\begin{document}

\title{Symplectic resolutions for conical symplectic varieties}

\begin{abstract}
We introduce the notion of a conical symplectic variety,
and show that any symplectic resolution of such a variety
is isomorphic to the Springer resolution of a nilpotent
orbit closure in a semisimple Lie algebra, composed with a linear projection.
\end{abstract}

\section{Introduction}
\begin{definition}\label{def}
A {\em conical symplectic variety} is an affine variety $W \subset {\mathbb C}^N$, smooth in codimension 1,
such that (i) $W$ is invariant under the dilation action of ${\mathbb C}^*$ on ${\mathbb C}^N$, (ii)
There exists a holomorphic symplectic form $\omega$ on $W_{reg}$ such that $\lambda^* \omega = \lambda \omega$ for all $\lambda \in {\mathbb C}^*$, (iii) For any
resolution $\pi:Z \to W$, the 2-form $\pi^* \omega$ (defined on $\pi^{-1}(W_{reg}))$ extends to a regular 2-form $\Omega$ on  $Z$.
 A resolution $\pi: Z \to W$ is called {\em symplectic} if $\Omega$ is a holomorphic symplectic form on the whole of $Z$.
\end{definition}

For a conical symplectic variety $W \subset {\mathbb C}^N$, 
its normalization $\tilde{W}$ becomes a symplectic variety in the sense of Beauville {\mathbb C}e{B}.
Typical examples of conical symplectic varieties are nilpotent orbit closures
$\bar{{\mathcal O}}$ in a semi-simple Lie algebra $\mathfrak{g}$ and their symplectic resolutions can be constructed as follows.  For a flag variety $G/P$, its cotangent bundle $T^*_{G/P}$ admits a natural Hamiltonian $G$-action, and the image of the moment map 
$T^*_{G/P} \to \mathfrak{g}^* \simeq \mathfrak{g}$ is a nilpotent orbit closure 
$\bar{{\mathcal O}}_P$ (the Richardson orbit of $P$). Hence we have a projective generically 
finite morphism $\mu: T^*_{G/P} \to \bar{{\mathcal O}}_P$, which is called the Springer map 
associated to $G/P$. If $\mu$ is birational, it becomes a symplectic resolution 
of $\bar{{\mathcal O}}_P$, which we call a Springer resolution. In {\mathbb C}e{F}, it is proven 
that symplectic resolutions of nilpotent orbit closures are exactly Springer 
resolutions.

Further examples of conical symplectic varieties can be constructed as follows: 
take a nilpotent orbit ${\mathcal O} \subset {\mathfrak g}$ and a linear subspace $L \subset {\mathfrak g}$ 
such that the natural projection ${\mathfrak g} \to {\mathfrak g}/L$ maps $\bar{{\mathcal O}}$ birationally 
to a subvariety $W$. 
Then $W$ is a conical symplectic variety provided it is smooth in codimension 1. 
This happens for example if $L$ is a line, not contained in
the secant variety of $\bar{{\mathcal O}}$.
We suspect that this construction gives all conical symplectic varieties. 

The main result of this note classifies symplectic resolutions for conical 
symplectic varieties. It generalizes the main theorem of {\mathbb C}e{F}, 
with a simpler proof.

\begin{theorem} \label{main}
Let $\pi: Z \to W$ be a symplectic resolution for a conical symplectic variety. Then 

(i)  $Z \simeq T^*_{G/P}$  for some flag variety $G/P$.

(ii) The Springer map $\mu: T^*_{G/P} \to \bar{{\mathcal O}}_P$ associated to $G/P$ 
is birational.

(iii) There exists a linear subspace $L \subset {\mathfrak g}$ such that 
$W \subset {\mathbb C}^N$ is the image of  $\bar{{\mathcal O}}_P \subset {\mathfrak g}$ under the 
projection $p: {\mathfrak g} \to {\mathfrak g}/L $. The induced map $\bar{{\mathcal O}}_P \to W$ is birational.

(iv) The map $\pi$ is isomorphic to the composition $p \circ \mu$.

(v) If $W$ is normal, then $W = \bar{{\mathcal O}}_P$.
\end{theorem}

Remark that our assumptions in Definition \ref{def} are optimal for Theorem \ref{main}.  
This theorem is inspired by the alternative proof of {\mathbb C}e{F} provided by Namikawa in 
Section 8 of {\mathbb C}e{N2}.  
As an application, we provide a simpler proof of the main theorem of {\mathbb C}e{N2}, Section 7. 

