The minimal model program and symplectic resolutions

Question

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 $L^+$ 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).

Answer 1

These things are quite involved and any reference that says something holds by [BCHM] without specifying the actual statement and how it needs to be applied is completely unfair. Also as you indicate [BCHM] is not the easiest read. Another account of most of the things that are in [BCHM] are also included in Hacon-K10, so you can at least try to look at two sources when you get stuck.

Also, whatever you do, in order for $X$ to admit a terminalization as stated, $X$ must have canonical singularities to begin with. Otherwise you cannot get a crepant morphism from something with terminal singularities. In other words, the existence of $Y$ implies that $X$ has canonical singularities. I assume that you get that for the particular $X$ this is applied to.

There are a few things one can say towards proving terminalization: For threefolds terminalization was proved by Reid in '83 and $\mathbb Q$-factorialization by Kawamata in '88. Both of these can be found in Kollár-Mori98, p.195. Often people refer to [BCHM] for the general fact that it provides a missing piece that had only been known up to dimension $3$ before and hence a lot of statements that had been only known up to dimension $3$ are now OK in arbitrary dimension. Of course, one should actually go through and check that everything works. In particular, if you look at Reid's proof of terminalization in dimension $3$ it uses some classification of $3$-dimensional canonical singularities, so the proof does not adapt to the arbitrary dimension right away.

One approach of trying to do both terminalization and $\mathbb Q$-factorialization is the following: The main theorem of [BCHM] is that the minimal model program can be run under fairly general conditions. One condition that is included in their statement that is often overlooked is that the models they get are all $\mathbb Q$-factorial. So the idea is this: run the minimal model program, then you end up with a klt $\mathbb Q$-factorial model. You probably don't even need this part since you should have an $X$ with canonical singularities as remarked above.

Anyway, once there you can take an arbitrary resolution and try to contract divisors that are non-crepant. The usual way to do this is to run a well-chosen mmp, that is, run the mmp with a well-chosen boundary divisor. This is vaguely explained in Hacon-K10, p.57. The relevant statement from [BCHM] is Corollary 1.4.3. See the paragraph following the statement. I suppose this may have been what Namikawa was referring to.

A variant of this idea is worked out in detail in Theorem 3.1 of Kollár-K10.

I don't know what's going on with the homotopy equivalence part of your question. I know that in general it is very hard to follow how the ample cone changes so I am guessing that in order to get what you want you will need to use some special properties of your situation.

Answer 2

The answer to the first question: if $X$ has klt singularities, then there exists a $\mathbb{Q}$-factorial variety $Y$ with a birational morphism $\pi: Y\to X$, such that if you write $\pi^*K_X=K_Y+\Delta$, then $\Delta$ is effective, and for any exceptional divisor $E$ of $Y$, we have the discrepancy $a(E,Y,\Delta)>0$, i.e., $(Y,\Delta)$ is terminal. This implies $Y$ itself is terminal. If you start with $X$ with only canonical singularities (This is stronger than klt singularities. But if $K_X$ is Cartier and $X$ is klt, then $X$ has canonical singularities. I mention this since I know in some cases from the representaion theory, indeed $K_X$ is trivial.), then $\Delta=0$. This follows from Corollary 1.4.3 of [BCHM]. In fact, it was known before that certain part of MMP would imply the existence of terminalization. The cases of MMP established in [BCHM] contain this part.