 Well, it's really a great honor to be giving this first talk at the meeting between Calabi's 90th birthday. So thank you for inviting me. What are we talking about is a topic which is, to my extent, been created by Calabi. The, briefly, you could say, the search for an optimal calimetric on a compact complex manifold. And there's a renowned work of Calabi going back about 60 years to the 1950s, particularly in Caler-Einstein metrics, which we'll discuss. And then more recent work, say, only about 30 years ago on what we call extremal metrics, which also we'll talk about. So I'll be trying to cover rather a lot of ground. And this is aimed to be a colloquium-style talk. So I obviously won't be able to miss out a lot. And I don't really want to say anything in proper detail. But the main point I want to convey is that these deep and seminal ideas of Calabi have given rise to lots of things that have been done by many people over the last 60 years or so, but also lead to lots of unsolved problems, which will likely give work for mathematicians for the next 60 years to come. So this is the overall point I'm trying to make. So let me try to make a start on this. Ambitious. Well, and just recall that in differential geometry, we have various notions of curvature and various notions of the sine of the curvature, depending upon the exact context. And let's just begin with the classical situation the case of two real dimensions, sine, where we have the Gauss curvature. And this is linked to topology by the Gauss-Ponnet formula. The integral of the Gauss curvature over a closed surface is given by a multiple-veiler characteristic. So we have the restriction of the oriented case, of simplicity. We have the positive situation with the two-sphere, positive eilocortristic, the torus, the zero eilocortristic, and surfaces of higher genus of negative eilocortristic. So this says that if we have a surface, depending on which type it's in, the average value of the curvature has got the appropriate sine. A bit of old theorem, it's actually a uniformization theorem, says that, in fact, if we have any Romanian metric on the surface, then we can change it conformally so that after that change, the metric has constant Gauss curvature with the constant determined by the sine given by this topological requirement. So moving on from this sort of very old paradigm, we come to discuss the case for higher dimensional x, compact complex manifolds, complex dimension n, compact. And we want to consider a k-limetric on x. Now, the various ways of defining this, one simple way is to say that, in Romanian geometry, we have a general Romanian metric. We have a notion of duty-sick coordinates at a given point. If we take a given point, then we can choose coordinates such the metric agrees with the Athenian metric up to first order. So it's plus order x squared, schematically. A way of defining a k-limetric is to say that we can do this, but we can also make it compatible with the complex structure so we can take our coordinates to be the real and imaginary parts of holomorphic coordinates from the point of view of the complex structure. So if we have a complex k-lim manifold, we have two basic topological things that we can consider. One is the first churn class of the manifold. And the other is the cohomology class of the k-limetric. This condition implies that the metric defines a closed two-fold. So we get two different classes in the real cohomology of x. So I realize this is a bit more technical for people who are not of geometry and topology specialists. But actually, the precise definition of these will not really enter too much. If you're not familiar with these, don't really worry because just assume that there are certain symbols that we can write down. In any case, we can consider among all these compact k-lim manifolds, we can consider ones where we have the property that these classes are proportional. So we consider, suppose, c1 of x is multiple of omega. So this may not be true, but for a general compact complex manifold, let's consider the cases where it is. So we then have then three cases where lambda is positive, 0, or negative. So what Colabi realized and pointed out back in the early 1950s was that in this situation, we have a good geometric problem for generalizing this classical situation derived from the fact that the Ricci curvature of a complex scalar manifold is a representative of this first-journ class. So c1 of x is represented by the Ricci form. So if we're in this situation, we have the necessary background to ask the question of whether we can find a scalar metric such that, in fact, the Ricci curvature is a multiple of the metric, as this is just the Einstein equation in the scalar situation. So can we find a metric? Can we find, can we solve, let's say, the Kepler-Einstein equation in this situation, such that, actually, this Ricci is equal to lambda of omega. So this precisely would generalize, in the case of complex dimension one, this would just, the Ricci tensor would boil down such that