{\em Acknowledgements:} 
We are grateful to Yoshinori Namikawa for pointing out an error in
a previous version, and to the referees for their helpful comments. 
Baohua Fu is supported by NSFC (11031008 and 11225106).

\section{Proof of Theorem \ref{main}}

For a smooth variety $Y$, we denote by $T_Y$ the tangent bundle, by $T^*_Y$ the cotangent bundle,
and by $K_Y$ the canonical bundle (the determinant of $T^*_Y$). 
A contact structure on $Y$  is a corank one subbundle $F \subset T_Y$ such that the bilinear form on $F$ with
values in the quotient line bundle $L = T_Y/F$ deduced from the Lie bracket on $T_Y$ is everywhere 
non-degenerate. This implies that $\dim Y = 2n-1$ is odd and $K_Y \simeq L^{-n}$.  We call $L$ the contact 
line bundle of the contact structure on $Y$. A typical example of contact manifold is the projectivised cotangent 
bundle ${\mathbb P}(T^*_M) := (T^*_M \setminus (\text{zero section}))/{\mathbb C}^*$ of a smooth variety $M$, where the contact 
line bundle is ${\mathcal O}_{{\mathbb P}(T^*_M)}(1)$.

\begin{proposition} \label{contact}
Let $X \subset {\mathbb P}^{N-1}$ be a closed singular subvariety 
and $f: Y \to X$ a resolution. Assume that $Y$ has a contact structure with the contact line bundle $f^* {\mathcal O}_X(1)$. 
Let $\tilde{X} \to X$ be the normalization map. Then $Y \simeq {\mathbb P}(T^*_{G/P})$ for some flag variety $G/P$ and 
the induced map $Y \to \tilde{X}$ yields the Stein factorization of the projectivised Springer map associated to $G/P$.
\end{proposition}
\begin{proof}
As $X$ is singular, we get $b_2(Y) \geq 2$. Note that  $K_Y = f^* {\mathcal O}_X(-n)$ with $n = (\dim X+1)/2$, 
hence $K_Y$ is not nef. By {\mathbb C}e{KPSW}, we deduce that 
$Y \simeq {\mathbb P}(T^*_M)$ for some smooth projective variety $M$ and 
${\mathcal O}_{{\mathbb P}(T^*_M)}(1) \simeq f^* {\mathcal O}_X(1)$. This implies that ${\mathcal O}_{{\mathbb P}(T^*_M)}(1)$,  hence
the tangent bundle $T_M$  is globally generated. 
We deduce that $M$ is a homogeneous variety, hence by {\mathbb C}e{BR} 
it is isomorphic to $G/P \times A$,
where $G/P$ is a flag variety for some semisimple algebraic group $G$ 
and $A$ is an abelian variety. 

Let $\hat{X} \subset {\mathbb C}^N$ be the affine cone of $X$. As 
$f^*{\mathcal O}_X(-1) = {\mathcal O}_{{\mathbb P}(T^*_M)}(-1)$, the map $f$ pulls back the ${\mathbb C}^*$-bundle $\hat{X} \setminus \{0\}  \to X$ to the ${\mathbb C}^*$-bundle 
$T^*_M \setminus \text{(zero section)} \to {\mathbb P}(T^*_M) $, which gives a birational map $\hat{f}: T^*_M \setminus \text{(zero section)} \to \hat{X} \setminus \{0\}$. Note that $T^*_M \simeq T^*_{G/P} \times {\mathbb C}^g \times A$, where $g = \dim A$. As $\hat{X}$ is affine, the map $\hat{f}$
contracts $\{x\} \times A$ to one point for any $x \in (T^*_{G/P}  \setminus \text{(zero section)}) \times ({\mathbb C}^g \setminus \{0\})$, hence $\hat{f}$ factors through $T^*_M \setminus \text{(zero section)} \to (T^*_{G/P}  \setminus \text{(zero section)}) \times ({\mathbb C}^g \setminus \{0\})$. As $\hat{f}$ is birational, we get $g=0$ and $M \simeq G/P$. If $\dim G/P =1$, then 
$G/P \simeq {\mathbb P}^1$ and $X$ is smooth, which contradicts our assumption. 
Thus, $\dim G/P \geq 2$.

