 0 & t_{2} & 0\\ 0& 0 & t_{3} \end{pmatrix}$ we obtain that each character $\lambda\in\mathfrak{X}(T)$ is given by $\lambda(t)=t_{1}^{l}t_{3}^{m}$ for some $(l,m)\in \mathbb{Z}^{2}$. For the Lie algebra $\mathfrak{g}=\operatorname{Lie}(G)$ we have the following root decomposition:
$$
\begin{equation*}
\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{g}_{\alpha_{12}}\oplus\mathfrak{g}_{\alpha_{21}},
\end{equation*}
\notag
$$
where
The root system of $G$ is $R=\{\alpha_{12},\alpha_{21}\}$. Since $\operatorname{Lie}(B)=\mathfrak{t}\oplus \mathbb{C}E_{12}$, it follows that the subset of simple roots with respect to $B$ is $S=\{\alpha_{12}\}$. The Weyl group of $G$ with respect to $T$ is $W=N_{G}(T)/T=\{e, r_{12}\}$, where $r_{12}$ is the reflection represented by the matrix $U=E_{12}-E_{21}+E_{33}\in N_{G}(T)$. Now we consider the homogeneous $G$-variety $G/U^{-}$. We have the isomorphism $\phi\colon G/U^{-}\cong \mathbb{C}^{2}\setminus\{(0,0)\}\times\mathbb{C}^{*}$ induced by the map
$$
\begin{equation*}
G\to \mathbb{C}^{2}\setminus\{(0,0)\}\times\mathbb{C}^{*}, \qquad \begin{pmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0& 0 & a_{33} \end{pmatrix}\mapsto (a_{12},a_{22},a_{33}).
\end{equation*}
\notag
$$
The isomorphism $\phi$ is $G$-equivariant with respect to the left multiplication action of $G$ on $G/U^{-}$ and the standard action of $G$ on $\mathbb{C}^{2}\setminus\{(0,0)\}\times\mathbb{C}^{*}$. Using $\phi$ we obtain that $B$-orbits of $G/U^{-}$ are of the following form: Now we calculate the weight lattice. The algebra of regular functions on $\Omega$ is
$$
\begin{equation*}
\mathbb{C}[\Omega]=\mathbb{C}[\mathbb{C}^{2}\setminus\{(0,0)\} \times\mathbb{C}^{*}]=\bigoplus_{(i,j,k)\in(\mathbb{Z}_{\geqslant0})^{2} \times\mathbb{Z}}\mathbb{C}x_{1}^{i}x_{2}^{j}w^{k}.
\end{equation*}
\notag
$$
It follows that $x_{2}^{l}w^{-m}\in\mathbb{C}(\Omega)^{(B)}_{(l,m)}$, where $\lambda=(l,m)\in \mathfrak{X}(T)=\mathbb{Z}^{2}$. Thus we see that $M=\mathfrak{X}(T)$ (of course, we can use Remark 3; in this case $P=B$, $H=U^{-}$ and $A\cong T$). The $B$-stable divisor $D_{12}$ defines the discrete valuation
$$
\begin{equation*}
v_{12}\colon \mathbb{C}(\Omega)\setminus\{0\}\to\mathbb{Z}
\end{equation*}
\notag
$$
and the point $a_{D_{12}}\in N$ by the formula $\langle a_{D_{12}},(l,m)\rangle:=v_{D_{12}}(x_{2}^{l}w^{-m})=l$. This means that $a_{D_{12}}=(1,0)$. Let $(\,\cdot\,{,}\,\cdot\,)$ be the standard scalar product in $\mathfrak{X}(T)\otimes\mathbb{R}$. The dual simple root $\alpha_{12}^{\vee}\in \mathfrak{X}^{*}(T)\otimes\mathbb{R}$ is given by the formula
$$
\begin{equation*}
\langle\alpha_{12}^{\vee},\lambda\rangle=\frac{2(\alpha_{12},\lambda)}{(\alpha_{12},\alpha_{12})}
\end{equation*}
\notag
$$
