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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:

1. 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.
2. 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}^*$action to keep everything nice). 2 added 1036 characters in body; added 8 characters in body; added 2 characters in body EDIT: Let me expand a little bit on what I am hoping for: In another paper of Namikawa (look in section (P.2)), he describes an approach to classifying symplectic resolutions which sounds a bit like my 2. above. You • start with a line bundle$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$is negative on, contract it, and then find a symplectic resolution of the contraction for which$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}^*$action to keep everything nice). 1 # The minimal model program and symplectic resolutions I've been reading some papers of Namikawa lately, and have on several occasions come across a claim I would really like someone to expand on. On page 4 of Poisson deformations of affine symplectic varieties, he says: According to Birkar-Cascini-Hacon-McKernan, we can take a crepant partial resolution$\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$is assumed to be affine and symplectic (as defined on the first page of Namikawa's paper). So, I have 2 questions: 1. Which results in this paper is this supposed to follow from? I can see some things along these lines, but with a lot of hypotheses I'm not used to dealing with (like "Kawamata log terminal"), and Namikawa doesn't say a word more than what is above for why this works. 2. How much control does one have on the ample divisors on the resolution? This is very vague, so let me lay out what I'm hoping for; Namikawa proves that$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.