The last few weeks have seen the appearance of two papers giving very different proofs of a quite important result in algebraic geometry, resolving a question that had been open for a very long time, and in the process helping to make progress in the classification of higher dimensional projective algebraic varieties. Readers should be warned that this doesn’t have anything to do with physics, and my knowledge of this kind of mathematics is highly shaky, so I’m relying largely on second-hand information from people much better informed than myself.

The theorem in question concerns the “canonical ring” of a smooth projective algebraic variety X, which is the graded ring R(X) defined by

$$R(X)=\oplus_{n=0}^\infty H^0(X, nK)$$

Here K is the canonical line bundle (top exterior power of the cotangent bundle) of X, nK is its n’th tensor power, and $H^0(X, nK)$ is the space of holomorphic sections of the bundle nK. This is also called the pluricanonical ring.

The new theorem says that this graded ring is finitely generated, and this implies quite a few facts about projective algebraic varieties of any dimension. In particular it implies the main goal of the “minimal model program” (also known as the Mori program) for classifying higher dimensional algebraic varieties.

A proof of this theorem was claimed back in 1999 by Hajime Tsuji, but it appears that there are problems with this proof. The arXiv preprint went through many revisions, but was never refereed and published. A couple weeks ago, a group of four algebraic geometers (Caucher Birkar, Paolo Cascini, Christopher Hacon and James McKernan) posted a preprint on the arXiv claiming a proof. Yesterday, Yum-Tong Siu, a well-known complex geometer from Harvard, posted another preprint, giving a very different, more analytical, proof of this theorem. Siu notes that he has been lecturing on this proof for over a year, first at last year’s Seattle conference on Algebraic Geometry.

The mathematicians involved in creating these two proofs are well-known experts, and it seems likely that both proofs are correct. Given that there are two of a quite different nature, it now seems extremely likely that this theorem has been proved.

For more detailed explanations of this result and its implications, I’m afraid that you’re likely to require someone who knows a lot more about algebraic geometry than I do. Perhaps some of my more expert readers here can help out.

I know next to nothing about this, but I believe that Proj of the canonical ring provides a nice model for the space.

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I’m planning on spending some time looking into this later, since it’s not my area, but it looks like this gives a minimal model for varieties of general type (i.e. most varieties given by high-degree polynomials). This is a big step forward for the minimal model program, which (in my understanding) splits birational classification into two steps: find minimal models for varieties, and then determine when varieties have multiple minimal models. I think that it’s hoped that varieties of general type only have one. But as I said, it’s not my area.

That is not quite true. There can be more than one minimal model, but only one canonical model (=Proj(R(X,K_X)).

Even for varieties of general type?

Yes. Why not? they are connected by flops and they are easy to construct. If a variety is of general type, then it follows that the # of minimal models is finite.