Let $\mu: T^*_{G/P} \to \bar{{\mathcal O}}$ be the Springer map associated to $G/P$;
this is a projective, generically finite morphism. By the Stein factorization, 
it follows that the algebra $A := H^0(T^*_{G/P}, {\mathcal O}_{T^*_{G/P}})$ is finitely generated, 
and $\mu$ factors as a birational morphism 
$\tilde{\mu} : T^*_{G/P} \to V := {\rm Sp}ec(A)$ followed by a finite
 morphism $\eta: V \to \bar{{\mathcal O}}$. 
Note that $\tilde{\mu}$ and $\eta$ are both $G \times {\mathbb C}^*$-equivariant;
also, $V^{{\mathbb C}^*}$ is a single point, say $o$.

Let ${\mathbb C}[\hat{X}]$ be the coordinate ring of the affine variety $\hat{X}$, then the birational morphism $\hat{f}: T^*_{G/P} \setminus \text{(zero section)} \to \hat{X} \setminus \{0\} \subset \hat{X}$ induces a homomorphism of ${\mathbb C}$-algebras ${\mathbb C}[\hat{X}] \to H^0(T^*_{G/P} \setminus \text{(zero section)}, {\mathcal O}) = A$, where the latter equality follows from $\dim (G/P) \geq 2$. This gives a morphism $V \to \hat{X}$ and the map $\tilde{\mu}$ (restricted to $T^*_{G/P} \setminus \text{(zero section)}$) yields the Stein factorization of $\hat{f}$. Hence we get that $V \setminus \{o\}$ is the normalization of 
$\hat{X} \setminus \{0\}$. By taking the projectivisation, we get our claim.
\end{proof}

We now classify conical symplectic varieties with only isolated singularities.
\begin{lemma}\label{iso}
Let $W$ be a conical symplectic variety with only isolated singularities. Then  $W = \bar{{\mathcal O}}_{min}$, where ${\mathcal O}_{min}$
is the minimal nilpotent orbit in a simple Lie algebra. 
\end{lemma}
\begin{proof}
As $W$ is invariant by the ${\mathbb C}^*$-action, it has a unique singular point, which is $\{0\}$.  As $W \setminus \{0\} \to {\mathbb P} W$ is a ${\mathbb C}^*$-bundle, we deduce that ${\mathbb P} W$ is smooth. Note that by 
Lemma 1.4 {\mathbb C}e{B1}, the symplectic form on $W \setminus \{0\}$ induces a contact structure on ${\mathbb P} W$ with contact line bundle ${\mathcal O}_{{\mathbb P} W}(1)$. As   ${\mathcal O}_{{\mathbb P} W}(1)$ is very ample, we deduce that ${\mathbb P} W \simeq {\mathbb P} {\mathcal O}_{min}$ by Cor. 1.8 {\mathbb C}e{B1}.
This gives that $W = \bar{{\mathcal O}}_{min}$.
\end{proof}

Now let us prove Theorem \ref{main}.
First note that the ${\mathbb C}^*$-action on $W$ lifts to $Z$ (see for example 
Prop. A.7 of {\mathbb C}e{N1}), which makes $\pi$ to be ${\mathbb C}^*$-equivariant.
Let $\Omega$ be the symplectic form on $Z$ extending $\pi^* \omega$, then  we have $\lambda^* \Omega = \lambda \Omega$. We denote by ${\mathbb P} Z$ the quotient
$(Z \setminus \pi^{-1}(0))/{\mathbb C}^*$. Then we get a morphism $\bar{\pi}: {\mathbb P} Z \to {\mathbb P} W$. 
By Sections 3 and 4 of {\mathbb C}e{N2}, we have
\begin{lemma} \label{l.Nam}
${\mathbb P} Z$ is a smooth contact projective variety with the contact line bundle
$L:=\bar{\pi}^* {\mathcal O}_{{\mathbb P} W}(1)$.
\end{lemma}