this Gauss curvature, and this would just be the generalization of the constant Gauss curvature condition that we mentioned before. Of course, while it's solving, as well as asking the question, Calabi also proved many basic results in this direction towards answering the question. But let me just recall a bit more of how this, the setup of this goes. A wonderful thing about scalar metrics is that they can locally be determined by a single function. So locally, the Kepler potential phi, so our metric can be expressed as the, in terms of the second derivatives of this function as follows. So if we work out the second derivative, we get at each point a Hermitian metric, a Hermitian matrix, and we're saying that in these complex coordinates, that is the metric tensor, the obvious adjustment between real and complex notation. And in fact, this Kepler-Einstein equation, although it apparently depends, because apparently the Ricci curvature depends upon two derivatives of the metric. So you expect that to depend upon four derivatives of this Kepler potential. In fact, it can be, we can integrate out essentially two derivatives and write it in a very comparatively explicit form, in this local version of the velocity. We take this, we take this Kepler potential phi, we take this matrix of second derivatives, and we take its determinant. And then this should be equal to e to the minus lambda times phi. We've got sine right ahead. So I'm not going to explain why I haven't explained what the Ricci curvature is. And I'm not going to explain how this equation leads to this more explicit equation, except to say that what we've got here is essentially the volume form of the metric. This is the metric tensor taking its determinant, gives the volume form. And the Ricci curvature is all to do with volume forms. So at one point of view, it gives the work in this sort of GDC coordinates at the point. And we ask how the volume form of the metric looks when we don't make this expansion. Then the second order term is given by the Ricci curvature. Or another way of saying it is that this c1 of x, this is essentially, obviously, it's the first-gen class of the canonical bundle of x. The algebraic geometry, I would say. And we can think of the volume, the Ricci curvature is the curvature of the canonical bundle minus the curvature. And the volume form, this is giving a volume form, is saying it's giving a metric on this canonical one. So apart from muttering, it's all to do with volumes. I'm not going to say anything more about this equation. But at least you can see it's an explicit kind of equation you can write down. It is a version of a complex Monge-Anpère equation because it involves this determinant of this matrix of second derivative and is much related to the real Monge-Anpère equation where you would take some, let's just consider, say, a convex function f. You take the Hessian matrix for second derivatives and that this should be some function of f in its first derivatives or something depending upon the context. So these complex Monge-Anpère equations are bound up with the study of the Kepler-Einstein condition. There's also a real analog to which also Calabi made many important contributions. Of course, these are highly nonlinear equations because the determinant is a very nonlinear function. And these things are important, particularly for example, in terms of affine differential geometry. To say, in classical differential geometry, say, of surfaces in R3, we consider properties that are invariant under Euclidean group of translations and rotations of R3 in affine differential geometry. We consider properties of surfaces invariant under all affine transformations of R3. We don't fix a metric on our three-dimensional space. OK, this was the area, research area, which Calabi's work this time initiated. And a huge amount of work has been done on it. So it turns out that the sign of lambda is very crucial in the study of this equation. The conventional approach, which was received for 20 years or so, was to study this equation by a prior estimates, or perhaps embedding this in a larger family of equations, and deriving a prior estimates for various quantities involving this Kepler potential by subtle and ingenious arguments. And then if lambda is negative, then all the signs work out in a good way when you want to do these estimates. If lambda is positive, they don't. So the case when lambda was solved by Oba and Yao, where lambda is equal to 0, was solved by Yao. It's both in the 1970s. And the case when lambda is positive is the hard case. Now, this would correspond to, in algebraic geometric terms, this corresponds to cases where the first-journ class is a positive class. It can be written as the class of a calometric. And then the corresponding manifolds in algebraic geometric terms would be called pharno-manifolds. So in these two, the negative and positive and zero case, there's a straightforward existence to it, actually. But in this pharno situation, it was realized for many years that that can't be the