 As the central fiber of a family, we put the C star action as in the definition of stability. And then we want to argue that the Fou-Tarke invariant for this limiting value of this thing is zero. I'm going to try to say something about these points. Before doing that, let's step back and say a few words about the kind of foundations of studying these metrics with cone singularities. Remember that these are things that are modeled transverse to a divisor on the basic cone dr squared plus theta squared. And in terms of a standard, if we take a standard complex coordinate, say z transverse to d and say w, a number of w's along d. So the divisor is locally defined by the equation z equals zero. Then in the basic model, we would take z, these complex coordinates and these cone coordinates will be related by a formula like this. So what I mean by the foundations is that essentially we want to set up an analog of the usual kind of linear elliptic theory. And in the usual way, just as when you do the usual analysis on manifolds, you start off with doing the flat case and then you treat the general case by perturbing about that. We can start off treating the sort of flat case where we take the standard cone times a Euclidean factor with w. So more precisely, what do we mean? We're not going to attempt to define elliptic theory for arbitrary operators. We want to consider really a very specific operator. Let's just say the crucial operator in sort of complex mongeampère theory, the crucial linear operator is the composite of d bar d with the inverse of the Laplacian. Why is this relevant? This is the thing that takes, supposing we want to have a given change in the volume form, which is what this complex mongeampère is all about, that's given by solving the Laplace equation. And then that gives the change in the Kepler potential. The change in the metric is given by taking d bar d. So this is the linearization of the operator. The change in the volume form goes to the change in the metric. And what we want is this to be a bounded operator with respect to essentially the same function space, roughly speaking. And the conventional thing you would use in the standard case would be, well, one conventional thing would be holder spaces, and that's what we want to do. It's all bounded. So it's not really quite clear what this means initially, because the output of this is going to be a one-one form. In this conical situation, what do we mean to compare the values of a one-one form at different places? It's not quite clear how to set it up. What we mean by that is that we'll express, we can write down a standard basis for the, which is an orthonormal basis for the one-one forms. If you write eta is dr plus i beta r d theta. Then we can write down a standard basis of the form of eta wedge eta bar, dw, schematically wedge eta bar, and dw, which is dw, and the schematic notation. So what we mean here is that we express the components of this with respect to this basis, and we want this to be holder in the ordinary sense. So for that, we want to understand the Green's function of this, Laplacian. So there are, I'm not an expert on this, there are at least two approaches to this. One is to say that this is essentially something which has been studied for hundreds of years, because in a slightly different context, it appears, I dare into it, if you're doing potential theory on a wedge in R3, it's essentially the same problem. So there are lots of fascinating papers from about 1910, from traditional mathematical physics papers, writing these things down with Bessel functions and all sorts of stuff. Alternatively, that after a modern kind of high technology theories, so one can alternatively try to read papers by Melrose and Matt Sayer and people like that. So they're both probably about equally difficult reading projects. Anyway, I'm following more than first, but there is a, all of this can probably be, but I think that is embedded in much bigger theories. This has all been developed in a more systematic and general way by the paper of Geoffrey's, Matt Sayer and Rubinstein. Anyway, all I want to say is that you have to be careful in doing this and I want to explain that things are not as completely straightforward as I might first think. The following reason, this is a function of two variables, but supposing we fix one variable to be some fixed point away from the divisor and we just think of g of p0, and we think it's a function of this one variable. Then this thing has an expansion about the divisor, which contains terms of the form r to the one over beta, say cos theta plus lots of stuff, but you get terms of this kind coming in. So what we have to do to understand this operator, if we put an arbitrary second-order operator in here, we'd have to differentiate this thing twice to see what we're going to get. But if we take two derivatives in the r direction, this will bring in an r to the one over beta minus 2 cos theta. So you see that if things get worse, this beta gets bigger. If beta is bigger than a half, this thing is not even bounded. You get a singularity. Is this a green function on the compact manifold or is this a big function on the component of the divisor with some boundary conditions? So I'm solving the model problem on... Let's bring in notation. Let's write c beta for the standard cone or the plane with the cross-body metric. So I'm taking the model problem on this flat space where one can write out the green's