If ${\mathbb P} W$ is smooth, then by Lemma \ref{iso}, $W = \bar{{\mathcal O}}_{min} \subset \mathfrak{g}$. As $W$ admits a symplectic resolution, this implies that $\mathfrak{g}$ is of type $A$ and $Z \simeq T^*_{{\mathbb P}^n}$ (cf. {\mathbb C}e{F} or Proposition \ref{sym} below). 
Assume now that ${\mathbb P} W$ is singular, then we can apply Lemma \ref{l.Nam} and Proposition \ref{contact} to conclude that $({\mathbb P} Z, L) \simeq ({\mathbb P} (T^*_{G/P}), {\mathcal O}_{{\mathbb P}(T^*_{G/P})}(1))$. Note that we may take $G = {\rm Aut}^0(G/P)$,
up to changing $G$ and $P$.  
 By the proof of Proposition \ref{contact}, 
we have the following diagram:
\[ {\mathbb C}D
Z @<<<  Z \setminus \pi^{-1}(0) @ >\simeq>> T^*_{G/P}\setminus \text{(zero section)}  @ >>> T^*_{G/P} \\ 
@V\pi VV  @VVV @VVV @VV\tilde{\mu}V \\
W @<<< W \setminus \{0\} @<\text{normalization}<< V\setminus \{o\} @>>> V\simeq \tilde{W}
\endCD \]

As ${\mathbb P} W$ is not smooth, we may assume $G/P \neq {\mathbb P}^n$. Now we can apply Lemma 5 {\mathbb C}e{N2} and the argument in 
{\em loc. cit.} (p. 183) to deduce that $Z \simeq T^*_{G/P}$ and this identifies $\tilde{\pi}: Z \to \tilde{W}$ with the map 
$\tilde{\mu}$ from the Stein factorization of the Springer map $\mu: T^*_{G/P} \to \bar{{\mathcal O}}$. 
As seen in the proof of Proposition  \ref{contact}, it follows that $\tilde{W}$ admits a $G \times {\mathbb C}^*$-action 
such that $\tilde{\pi}$ is equivariant.

As $W \subset {\mathbb C}^N$ and ${\mathbb C}^*$ acts on ${\mathbb C}^N$ by dilations, 
the coordinate ring ${\mathbb C}[W]$ is a subalgebra of ${\mathbb C}[\tilde{W}]$ 
generated by elements of degree 1. On the other hand, we have  
${\mathbb C}[\tilde{W}] = H^0(T^*_{G/P}, {\mathcal O}_{T^*_{G/P}})$. The space of degree 1 
elements in $H^0(T^*_{G/P}, {\mathcal O}_{T^*_{G/P}})$ is $H^0(G/P, T_{G/P})$, 
which is $\mathfrak{aut}(G/P) = {\mathfrak g}$. Thus, the algebra ${\mathbb C}[W]$ 
is generated by a linear subspace of ${\mathfrak g}$, that we view as the orthogonal 
of a linear subspace $L \subset {\mathfrak g}$. So the normalization map 
$\tilde{W} \to W$ factors as the map 
$\tilde{W} \to \bar{{\mathcal O}} \subset {\mathfrak g} \cong {\mathfrak g}^*$ corresponding 
to the inclusion ${\mathfrak g} \subset H^0(T^*_{G/P}, {\mathcal O}_{T^*_{G/P}})$, 
followed by a map $p: \bar{{\mathcal O}} \to W$, the restriction of the quotient 
map ${\mathfrak g} \to {\mathfrak g}/L$. Thus $p$ is birational. If $W$ is normal, then  
$W = \tilde{W}$, hence we get $W = \bar{{\mathcal O}}$.
This finishes the proof of Theorem \ref{main}. \\

In the proof of Theorem \ref{main}, we used {\mathbb C}e{F} to deduce that among minimal nilpotent orbit closures, only that of type $A$ admits a symplectic resolution. One can in fact prove a stronger result.
\begin{proposition} \label{sym}
Let $W$ be a normal variety of dimension $2n \geq 4$ with an isolated singularity $0$ and a contracting ${\mathbb C}^*$-action.  Assume that $W \setminus \{0\}$ admits a symplectic form $\omega$ satisfying $\lambda^* \omega = \lambda \omega, \forall \lambda \in {\mathbb C}^*$.
If $W$ admits  a symplectic resolution $\pi: Z \to W$, then $Z \simeq T^*_{{\mathbb P}^n}$ and $W \simeq \bar{{\mathcal O}}_{min} \subset \mathfrak{sl}_{n+1}$.
\end{proposition}
\begin{proof}
Recall that the action of ${\mathbb C}^*$ on $W$ lifts to an action on $Z$ (see Prop. A.7 of {\mathbb C}e{N1}).
Let $Z^{{\mathbb C}^*}$ be the subvariety of $Z$ consisting of ${\mathbb C}^*$-fixed points, which is a disjoint union of smooth subvarieties 
since $Z$ is smooth. As $W^{{\mathbb C}^*} = \{0\}$, we have that $Z^{{\mathbb C}^*}$ is contained
in the fiber $\pi^{-1}(0)$, hence it is a union of projective manifolds.