case. There's an old term, a Matsushima, who says that if we have a Kepler-Einstein metric and the automorphism group of x, the holomorphic automorphism group is reductively group. That's to say, this is a complexity group, but it must be the complexification of a compact group. Some groups, some of the groups are, and some aren't. So in particular, if you take, say, the projective plane and you blow up a point, then its automorphism group is not reductive. So this has got no Kepler-Einstein metric. In fact, you can see in a more basic way, you can see why having automorphisms can show that there's no possibility of just a naive way deriving a-power estimates. Because if you have one of these holomorphic automorphisms, f from x to x, then if you have one solution omega, you have another solution by pulling back by f. If this is a non-compact group, then you can take a sequence of f's to diverging. And so this sequence of pullbacks will diverge in a reasonable sense. They can't satisfy any fixed uniform estimate. So the simplest case, for example, of the two-sphere, we can write down a round metric on the two-sphere, but we can apply conformal transformations of the two-sphere to get many more, in which we squash all the volume all the air up into an arbitrary small neighborhood of a point or something. In a sense, it's much from kind of a PDE point of view, in a sense, it's much easier to prove the existence of a Kepler-Einstein metric on some complicated high-dimensional variety of negative curvature than to prove the existence of the round metric on the two-sphere if you didn't happen to know that one existed. Of course, you do. So following this renowned work of Yao and Obe in the 1960s, 1970s, this was a big question in the field of just when should the DeFano manifolds have Kepler-Einstein metrics. But as we said, going beyond that, in an important paper in 1983, Calabi introduced a more general question of first defining and asking about the existence of what's called extremal metrics. So for this, what we can do is we can take, if we have any Kepler-Einstein manifold, we can take the L2 norm of the curvature tensor and think of this as a functional on the space of Kepler-Einstein metrics, on the space of Kepler metrics. Or more precisely, I mean, the space of Kepler metrics in a fixed-comology class, the class of omega fixed. So I believe this idea actually entered also into his earlier work as well. Of course, around 1980, it was particularly, there were clear analogies then with the study of the Yang-Mills functional, which is study the L2 norm of the curvature of a more general connection, a different setting. In any case, we can ask, what are the Euler-Lagrange equations of this functional, if we consider it on the space of metrics in a fixed class? The Euler-Lagrange equations. These can be expressed by saying that if we take the gradient of the scalar curvature, so it's right in the s, so this is the trace of the Ricci curvature, just as a function, we take the gradient in the usual Romanian geometry sense. This is a vector field. If we multiply that by i using the complex structure, then this should be a whole, this is a holomorphic killing field. That's to say, both the one parameter group generated by this vector field is both a complex automorphisms that Kami were considering here, but also Romanian automorphisms, isometries of the Roman model. And a particularly important case here is when the scalar curvature is constant. So of course, this vector field is 0 in particular. And so these things, this is the definition of an extremal metric, but a particularly important class is the case of constant scalar curvature. For example, you might be studying a complex manifold, which has got known on trivial holomorphic vector fields. And so any extremal metric is forced to be constant scalar curvature. So again, it's plausible that such things should exist, because we might feel that if we try to minimize this norm of the Romanian curvature, we might feel there ought to be a subminimum, optimal metric, which will give us this extremal metric. But again, there are examples. Going back to Levine originally, exploiting this automorphism group to show that, again, there's no simple existence theorem. So we can find a manifold x, but the holomorphic automorphism group of x is equal to c. Therefore, there's no extremal metric. Why is that? Because this vector field, if we had one, is supposed to be a killing field, that would mean that we have to have some, this thing would have to generate a compact subgroup with the automorphism group. But c has got no non-trivial compact subgroup. So that means that if it had an extremal metric, the scalar curvature would have to be constant. But in fact, this Matsushima theorem that we stated here in the Keiler-Einstein case applies equally well to constant scalar curvature. So the automorphism group would have to be reductive, but c is not reductive. So by such arguments, you see that this manifold, which you can write down explicitly, has no extremal