function explicitly in terms of... But the boundary conditions near the wedge or something like that? No, I forget the wedge. I was just saying that the same things do appear. But in my version, I'm just saying... I'm considering a little plus operator for this conical metric on this sort of flat model where one can write down essentially... That's what I'm doing. Then later on, of course, you paste that into a compact manifold to do things there. But the crucial thing is this is bad if beta is bigger than one half. So if you're too ambitious and you just put a general second-order operator here, you won't get a bound on holder spaces in general. But fortunately, we don't have a general operator here. We have this particular one. So essentially, this second derivative with respect to r only occurs in... So we can write delta as, say, delta r theta plus delta w, schematically, that is taking the plus in different direction. The only place this operator occurs is inside here. So we can express... If we take the derivative with respect to w, we have no problem. We can express this in terms of the Laplacian, which is what we already know. Beyond that, we only get terms in d by dr of g. We'll end up with d by dw. That doesn't happen. That will just involve things like 1 over r to the beta minus one, cos theta, and there we're just okay because beta is less than one, so this is slightly positive. So this will be in c alpha if alpha is less than 1 over beta minus one. So this is what I'm going to take it. But what you conclude, essentially, by... Once you've done this, you can follow the usual proof of the Schouder estimates with some modifications to obtain these estimates, but you're only going to get it in this range of holder spaces depending upon beta. And it's only worked because you weren't considering general derivatives, you're only considering these particular derivatives. So one application of having that theory is that once you have that linear theory in place, you can relatively easily solve the deformation problem to say if you have a solution for one angle, you can slightly increase the angle. In fact, let's say something about that. But for the deformation problem, let's say something a bit more precise than what I just said, if you just write it down, and you use the implicit function theorem to linearize the problem, you apparently might have a problem given by obstruction, given by holomorphic vector fields on x, which are tangent to d. If you have one of these, then there will be a co-colonel of your operator which you would potentially obstruct this deformation. But it's quite an easy fact. This is done by Jansson and Wang, but there are no such things. This space is zero in general. So although you apparently have this obstruction, actually when you look more carefully, it's always vanishing. So you can always do this deformation. Chlorianthicanon. This is assuming V is chlorianthicanon. In other situations, as I said, d is in some fixed multiple... So before diving into the technical things, let's give an example where maybe what I'm going to say is not absolute, perhaps one doesn't actually know precise proofs of every point, but it's only an example where one really understands how things should work if you try this. So let's try to do this program in a case where we know this is not a solution. Let's take Cp2 and blow up at two points. There's no Kepler-Unstein metric on this thing. So let's see what happens. We'll do this. So let's take homogeneous coordinates x, y, z and blow up two points on the line at infinity. This is also a line at infinity. These are our two points. Let's just take a divisor in minus k. That's just to give them a cubic curve passing through these two points. So... I can't quite draw it. So a cubic through the two points. So the generic case will be then when this cubic meets the line at infinity and one further distinct point. So first we'll consider that. You can essentially see what happens is that supposing our cubic is given by the equation p of x, y, z equals zero, then as you approach the limiting value where you're going to stop, this cubic is going to... In this case, the ambient space will stay the same. It's going to be projective space. But what will happen is that the cubic will degenerate into a singular curve according to the fashion p of, let's see, x, y of silent z equals zero, silent z equals zero. So what will happen is that our curve will degenerate by applying the one parameter subframe that squashes everything towards x equals y equals zero. In the limit, we'll get three lines. And that's where we'll stop because when we said this was a smooth D there were no singularities. Now when we've got this singular situation we do have a construction, and so we do... That sort of fits in that we do meet this obstruction. That's why we stop. Plus, of course, the fact that we've got some... We'll have some nasty singularity in our metric here as well. Moreover, by calculations of Gavel, Zecalidi and Chilly, one can work out exactly where we stop in the situation. And it's at... It's written down somewhere. So beta max is equal to 21 over 25. Yep. We know exactly that. We'll go that far and no further. And then we'll get this picture here. On the other hand, according to calculations of Zecalidi, if we took a special case where one of these was a double point so we had a double intersection point, then we would know not quite so far. Seven over nine, which