By the Bialynicki-Birula decomposition (see {\mathbb C}e{BB} which applies in this non-projective setting, since $Z$ is proper over 
$W$ which is contracted to $0$ by the $ {\mathbb C}^*$-action), there exists an irreducible component $M$ of $Z^{{\mathbb C}^*}$ 
such that the set $U:=\{z \in Z| \lim_{\lambda\to 0} \lambda \cdot z \in M\}$
is open in $Z$. Let $\Omega$ be the symplectic form on $Z$, which is the extension of $\pi^* \omega$.
 As $\lambda^* \Omega = \lambda \Omega$ for all $ \lambda \in {\mathbb C}^*$, we obtain that $(U, \Omega) \simeq (T^*_M, \omega_{can})$ as ${\mathbb C}^*$-varieties (cf. Lemma 3.7 {\mathbb C}e{F}), where $\omega_{can}$ is the canonical symplectic form on $T^*_M$.
On the other hand,  we have $\dim \pi^{-1}(0) \leq \cfrac{1}{2} \dim W = \dim M$ (Cor. 8.5 {\mathbb C}e{CMSB}). This implies that $M$  is an irreducible component of $\pi^{-1}(0)$. As $M$ is smooth, by Cor. 8.7 {\mathbb C}e{CMSB},
we get that $M \simeq {\mathbb P}^n$. It follows (for example by the arguments above) that the map $U \to W$ is the Springer map
$\mu: T^*_{{\mathbb P}^n} \to \bar{{\mathcal O}}_{min} \subset \mathfrak{sl}_{n+1}$.  As $\mu$ is projective, we get $U=Z$. 
\end{proof}

\section{An application}
A {\em symplectic variety} is a normal variety $W$ with a symplectic form
$\omega$ on its smooth locus such that for any resolution $\phi: Z \to W$, 
the 2-form $\phi^* \omega$ defined on $\phi^{-1}(W_{reg})$ extends 
to a regular 2-form on $Z$. 

As an application of Theorem \ref{main}, we provide an alternative proof 
of the following main result of {\mathbb C}e{N2}.
\begin{theorem}[Namikawa]
Let $(W, \omega)$ be a singular symplectic variety embedded in ${\mathbb C}^N$ 
as a complete intersection of hypersurfaces defined by homogeneous polynomials.
Assume that the symplectic form $\omega$ satisfies 
$\lambda^* \omega = \lambda^k \omega, \forall \lambda \in {\mathbb C}^*$ for some 
$k$. Then $(W, \omega)$ is isomorphic to the nilpotent cone
$(\mathcal{N}, \omega_{KK})$ of a semisimple complex Lie algebra $\mathfrak{g}$ 
together with the Kirillov- Kostant form $\omega_{KK}$.
\end{theorem}

\begin{proof}
By Section 2 of {\mathbb C}e{N2}, $W$ is a conical symplectic variety with 
a symplectic resolution, hence by Theorem \ref{main}, we get that 
$(W, \omega)$ is a nilpotent orbit closure in a semi-simple Lie algebra 
${\mathfrak g}$. But every nilpotent orbit closure $\bar{{\mathcal O}} \subset {\mathfrak g}$ 
which is a complete intersection in $\mathfrak{g}$ 
must be the full nilpotent cone, by the main result of Section 7 
of {\mathbb C}e{N2}. 
We now provide a proof of that result, which is somehow shorter and
more uniform than the original one.