for all $\lambda\in\mathfrak{X}(T)$. It follows that $\alpha_{12}^{\vee}=(1,0)$. Thus, $\alpha_{12}^{\vee}=a_{D_{12}}$ (which actually follows from Remark 3). 7.2. $(\mathrm{SL}(2)\times {\mathbb{C}}^{*})/U^{-}$-embeddingsUsing Corollary 7 we obtain the following criterion. Corollary 8. Let $X_{\Sigma}$ be a horospherical variety with open orbit $(\mathrm{SL}(2)\times\mathbb{C}^{*})/U^{-}$ and coloured fan $\Sigma$ such that $\mathbb{R}^{2}\setminus |\Sigma|$ is connected. Then $X_{\Sigma}$ admits the Hartogs phenomenon if and only if Now we construct examples of horospherical varieties and the corresponding coloured fans. Remark 6. The standard action of $G=\mathrm{SL}(2)$ on $\mathbb{C}^{2}$ induces the action of $\mathrm{SL}(2)$ on the projective space $\mathbb{P}^{2}=\mathbb{P}(\mathbb{C}\oplus\mathbb{C}^{2})$ by the formula We denote the subgroup of $G$ consisting of upper triangular matrices by $B$. Note that $G$-orbits of this action are as follows:
We also have a $B$-stable but not $G$-stable divisor
$$
\begin{equation*}
H_{12}=\bigl\{[z_{0}:z_{1}:z_2]\in\mathbb{P}^{2}\mid z_{2}=0\bigr\}.
\end{equation*}
\notag
$$
Note that the open $B$-orbit is $\mathbb{P}^{2}\setminus(H_{\infty}\cup H_{0}\cup H_{12})\cong\mathbb{C}^{2}\setminus\{x_{2}=0\}$ (where $x_{2}=z_2/z_0$). Remark 7. Consider the toric action of $\mathbb{C}^{*}$ on $\mathbb{P}^{1}$ by the formula In this case we have the following $\mathbb{C}^{*}$-orbits: Now we consider the compact variety $X'=\mathbb{P}^{2}\times\mathbb{P}^{1}$ with the action of $\mathrm{SL}(2)\times\mathbb{C}^{*}$ given by the formula
$$
\begin{equation*}
\begin{aligned} \, &\begin{pmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & a_{33} \end{pmatrix}\!.\bigl([z_{0}:z_{1}:z_{2}],[w_{0}:w_{1}]\bigr) \\ &\qquad =\bigl([z_{0}:(a_{11}z_{1}+a_{12}z_{2}):(a_{21}z_{1}+a_{22}z_{2})],[w_{0}:a_{33}w_{1}]\bigr). \end{aligned}
\end{equation*}
\notag
$$
In this case the open $\mathrm{SL}(2)\times\mathbb{C}^{*}$-orbit is $\mathbb{C}^{2}\setminus\{(0,0)\}\times\mathbb{C}^{*}\cong \mathrm{SL}(2)\times\mathbb{C}^{*}/U^{-}$, and the open $B$-orbit is $\mathbb{C}^{2}\setminus\{(x_{2}=0)\}\times\mathbb{C}^{*}$. We have the following $\mathrm{SL}(2)\times\mathbb{C}^{*}$-orbits that are not open:
We also have the following $B$-stable divisors:
Now we calculate the points $a_{D_\infty}$, $a_{D_{10}}$, $a_{D_{01}}$ and $a_{D_{12}}$ in $N=\mathbb{Z}^{2}$ corresponding to the $B$-stable divisors $D_{\infty}$, $D_{10}$, $D_{01}$ and $D_{12}$. Consider a character $\lambda=(l,m)\in M=\mathbb{Z}^{2}$. We have the following:
So we obtain the coloured fan $\Sigma'$ of $X'$, which consists of the following coloured cones (see Figure 1):
Now we remove some $\mathrm{SL}(2)\times\mathbb{C}^{*}$-stable divisors from $X'$ to obtain a noncompact horospherical variety $X$, and we apply to $X$ the convex geometric criterion from Corollary 8.
Citation in format AMSBIB
\Bibitem{Fek22} |
|
||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|