is slightly smaller in the case of the special case. And then... Actually, I don't quite... I'm less certain about what the picture would be if it degenerates, but presumably one could figure out how it should be going on. So are we seeing the destabilizing test configuration? Are we seeing the destabilizing test configuration? That's right. This thing will be the... The way you work this out, if you work out when this futaki invariant is zero for this thing. But then that's... For this one parameter, so brute. The kind of the special case is another one parameter subgroup that stops you a bit before that, because that's why you stop at a slightly smaller angle. Yeah. So this has nothing to do with the proofs in general, but it's comforting to see a case where one really, at least feels but understands exactly what's happening. So let's now turn to the... Not... For me it's the central problem, which is to extend what we did in the first lecture and a half to the case when we have these divisors. So remember that was all based upon... We had the drum-off-house-storff limit, and then we went to relate that to algebra. So one other kind of foundational thing, which I will say almost nothing about, is to show that, in fact, we can approximate these metrics with cone singularities by metrics of positive Ricci curvature. In fact, we can keep the... We can keep the... Arbitrary close to being a Kaler-Einstein metric away from the divisor and slightly smooth out this delta function of curvature to a big lump of something smooth with a very large positive Ricci curvature. But just a... All that is not too surprising, but that requires substantial proof. But one point to make is that when we were... In our previous discussion of these L2 techniques and so on, we were looking at formulae for involving the Ricci curvature. There we said we had a hypothesis that the Ricci was bounded, and that was all we needed to think around it. In this case, it's not going to be... In this family, it's not going to be bounded, but the crucial thing is it always appears with the right sign. So for those purposes, you don't care about very big... Positive Ricci curvature only helps you with those kind of... Vights and Bot formulae arguments. So this approximation, in fact, means that we can fit into, without any more ado, into the general kind of Chica-Colding theory that our limit, our Gromov-Hausdorff limit, will be a limit of Romanian manifolds of positive Ricci curvature. Then we have a GH limit. So, not sure what notation we use for this, but let me say something more about the singular set. S is the singular set. About the general Chica-Colding theory, or sigma we call it. The one divides this up, and it quantifies this to subsets, let's say, sigma i. These are the points where... So no tangent cone splits off a c n minus i plus 1 factor. If i be equal to 1. So in fact, what this means, the whole singular set is equal to sigma 1. All that's saying is that if we have a tangent cone, if i is 1, this is cn, if we have a tangent cone which is cn, in fact, we're a regular point. We're not a singular point. That's not a basic fact. But then there are subsets, sigma 2 and so on. So the essential difference in what we're doing compared with your ordinary case, the first most obvious part of the difference, in the standard case, we in fact have sigma is equal to sigma 2. That's to say there are no points here, which aren't in here. There are no points which split off a cn minus 1 in that tangent cone. But now clearly there will be there will be points where the tangent cone... So in our case, we will expect to find tangent cones which are just the model we wrote down, cn minus 1 times c beta. Various angles beta. So the fact that these don't occur as limits in the ordinary situation is... In our case, under the hypothesis we're working with is a relatively deep theorem of Chiga. But here we do. We're going to encounter these. So the other thing to say about these things is that the Hausdorff co-dimension of the sigma i is at least 2i, the real co-dimension. So that's why before we had things of co-dimension 4 at least, as we only hit this sigma 2. But now we're going to get some co-dimension 2, singularities obviously, with co-dimension... The whole story is about real co-dimension 2 singularities. So let's suppose we're at a point in our limit, let's call it z denote our Gromov-Hausdorff limit, with such a tangent cone. We'd like to do just what we did before around such a point. We wanted to construct these localised holomorphic sections really on the smooth, before we take the limit. We want to have the same story. So what we need is a good cut-off function. This is our singular set in the tangent cone. We're going to work on a big ball. What we want to do is we need a cut-off function which vanishes... Which way you're going? You want a function which is equal to 1 on the singular set, sported in a small neighbourhood, and with the integral of the small as we like. That's what we need to do. So, at first sight, this is a difficulty if you're meeting such a thing for the first time. Because if you do the obvious thing as in our discussion before, you would find that if you take the obvious thing where you just take a standard function and scale it, this thing is scaled in variance. So you won't scale this down. But a second sight, or because you've seen something like it before, you can do a slightly more complicated thing to write down a suitable function based upon... So let's say r to be the... Essentially it's a two-variable problem because nothing's really happening so if we define... If we define in terms of a pair of parameters q and delta, if I define g of r is 0 for r... I'm taking it... This is going to be 1 minus the function I talked about before. r less than delta is equal to 1 over log q times log r over delta if delta is less than or equal to r is less than q delta. And then when r is q delta, this is 1. Now if we want to work out the l2 norm of the derivative, that's going to be the integral from delta to the q delta. 