In the next days I have to give a talk in which I need to explain some of the usual singularities of pairs that one meets when dealing with the minimal model program: KLT, DLT and LC pairs. In particular I would like to give an intuition (also to non specialists) of the reason why an LC pair is much more difficult to treat than a DLT pair, for example. In the same context I would also like to give an intuition of what the LC centers of a pair are and why they are "special" subvarieties for a pair. Note that I'm always considering normal (possibly non-smooth) projective varieties. Do you have any ideas?
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3 Answers
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For references, probably you already know things like Kollar-Mori and Kollar's "Singularities of pairs", for LC-centers and subadjunction, Kollar has some notes on that as well. There's some sections on this in the recent book by Hacon-Kovacs too that I've looked at recently.
With regards to your specific questions I have some comments but probably VA will have better comments:
1. an LC pair is much more difficult to treat than a DLT pair
The archetypical LC pair is probably $(\mathbb{A^2}, Div(x) + Div(y))$, the two coordinate axes (EDIT: this example is also DLT, and doesn't really represent the most poorly-behaved properties of LC-pairs). This is a simple normal crossings (SNC) pair and we understand how they behave pretty well with respect to numerous operations. On the other hand, if you have a pair $(X, D)$ where the pair is KLT, then even if the singularities of $D$ are bad, the fact that $(X, D)$ is klt means that you can perturb the coefficients of $D$ in many ways. This gives one a great amount of flexibility in numerous situations.
DLT, by Szabo's criterion, is a combination of KLT and SNC with coefficients $\leq 1$. Basically, a DLT pair is LC and where it is not KLT, it is SNC (simply normal crossings), which we understand. Dually, where it's not SNC, it's KLT, and we can perturb things and do those other tricks.
2. what the LC centers of a pair are and why they are "special" subvarieties for a pair.
The most basic example of a LC-center of a pair $(X, D)$ is a prime divisor $D_i$ which is a component of $D$ and such that the $D_i$-coefficient of $D$ is $1$. For example, if $X$ is smooth, and $D$ is a prime divisor, then $D$ is a LC-center of $(X, D)$. Why is this nice? This lets us relate the canonical divisor $K_X + D$ of $(X, D)$ and the canonical divisor of $D$.
Explicitly, in this smooth case, $(K_X + D)|_D = K_D$. Because of this, you can translate many properties of the pair $(X, D)$ to things on $D$ (look up extension theorems for example, or adjunction/inversion of adjunction, variants of this hold without the smooth hypothesis). This is very very useful for induction on dimension. LC-centers are a way to generalize this to higher codimensional subvarieties.
By definition, LC-centers are exactly the images of divisors $D_i'$ where $D_i'$ is a component of $D'$ with coefficient $1$, and $(X', D')$ is a pair obtained by taking a log resolution $\pi$ of $(X, D)$ and setting $D' = \pi^*(K_X + D) - K_{X'}$.
In particular, if $W$ is an LC-center of a pair $(X, D)$, then one has $$(K_X + D)|_W = K_W + (\text{some correction terms}).$$ Again, many properties of the pair $(X, D)$ can be translated to properties of $(W, (\text{correction terms}) )$, this isn't trivial to show.
One should note that even if you have a 1-dimensional LC-center, then there can still be a correction factor (just not in the smooth example I described above).