Let $d_1, \cdots, d_r \geq 2$ be the degrees of defining equations of 
$\bar{{\mathcal O}}$ in ${\mathfrak g}$. Then $ r = \mathop{\rm codim}\nolimits_{\mathfrak g}({\mathcal O})$ and by {\mathbb C}e{N2} 
(Section 2, p. 160), we have 
\begin{equation}\label{eq}
\sum_{i=1}^r d_i  = \cfrac{1}{2} \dim {\mathcal O} + \mathop{\rm codim}\nolimits_{\mathfrak g}({\mathcal O}),  
\quad \text{and} \quad \mathop{\rm codim}\nolimits_{\mathfrak g}({\mathcal O}) \leq \cfrac{1}{3} \dim {\mathfrak g}. 
\end{equation}
We may assume that ${\mathfrak g}$ is simple. We denote by $I$ the ideal of 
$\bar{{\mathcal O}}$ in the coordinate ring ${\mathbb C}[{\mathfrak g}]$, and by ${\mathfrak m}$ the maximal 
ideal of $0$ in ${\mathbb C}[{\mathfrak g}]$. Let $G$ be the adjoint group of ${\mathfrak g}$; then 
$G \times {\mathbb C}^*$ acts on ${\mathfrak g}$ (where ${\mathbb C}^*$ acts by dilations) 
and stabilizes $\bar{{\mathcal O}}$ and $0$. Hence 
$G \times {\mathbb C}^*$ acts on ${\mathbb C}[{\mathfrak g}]$ and stabilizes $I \subset {\mathfrak m}$. 
Since $G \times {\mathbb C}^*$ is reductive, we may find a submodule $M \subset I$ 
which is mapped isomorphically to $I/{\mathfrak m} I$ under the quotient map 
$I \to I/{\mathfrak m} I$. By the graded Nakayama lemma, a homogeneous basis 
of $M$ yields a minimal generating system of the ideal $I$, 
and hence a regular sequence in ${\mathbb C}[{\mathfrak g}]$ since $\bar{{\mathcal O}}$ 
is a complete intersection. In geometric terms, the morphism
\[ f : {\mathfrak g} \longrightarrow M^*=: V \] 
corresponding to the inclusion $M \subset {\mathbb C}[{\mathfrak g}]$ is flat and 
its (scheme-theoretic) fiber $f^{-1}(0)$ equals $\bar{{\mathcal O}}$ 
(this is a slightly more precise version of Lemma 3 in {\mathbb C}e{N2}). 
Thus, $f$ is open, and hence surjective by ${\mathbb C}^*$-equivariance. 

Choose a maximal torus $T \subset G$ and denote by ${\mathfrak t}$ its Lie algebra; 
this is a Cartan subalgebra of ${\mathfrak g}$. Since $G {\mathfrak t}$ is dense in ${\mathfrak g}$, 
there exists $x \in {\mathfrak t}$ such that the differential $df_x : {\mathfrak g} \to V$ 
is surjective. As $df_x$ is linear and $T$-equivariant, it follows that 
each weight of the $T$-module $V$ is also a weight of ${\mathfrak g}$. In particular, 
the highest weight of any simple summand of the $G$-module $V$ is either the
highest root, or the highest short root $\lambda$ (if $G$ is not simply laced), 
or $0$. Moreover, the highest root cannot occur, since the corresponding 
simple module is just ${\mathfrak g}$, and 
$\dim(V) = \dim({\mathfrak g}) - \dim(\bar{{\mathcal O}}) < \dim({\mathfrak g})$.

If $G$ is simply laced, then $V$ must be the trivial $G$-module, 
and hence $\bar{{\mathcal O}}$ contains the nilpotent cone $\mathcal{N}$; so 
we conclude that $\bar{{\mathcal O}} = \mathcal{N}$. Thus, we may assume 
that $G$ is not simply laced; then we have an isomorphism of $G$-modules 
\[ V \cong p V(0) \oplus q V(\lambda), \] 
where $p$, $q$ are non-negative integers, $V(0)$ denotes the trivial 
$G$-module ${\mathbb C}$, and $V(\lambda)$, the simple $G$-module with highest 
weight $\lambda$. (The latter module is called the ``little adjoint module'' 
in {\mathbb C}e{P}, where its invariant theoretical properties are investigated.)
If $q = 0$ then we conclude as above that $\bar{{\mathcal O}} = \mathcal{N}$; thus, 
we may further assume that $q \geq 1$. 

If $G$ of type $C_n$ (resp. $F_4$, $G_2$), then $V(\lambda)$
has dimension   $2 n^2 - n - 1$ (resp. $26$, $7$),
which contradicts  the inequality 
$\mathop{\rm codim}\nolimits_{\mathfrak g} ({\mathcal O}) = \dim V \leq \frac{1}{3} \dim({\mathfrak g})$ in \eqref{eq}. 
Thus, we may assume that $G = {\rm SO}(2n + 1)$; then 
$V(\lambda)$ is the natural $G$-module ${\mathbb C}^{2n+1}$.