metric. So what we find then is that when we ask these natural questions about how to find an optimal metric on a Keiler manifold in somewhat different contexts, then the answers have to be quite difficult. It's hard to even know what is true, let alone to prove anything. So the general idea of what's believed, in some cases proved to be true, is that the criterion for existence in these kind of problems should be related to stability. Conjecture? Is that the existence related to stability? So what we want is a criteria. We have this complex manifold, a complex algebraic manifold. We want some algebra geometric criteria, which we can state, but under which we can solve this differential geometric problem. And this is what one expects to be true. So this idea was wrote initially in the Keiler-Einstein case by Yao many years ago, and has been refined by various people over the years in different directions. But let me just, and I'm not going to try to define this precisely, but just to give a general idea, is that stability has to do with studying degenerations of our manifold x. So let's just say everything is complex here, or we've now due break. We consider a family, an n plus one dimensional family with a map, we call it a flat family, we should say technically, such that the fibers for non-zero t are isomorphic to the manifold x we started with, whereas the fiber over zero is some other object, not necessarily, in the case of interest not, isomorphic to x. And in fact, this thing could actually be, this could be some kind of singular variety, or even, in principle, a scheme or something like that. So we should study degenerations of this kind. And the notion of stability is saying that x is stable if, for all such degenerations, is a certain numerical invariant, should have the right signs. So I'm not going to have time to explain. In an example, hopefully, we'll see what this is. But I'm not going to try to explain it in detail. But if we have such a generation, then we can assign a numerical invariant to it. The notion of stability is saying that x is stable if any time you degenerate it, this futarci invariant is positive. So, and so there are versions of such conjectures for extremal metrics, constant scalar curvature metrics, and these Kaler-Einstein metrics are far no manifolds. And in fact, in the last case, this is actually a, no longer conjecture, but a theorem. And if luck, I shall have time at the end of the talk to say something about that. This is, say, with advance, say this last part will be some joint work with Shen and Sun. But in the more detailed talks which Scott mentioned on Monday, I should be concentrating on that. So if I don't get to say really anything sensible about it today, I will try to make up for that on Monday. What I want to try to do for the next section of the talk, that's caught for an hour or so, is to illustrate how this works in the case of toric manifolds. Because I think that's a good way of getting a point one can make the pots going on fairly accessible. So, in the case of the toric manifolds, so let's say I want to consider a manifold X of complex dimension N with an action of the n-dimensional torus acts. So, we want to consider this to act holomorphically, we want to consider metrics that are invariant under this action. Then there's a big theory about these things which says that they're essentially determined via a polytope P in Rn, a convex polytope in Rn, which you can think of as being obtained by just taking, oh my god, you just take the quotient of X over T. So, that's the basic case, say, is to say X is S2. So, N is one, the one-dimensional torus is the circle. We're taking that as acting on the sphere by rotating about an axis. If we take the quotient space, we get an interval. Let's say we take any point in the interior of the interval, the fiber, close by the orbit, it's a circle, but when we go to the boundary, these circles collapse down to points. And the general case is similar. If we take, say, some polygon in R2, then we should think of a two-complex dimensional manifold of a four-real dimensional manifold which is built up by taking a fringe point in the interior of the polygon, two-dimensional torus. We go to the boundary, this part of the boundary, it collapses to one-dimensional torus, with the vertices, it collapses to points. In any case, what one then gets, if we go to a differential geometry on these toric manifolds, we can, if we like, express everything in more elementary terms as working on a polygon in Rn. And N is leopard again to become equal to two. So, it turns out that we can express a scalar metric, can be expressed in terms of a certain convex function on P, which is, in fact, it's called the Legendre transform of the scalar potential we were considering before, which satisfies such suitable boundary conditions on the boundary. So, what is the extremal, we just asked for extremal metrics in this toric situation. They can be described in the following way. Consider a functional, f of u, which has got a