# Symplectic resolutions amongst cotangent bundles

It is known that a generalized flag variety $$X=T^*(G/P)$$ is a (symplectic) resolution of singularities of its affinization $$X^\text{aff}\mathrel{:=}\operatorname{Spec}(H^0(X,\mathcal{O}_X))$$. In type A, this affinization is the closure of a nilpotent orbit, and in other types, it is finite over it. Moreover, it is conjectured in Kaledin's paper (Geometry and topology of symplectic resolutions, Conj. 1.3) that these are the only cotangent bundles of smooth projective varieties which are resolutions of their affinization.

As this paper was written some time ago, I am interested in whether any progress has been made towards proving this conjecture?

NB Everything is assumed under complex numbers, the flag variety, and the symplectic structure.

• What is "this property"? For example, is the question which cotangent bundles $T^*(M)$ have the property that their affinisation is the normalisation of the closure of a nilpotent orbit for some reductive group $G$? Commented Mar 26, 2021 at 20:18
• Yes, exactly. I have changed the question to make it more explicit now. Commented Mar 26, 2021 at 21:29

I think @Joel is saying the following (this was too long for a comment, so I put it as an answer):

Given a conical symplectic resolution $$X\mathrel{:=}T^*M \rightarrow X^\text{aff}$$ whose $$\mathbb{C}^*$$-action contracts the fibres of $$X$$ with weight 1, the ring of global functions $$R=H^0(X,\mathcal{O}_X)$$ has a set of homogenous generators $$\{x_1,\dotsc,x_n\}$$ with maximal weight 1. (**)

The last fact, together with the normality of the ring $$R$$, is precisely saying that $$X^\text{aff}$$ is a conical symplectic singularity with maximal weight 1, thus the Namikawa's theorem applies, stating that $$X^\text{aff}$$ must be a closure of a normal nilpotent orbit in a semisimple $$\mathfrak{g}$$. Then, by Fu_Symplectic Resolutions for Nilpotent Orbits (Theorem 0.1), we indeed have $$X \cong T^*(G/P),$$ for some parabolic subgroup $$P$$ of $$G$$ (and $$\operatorname{Lie}(G)=\mathfrak{g}$$). Moreover, $$X^\text{aff}=\overline{O}_P\mathrel{:=}\operatorname{Im}(\mu),$$ where $$\mu:T^*(G/P)\rightarrow \mathfrak{g}^*\cong \mathfrak{g}$$ is the moment map of the $$G$$-action (and $$\mathfrak{g}^*\cong \mathfrak{g}$$ is obtained via Killing form).

Though, what still worries me is that the argument above does not cover resolutions $$X=T^*(G/P) \rightarrow X^\text{aff},$$ where the induced map $$X^\text{aff}\rightarrow \overline{\mathcal{O}_P}$$ is not an isomorphism, but rather a finite cover. This is a general setup, and in some cases (e.g. when $$G=\operatorname{SL}_n$$), this map is indeed an isomorphism.

Having this in mind, perhaps the statement (**) about the weights of global functions is wrong?

In A characterization of nilpotent orbit closures among symplectic singularities, Namikawa proved that a weight 1 conical symplectic singularity must be a nilpotent orbit closure.

So I guess that this implies that a weight 1 conical symplectic resolution (which just means that the affinization is a weight 1 conical symplectic singularity) must be the cotangent bundle of a partial flag variety. (I think that it is known that if a nilpotent orbit closure admits a symplectic resolution, then this symplectic resolution must be a cotangent bundle.)

• Joel, could you please check whether what I am saying in the answer below is what you are saying. In particular, there is an issue (see the question at the end). Commented Apr 6, 2021 at 9:53