Since $f$ is surjective and $G {\mathfrak g}^T$ is dense in ${\mathfrak g}$, it follows
that $G V^T$ is dense in $V$. But this does not hold for 
$V = 2 V(\lambda)$ (as $V(\lambda)^T$ is a line), and hence $q \leq 1$. 
So we may take $V = p V(0) \oplus V(\lambda)$. Since the algebra
of invariant functions ${\mathbb C}[V(\lambda)]^G$ is generated by the
quadratic form defining $G$, it follows that the categorical quotient 
$V/\!/G := {\rm Sp}ec \, {\mathbb C}[V]^G$ is an affine space, and the quotient morphism
\[ q_V : V \longrightarrow V/\!/G \]
is flat with reduced fibers. Since $q_V$ sits in a commutative square
\[ {\mathbb C}D
{\mathfrak g} @>{f}>> V \\
@V{q_{{\mathfrak g}}}VV @V{q_V}VV \\
{\mathfrak g}/\!/G @>{f/\!/G}>> V/\!/G, \\
\endCD \]
where $f$, $q_{{\mathfrak g}}$ are flat with reduced fibers, we see that 
$f/\!/G$ is flat with reduced fibers, too.

We claim that $f/\!/G$ is also finite. Consider indeed the restriction
$f^T : {\mathfrak g}^T = {\mathfrak t} \to V^T$. Then the fiber of $f^T$ at $0$ equals
$\bar{{\mathcal O}} \cap {\mathfrak t} = \{ 0 \}$ (as a set). 
Since $f^T$ is equivariant for the natural
actions of ${\mathbb C}^*$, it follows that $f^T$ is finite. Hence so is
$f^T/W : {\mathfrak g}^T/W \to V^T/W$, where $W$ denotes the Weyl group of
$(G,T)$. But the natural map ${\mathfrak g}^T/W \to {\mathfrak g}/\!/G$ is an isomorphism
by the Chevalley restriction theorem; moreover, the analogous map
$V^T/W \to V/\!/G$ is finite, and dominant since $GV^T$ is dense in $V$. 
The finiteness of  $f/\!/G$ follows from this in view of the commutative square
\[ {\mathbb C}D
{\mathfrak g}^T/W @>{f^T/W}>> V^T/W \\
@VVV @VVV \\
{\mathfrak g}/\!/G @>{f/\!/G}>> V/\!/G. \\
\endCD \]

Since $f/\!/G$ is also flat with reduced fibers, it is a finite \'etale cover, 
and hence an isomorphism as $V/\!/G$ is an affine space. In other words,
the algebra ${\mathbb C}[{\mathfrak g}]^G$ is freely generated by the pull-backs 
under $f$ of the $p$ projections $V \to V(0)$ and of the basic 
invariant of $V(\lambda)$. The degrees of these invariants are
$a_1,\ldots,a_p,2 a_{p+1}$, where $a_1,\ldots,a_p$ denote the
weights of the action of ${\mathbb C}^*$ on $p V(0)$, and $a_{p+1}$
the weight of that action on $V(\lambda)$. On the other hand,
the degrees of the basic invariants of ${\mathbb C}[{\mathfrak g}]^G$ are 
$2,4,\ldots,2n$. In particular, we have $n = p + 1$ and
\[ a_1 + \cdots + a_{n-1} + 2 a_n = n^2 + n. \] 

This implies that $r = \dim V = 3n$ and 
$\dim {\mathcal O} = \dim \mathfrak{g}- 3n = 2n^2-2n$.
Moreover, we may assume that the degrees of defining equations of $\bar{{\mathcal O}}$
satisfy $d_i = a_i$ for $i = 1, \cdots, n-1$, and $d_n = \cdots = d_{3n} = a_n$.
By \eqref{eq}, we have 
$d_1 + \cdots + d_{3n} = (n^2 - n) + 3 n$,
which gives that 
\[ a_1 + \cdots + a_{n-1} + (2 n + 1) a_n = n^2 + 2 n. \]
Combining the two displayed equalities yields
$(2n - 1) a_n = n$ which is not possible since $a_n = d_n \geq 2$.

\end{proof}

Michel Brion 

Institut Fourier, Universit\'e de Grenoble, France 

e-mail: Michel.Brion@ujf-grenoble.fr 

Baohua Fu

Institute of Mathematics, AMSS, 55 ZhongGuanCun East Road,

Beijing, 100190, China

e-mail: bhfu@math.ac.cn

\end{document}