non-linear term. So, we take this determinant of the Hessian, we wrote this, it's right there, that is Uij, say this matrix of second derivatives. We take its logarithm, we integrate that over P, and then we add on a, apparently innocuous linear term, so that's why it's La of U, La of U is given by the integral of U over the boundary of the polygon with respect to a certain measure that comes with these, comes into these boundaries. So, there's a certain measure on the boundary of just a multiple of the big measure on each face, which is given by the geometry, and we take that integral of U with respect to that measure, and then we subtract off the integral of a times U over P, where a is a certain affine linear function, fixed affine linear function. In the case of constant scalar curvature, if that a would be a constant, a general extremometric, it's not quite constant, but it only varies linearly over the polygon. So, an extremometric is given by finding a critical point of this function, or in fact a minimum of a function, among all convex functions on the polygon P. So, you get a sort of an idea about, without trying to prove anything, you get a kind of physical intuition about what this is saying, but we're trying to minimize F, so we're trying to make the determinant, we'd like to make this determinant of the Hessian large. We'd like, for that point of view, we'd like to make U grow rapidly as we approach the boundary. On the hand, we all have this linear function L, which involves the integral of U over the boundary. So, if we make our function grow too rapidly at the boundary, this term will become bad, so that will stop F becoming large. So, minimizing the functional has to do with balancing this term against this term, roughly speaking. And, in fact, when N is two, it's more or less a complete theory of this problem. First, in the case when A is a constant, this was done by me a few years ago, and it extended to almost the general case by three authors, B, Chen, Li, and Shen. The general case. And I can tell you what the answer is, and say, you should have some intuition as to why it should be correct. So, what it says is there's a solution, exists, a solution of this minimizing problem, if and only if this linear function has the following property, that LA of F is weakly positive for all convex F on B, and strictly positive unless F is affine linear. But this, I didn't, I'm explaining a bit, this affine linear function, A, is determined precisely by the condition that this functional should vanish on the affine linear functions, not the necessary condition. So, this is the, sounds like to be the criteria for this solving of this nonlinear equation. Minimizing this problem. Actually, I forgot to put in, little point, which I mentioned in the abstract, so let me put it in now. This is, we could consider functionals more generally, we could consider functionals which are given by the integrating some function of the determinant of the Hessian. And we get a family of equations of, more generally, let's see, in particular, if we take the function to be a power, if you take the interval of debt, UIJ to the one over N plus two, then we get an interesting equation that the Euler-Rodrange equation is functional. It's an interesting equation, which is studied in a broad way by Calabi, called the affine maximal equation. As to say, the Euler-Rodrange equations, if we think of the function U as the graph of U as a surface in our three, then the Euler-Rodrange equations are actually affine invariant. This is actually an affine invariant functional. And so this is a natural topic in affine differential geometry. So, which is studied by Calabi. And in fact, the kind of PDE techniques, which are developed by Trudering and Wang for studying this problem are also very useful for studying this extremal problem. What we get is a complicated family of fourth order, not only a fourth order, E to E, arising from these kind of equations, Euler-Rodrange equations. So I should have said that a bit earlier in the discussion. Any case, what we get is in the two-dimensional case, we do, in the end, get a definite criteria for solving this problem. Roughly speaking, supposing we had, say, LA of F is, say, positive, that if we take some U naught and we take some large muzzle of F, so T, big and small, then we can make this functional as small as we like because we get a little bit, sorry, this is less than zero, I would say, because this thing, we can make, this is linear, so this is a model of T, a negative model of T, and this has got a log in, so it'll only grow slowly with respect to T, so it'll grow a log of T, it'll grow slowly. So if you have such a function in which this linear LA of F is negative, then you can write down, you can see that this functional can't be that bounded below, and what the upshot of the theory is that the converse is also true. So this maybe doesn't look very explicit because we have to study all possible convex functions F on P, but in fact you can, so maybe strictly for simplicity, I