I've been reading some papers of Namikawa lately, and have on several occasions come across a claim I would really like someone to expand on.
On page 4 of Poisson deformations of affine symplectic varieties, he says:
According to Birkar-Cascini-Hacon-McKernan, we can take a crepant partial resolution $\pi: Y \to X$ in such a way that $Y$ has only $\mathbb{Q}$-factorial terminal singularities. This $Y$ is called a $\mathbb{Q}$-factorial terminalization of $X$.
Here $X$ is assumed to be affine and symplectic (as defined on the first page of Namikawa's paper).
So, I have 2 questions:
- Which results in this paper is this supposed to follow from? I can see some things along these lines, but with a lot of hypotheses I'm not used to dealing with (like "Kawamata log terminal"), and Namikawa doesn't say a word more than what is above for why this works.
- How much control does one have on the ample divisors on the resolution? This is very vague, so let me lay out what I'm hoping for; Namikawa proves that $X$ is homotopy equivalent to a generic deformation $Y'$ of $Y$ in a reasonably canonical way (there are some choices involved, but they're controlled). For any isomorphism, you can ask if a class in $H^2(Y';\mathbb{R})$ is in the nef cone of $NS(X)$ under the induced isomorphism on cohomology. What I'm hoping is that there's a way of resolving $X$ and then choosing a homotopy equivalence to $Y$ that makes this so, which is unique if the class is in the interior.
Is there any hope of such a picture existing? I'm having too much trouble parsing the BCHM paper to tell whether such a story is in there or not.
EDIT: Let me expand a little bit on what I am hoping for: In another paper of Namikawa (look in section (P.2)), he describes an approach to classifying symplectic resolutions which sounds a bit like my 2. above. You
- start with a line bundle $L$, which you want to make into the ample line bundle on a different resolution (you imagine it is the proper transform of that line bundle).
- attempt to do a flop which makes this line bundle closer to being nef; that is, you find a curve $L$ is negative on, contract it, and then find a symplectic resolution of the contraction for which the proper transform $L$ L^+$ is relatively ample.
- rinse and repeat until $L$ is genuinely ample.
In the paper mentioned above, this is done for nilpotent orbits, using very specific known facts about how these orbits and their resolutions work. What I was really hoping for was some indication of whether this story can be run on a general symplectic resolution (probably with a $\mathbb{C}^*$ action to keep everything nice).
