Open problems in Birational Geometry, after BCHM

Rencently a breakthrough was made in the context of the Minimal Model Program by the work of Birkar-Cascini-Hacon-McKernan. They proved that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.

Since I'm a master student and so I have no a wide view of the subject (I'm not an expert), I would like to know what are the main open problems in this direction (I mean, in the framework of the Mori Program). More generally, right now what are the driving forces, the big open questions in birational geometry?

Feel free to close this question, if too generic for the purposes of the site. Thanks in advance.

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Finite generation of the canonical ring of a compact Kahler manifold is still open (you know, if you believe in things like Kahler manifolds). – Gunnar Þór Magnússon Nov 30 '12 at 18:06
This is a bit off-topic, but lest the casual reader get the wrong impression, let me also mention that there has been plenty more progress in birational geometry since BCHM. Some examples: the proof of ACC for log canonical thresholds (first de Fernex--Ein--Mustata, then Hacon--McKernan--Xu); boundedness of birational automorphisms for varieties of general type (Hacon-McKernan-Xu); an alternative proof of finite generation of the canonical ring (Cascini--Lazic). – user5117 Dec 1 '12 at 18:03
No, it's not off-topic :), thanks for pointing out this; your comment is very useful! – Emanuele Ventura Dec 1 '12 at 18:19
I think that your question (and the answers) are interesting. I just to give a small comment. It seems here that people assume that "Birational geometry" = "Minimal model program". The question here is only about existence of MMP and related questions, but there are plenty of other open questions in birational geometry. For example, we do not know if a general smooth cubic of dimension $4$ is rational, and we do not know how to describe the group of birational transformations of $\mathbb{P}^n$ for $n\ge 3$ or of a cubic of dimension $3$. – Jérémy Blanc Dec 3 '12 at 8:53
Yes, there are many explicit rationally connected varieties with large finite group of automorphisms and we do not know if these are rational or not, so we do not know if the corresponding finite groups are subgroup of the Cremona group (group of birational transformations of $\mathbb{P}^n$). – Jérémy Blanc Dec 6 '12 at 13:43

[Just 'cause Artie asked:] :)

Many parts of the mmp are not know for log canonical pairs. There are many results in that direction, but also many questions are open. In some sense log canonical is a more natural class than klt or even dlt and it is very important from the point of view of applications to moduli theory because semi log canonical (the non-normal version of lc) singularities are closed under stable degeneration while klt singularities are not. A major difficulty stems from the fact that lc singularities are not rational.

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Thank you for the nice answer! :) – Emanuele Ventura Dec 1 '12 at 11:04
Thanks Sándor!. – user5117 Dec 1 '12 at 15:58
• The Abundance conjecture.

In its simplest form it says: If $X$ is a minimal variety (that is, the canonical divisor $K_X$ is nef and $X$ has terminal singularities) then some multiple $mK_X$ is base-point free. Thus sections of some power of the canonical bundle give a morphisms to projective space. In this case, it is straightforward to prove that the canonical ring $R(X,K_X)$ is finitely generated.

• Termination of log flips.

I think both of these conjectures are open in dimension $\ge 4$.

Edit: As Artie points out, existence of flips is known in all dimensions by the work of BCHM. There are also partial results on termination in dimension 4 by, Birkar, Fujino, Alexeev-Hacon-Kawamata,..

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BCHM proves existence of (klt) flips: Corollary 1.4.1. – user5117 Nov 30 '12 at 17:46
Many parts of the mmp are not know for log canonical pairs. There are many results in that direction, but also many questions are open. In some sense log canonical is a more natural class than klt or even dlt and it is very important from the point of view of applications to moduli theory because semi log canonical (the non-normal version of lc) singularities are closed under stable degeneration while klt singularities are not. A major difficulty stems from the fact that lc singularities are not rational. – Sándor Kovács Nov 30 '12 at 18:08
Sándor: you should write your comment as an answer! – user5117 Nov 30 '12 at 18:43
Done. ......... – Sándor Kovács Dec 1 '12 at 7:08

In characteristic 0.

1. Abundance conjecture. This is probably universally accepted as the most important question for the current minimal model program after BHCM's proof of finite generation. I want to remark that n-dimensional log canonical MMP follows from n-dimensional klt MMP and n-1 dimensional log canonical MMP. So there is no new difficulty for log canonical pairs at least for MMP other than abundance for klt.

2. General type. In fact, after BCHM, Koll\'ar and many other people's work, I think we have a pretty good understanding of the rough classification of general types, namely, we know they form a moduli space which has a pretty reasonable compactification.

3. Fano and singularities. The boundedness of Fano, namely the BAB-conjecture is one of the most fundamental question about singular Fano. We only know special cases. In philosophy, there is a local-to-global principle, and correspondingly, we should consider certain boundedness of singularities. The famous challenge there is Shokurov's ACC of mld conjecture. Note this should be substantially harder than ACC of log canonical thresholds which now is a theorem. Another one is the semi-continuity of mld conjecture. And Shokurov proved these two conjectures together imply termination of flips.

4. Calabi-Yau. There are two fundamental problems there to me. One is the finiteness of the topological type, Another one is Kawamata-Morrison Cone conjecture. Each of them is still out of reach even in dimension 3. This is a not a big surprise, since we are still quite lack of understanding of CY in dimension 3. On the other hand, if you apply local-to-global principle here, then we should consider semi-log-canonical CY. And Koll\'ar's recent examples point out that the slc picture is even more complicated.

In characteristic p.

In char p, the most important question to me is the resolution of singularity. The next thing is how many results in characteristic 0 MMP still hold in characteristic p. WIthout vanishing type theorem, we can still formulate most of those results, but then it really puts a question mark on whether we should still believe them.

My feeling is although there are many substantial work there recently, it's still not clear what the picture should be.

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About 4. there is also the long-standing question of whether every Calabi-Yau (in the strict sense, of any dimension) contains a rational curve (the expected answer is yes). – YangMills Dec 2 '12 at 4:29
CX, thanks for the detailed answer :) – Emanuele Ventura Dec 6 '12 at 8:27

I'm not an expert, but my understanding is that BCHM assumes characteristic zero. What about positive characteristic?

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Very good point. A huge effort is now directed towards positive characteristics. A big problem is that Kodaira vanishing fails, so most of the methods need new ideas. And of course there is the issue of (the lack of knowing the) existence of resolution of singularities, but that is a problem that can be set aside if one starts with smooth varieties. It is still a problem, because for instance one cannot simply take a common resolution of two different ones, which we do in characteristic 0 without thinking much about it. So this is definitely an area where there is lots to do. – Sándor Kovács Nov 30 '12 at 18:02