should talk about, for exactly, precisely true, I should talk about the constant state of curvature case. In fact, it suffices to consider F of the following form, if alpha is an affine linear function, let's try and say F alpha is the maximum of alpha on zero. So in the one-dimensional case, here's the graph of alpha, we had a function with a single discontinuity, it's derivative, so it's the fact it suffices not to look at all the general convex functions, but these special small family, you can skip them by an affine linear. So what does all this have to do with stability? In the form that I outlined, it's basically we have, it's basically in the bad case that we don't have a solution. So our picture of our polygon, so remember this is always just known for the two-dimensional situation. Toric manifolds of complex dimension two. So we have such a violating F alpha, so we can think of that as, if we look at the discontinuity in F alpha, it will be some line dividing our polygon into two pieces. So it turns out that these two pieces also satisfy the condition that they also define Toric manifolds. So this picture in Rn corresponds in complex geometry to degenerating complex manifold into the union of two pieces. The simplest case will be to say, so for example, if we take n equals one, then we can take a iconic, say x, y equals epsilon. So if you think of taking the corresponding projective curve, that's a copy of the two-sphere, but if epsilon is not zero, then we can degenerate that. So let's say one, so this is a conic, this is a non-singular conic, isomorphic to S2, which we take the corresponding, we add points to infinity. If we take x, y equals zero, then we get a union of two pieces. So this is the simplest kind of way in algebraic geometry in which you can degenerate one thing to another one just by in the limit getting something that breaks up into two pieces. So in other words, this picture, what the conclusion is, but if there's no solution, there's no solution, and then we can construct an x, an x naught, which is reducible, I give them by a union of two pieces as in our discussion. And this, the sign of this, this LA of u, this is a change that put at the futakian variant we talked about. So the sign of this thing, because the crucial thing that we saw in the existence problem, this is essentially corresponds to the futakian variant. Most of what I've said has analogs in the general story. For example, this functional here, there's what's called, in the general situation, you have what's called the Mabuchi functional, this is the sort of Mabuchi functional written down in a special case. But rather than trying to explain that, I've taken the point if you're trying to write down this situation where we've written down a rather elementary form and stated explicitly. What I should emphasize, although in the end, the statements can be made quite simple and easy to grasp, the proofs are all very long and difficult because they involve grappling with this complicated fourth order PDE which arises from this function. So now I said, as promised, go on to the last part of the talk, which I'll talk about this, it's another older problem. But before that, I should say is that beyond this, if we ask about the general question about these existence of these extremal metrics or constant-scaled curvature metrics, no one really has very much idea at all. The great effort we can solve was the simplest case, where we both imposed this large symmetry, this Toric symmetry, which makes things much easier. And also we work in the smallest interesting dimension, A equals two. So kind of the simplest possible case we more or less have a good understanding of. But beyond that, no one really has any idea very much at all. So this is what I mean when I say that this work for mathematicians for many years to come in trying to develop these ideas introduced by Calarbi. So the last few minutes, you maybe have about 10 minutes or so. I should talk about, go back to the case of Fano Manifolds. So I say this is joint work with Sushong Chen, one of Calarbi's former students and Song Sun, who I suppose is a Calarbi grand student. Song is a student of Chen. So we haven't really, I didn't really properly define this stability. So that's, I'm not gonna start trying to do that now. But the theorem is, the statement is that we have a Calarbi's Dimetric. So a Calarbi's Dimetric exists. This is as conjectured by Yao many, many years ago to some form if and only if X is stable. Given that I haven't properly precisely defined this notion. So let me just say something about this. The approach we use is to use to embed the problem in a family that's somewhat different to the families that have been considered before. If we take a sort of divisor in X, a complex co-dimension one submanifold of simple kind, then we consider metrics with cone singularities transverse to D. So in a complex co-dimension one, real dimension two, in the transverse direction, we see a real two-dimensional picture. What we're saying is the metric