1 over log q squared times what are we going to get? What are we going to get? We're going to get 1 over r squared r dr. Essentially we're going to get... That's 1 over r delta to the q that gives log q over log q squared is 1 over log q. So we can make q as large as we like to make this as small as we like and then we make delta sufficiently small the function is still so it's still equal to 1 on an arbitrarily small neighbourhood. So with a bit more thought we can construct these good cut-off functions in the basic co-dimension 2 situation. And this is essentially the same as the fact in dimension 2 you don't have a sobblef embedding in C0. If there were no such function then you would have so this is maybe kind of the first step towards extending what we did to this case when we have these two-dimensional singularities but there's a lot more to it than that but I'm not I'm not going to try to go into very much of that at all beyond to say a few more words we've got to. So more systematically what we want to say is that we're going to consider tangent cones of our limit, the cone on y and they're going to have a singular set sigma and we'll say that we have a good tangent cone if there is a good cut-off function so let's not write it a function equal to 1 on sigma so that way around this is 1 minus the g we had before sported in an arbitrarily small neighborhood with the integral of mod grad g squared as small as we like a family of functions, however small we want to make this we can make it small so clearly if this was a a co-dimension two sub-manifold for example, clearly we'd be alright by the same argument but as we said potentially these singular sets are completely bizarre things at the outset we know very little about these sets so that's why there is work to do but the basic of technical fact technical result is that all tangent cones are good so once you know that everything we did goes through without shame all we used with regard to the tangent cone was that you had this cut-off function everything goes through but this takes a lot of work let's just mention a relevant notion which I call it Minkowski measure but I'm not sure I call it this because looking on the internet there seem to be vaguely related notions that might be exactly the right language but this is a useful notion for what we want to do supposing we have a set some general context we have a notion of house-storff measure which is given by covering by arbitrary collections of balls with different radii but that doesn't work very well for these purposes what works better is to consider balls of a fixed radius so we'll define this by saying M of A is less than equal to some number M if and only if there's a way of saying it there'll be more if and only if for all epsilon, small numbers there exists a cover of A by fewer than M 2N minus 2 is what we're doing epsilon balls so this is if A is in some two-end dimensional ambient space this is the number you'd this is the number you'd expect for a kind of a real real co-dimension two Minkowski measure co-dimension this implies that the volume of epsilon neighborhood in a context where the volume of balls is essentially like the Euclidean volume up to bounded factors the volume of an epsilon neighborhood is less than some universal constant times N times epsilon squared if you just think about because you cover an epsilon neighborhood taking these balls but twice the size they've got volume so it should be 2 minus 2N the volume of each ball is roughly epsilon for 2N this is the number of balls so you get epsilon squared so it's a kind of a good notion for a co-dimension two set it gives you control of the volume of the neighborhoods and if you have this condition then you can write down essentially this function this is the distance function to the set using the this bound on the volume of the neighborhoods it's very easy to prove that you have these good good functions so the arguments are actually long and complicated but roughly we write the singular set as something of Hasdorf co-dimension bigger than 2 something of bounded Minkowski measure roughly it is not exactly roughly speaking that's the kind of the idea the arguments revolve around controlling this quantity for the singular set ok so that's all I'm proposing to say about this part I will say something not quite actually the cases when what we're indicating in the picture is the case when the limiting angle is strictly less than 1 or strictly less than 2 pi is another case when the limiting angle is 2 pi and those require different proofs the second one though in a sense you think it should be better because you sort of got that in fact the proofs are in some ways harder the the problem is when you look at a small ball these these divisors have a fixed volume fixed homology class we may have a fixed volume whatever else we're doing