answered Oct 13, 2010 at 18:15
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2$\begingroup$ Your "archetypical LC pair" is actually snc. Perhaps a better example is the pair of a quadric cone with two rays intersecting in the vertex. $\endgroup$ Oct 13, 2010 at 18:48
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$\begingroup$ Krampusz, from the point of view of this question, that's a very good point. The SNC pair (with coefficients $\leq 1$) is DLT and so doesn't distinguish the two notions at all. The pair you described does, as would $(\mathbb{A}^3, V(x^3+y^3+z^3))$. But I always thought SNC pairs (with coefficients $\leq 1$) were the typical LC pairs (at least the first example that one runs into) due to resolution of singularities. Of course, they are also DLT pairs. The quadric cone example turns into a SNC example (all divisors with coefficient 1) after blowing up the origin, right? $\endgroup$ Oct 14, 2010 at 1:53
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$\begingroup$ @Karl: yes, the cone example is resolved to an snc pair with one blow up. $\endgroup$ Oct 14, 2010 at 4:03
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$\begingroup$ When you speak about LC center in the general case and you say that the pair $(W,correction term)$ inherits some properties of $(X,D)$ are you thinking to subadjunction? For example to the very recent Fujino-Gongyo paper? Or do you have other issues in mind? $\endgroup$ Oct 14, 2010 at 9:06
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$\begingroup$ Yes, that's what I have in mind, but in the one dimensional case, this is just adjunction / inversion of adjunction. $\endgroup$ Oct 14, 2010 at 14:27
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I would add an important distinction of LC singularities to Karl's excellent answer above: DLT (and in particular KLT) singularities are rational, while LC are not. This already appears when the boundary is empty. LC singularities are not even necessarily Cohen-Macaulay, such as for example a cone over an abelian variety of dimension at least 2. I think that this is the main reason for LC pairs being so much harder. For LC centers, it's also a good idea to look at Ambro's work on quasi-log varieties. That seems to be a very effective way of handling LC centers. This is evidenced by this recent paper in JAMS and Kollár's local Kawamata-Viehweg vanishing result (on the arXiv).
Another comment on LC centers: this is probably obvious for most people, but the point of an LC-center is that that's why a pair is not KLT. They are also called "non-klt" centers. I believe Christopher Hacon prefers this terminology. As far as I understand, LC centers are similar in spirit to associated primes. The simplest thing about a module is its support and in particular its irreducible components. However, considering associated primes gives a better understanding as there are components that kind of should be considered part of the support, but they are embedded, sort of "shadowed" by other components. The union of LC centers is exactly the non-klt locus, but just like with associated primes, there may be some that are embedded, so knowing the LC centers gives more refined information about the failure of the pair to be klt.
answered Oct 13, 2010 at 19:17
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1$\begingroup$ I would also agree that LC-centers are probably the wrong choice of words (especially when the pair in question is not log canonical). My personal preference is that $W \subseteq X$ is called an LC-center if $(X, \Delta)$ is LC at the generic point of $W$ but not KLT at that same generic point of $\Delta$. $\endgroup$ Oct 14, 2010 at 1:32
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$\begingroup$ Karl, it seems that something is wrong with $W$ and $\Delta$ in your comment. $\endgroup$– SashaOct 14, 2010 at 10:14
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$\begingroup$ You are right, that last $\Delta$ should be a $W$. $\endgroup$ Oct 14, 2010 at 14:28
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$\begingroup$ I also should have been clear that $W$ is still the image of some $E$ (a divisor on some birational model) whose associated discrepancy is $-1$. $\endgroup$ Oct 17, 2010 at 21:04
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I know very little of this subject, but I can attest to the value of it, since many of us studied the singularities of theta divisors for years without noticing or even conjecturing some basic results that followed from Kolla'r's observation that (A,D) is a log canonical pair if D is a theta divisor on a principally polarized abelian variety A. As Kolla'r put it with extreme simplicity and clarity: "we know a highly singular divisor moves, but a theta divisor doesn't move, so it should not be very singular." He deduced that the points of D of multiplicity ≥ k have codimension ≥ k in A. I particular there are no points on D of multiplicity > dim(A). The generalization of this result by Ein and Lazarsfeld that the extreme case occurs only for a product p.p.a.v., established the normality of an irreducible theta divisor. To me, this is a beautiful example of the concept of l.c. pairs which allowed the characterization of indecomposable p.p.a.v's in terms of the geometry of their theta divisors. This is one of the first cases of a technique that allowed the analysis of theta divisors that could not be understood in terms of curves, e.g. as Jacobians or Pryms.