is modeled on a cone of the kind we're familiar with. So the length of this thing will not be two pi, it will not be two pi beta, where naught, there's some beta, one to one. And then the idea is to start in a regime for a small beta, where you can easily prove existence of a Calarbi's Dimetric with this cone singularity, then see that we can deform beta. If we have a solution for one cone angle, we can slightly increase the cone angle into that solution. And then say either as we keep trying to deform and deform and deform, either we can go all the way up to cone angle two pi, which is to say we found the smooth Calarbi's Dimetric that we wanted, or we must get stuck at some point. And the way we get stuck precisely allows us to construct this degeneration, which is going to contradict stability. This is the strategy. Either in this case, i.e. smooth Calarbi's Dimetric, or construct degeneration X as we discussed in outline before. So there's a lot of... So in my lectures on Monday, for those who are not everyone will even be here on Monday, but those who are in my lectures on Monday, and possibly even later in the week, I will be talking more technically about all these things. But the crucial thing we need to do is supposing we have some sequence of angles, beta i, say increasing to some limit, beta infinity. And so supposing we have our solutions, omega i for these angles, then what we want to do is to find a limit as an algebraic variety. That's to say we want this limit, this is going to be, after substantial more work, precisely the fibre over zero of our degeneration. But the crucial thing is can we take a sequence of these K-Rheinstein metrics, or with these singularities, but that's sort of technicality, and get a limit which again has got an actual algebraic structure. So there is a well-established theory of taking limits of Romanian manifolds in some sense, in some simple weak sense. That's to say we can take Gromov-Hausdorff limits. So this is really a notion from metric space geometry. If we have A and B metric spaces, then we define the Gromov-Hausdorff distance, we call the definition. This is the infimum of the numbers epsilon, so it exists a metric on the disjoint union, which, as I said in words, agrees with the given metrics on the two factors, and such that A and B are both epsilon-dense. So we have these two things. We want to put a metric on the disjoint union agreeing with the given metric on the two pieces such that for each point of A, there's a point of B within a distance, epsilon of it, and similarly for each point of B, there's a point of A within epsilon. So if you can think of this as a kind of an approximate map, if you think of we don't have an exact map from A to B, but to each point of A, we can associate a set in B of a small diameter which will approximately correspond to A and vice versa. So it's kind of a smudged map between A and B. In any case, this is the famous definition introduced by Gromov, and we can apply it to Riemannian manifolds, the metrics defined in the usual way by Riemannian manifolds. And under very general conditions, if we have a sequence of Riemannian manifolds, we can take a sub-sequence, we've got a Gromov-Hausdorff limit. We can get some, initially, a metric space to which our metrics converge in this sense. So there's no real problem in getting a limit in some sense in this Gromov-Hausdorff sense as a metric space. The essential point, which I think is no value of my trying to say anything more about it now, is to say we want this Gromov-Hausdorff limit, which starts life as a metric space, actually to be an algebraic variety, and then they say with more work, we fit it in one of these degenerations, of course the sign of the futark invariant that's got to come into the picture. But that's in the end how this strategy for proving this theorem goes. Yeah, D, D. Right, that's a good question. So this could be, we can take, we can take, essentially, any multiple of the anti-canonical system. So this is a smooth beat. So D is going to be in some positive lander. A smooth divide. So by general, if you have a far no manifold, we can choose a multiple such that there are smooth divisors in this class. Take any one. Yeah, well, there was a natural notion of superiority effect there, essentially what we do is make it an ocean of, that's right, it's very much modeled on the corresponding things for parabolic bundles and things of that kind where you, again, Is there a reason that had a time to expect that you can go from what's really a real family because you have the beta parameter, which is real complex. There's a lot of work involved in that. Yeah, and in fact, there's something I just started saying, which is that, more technically, you do this, you consider things with C star actions. So there's a lot of, precisely, a lot of the technical work goes into that issue. They involve, it involves using a famous Hilbert Mumford. One needs to show the automorphism group of this limit is reductive. Then you can apply a famous Hilbert Mumford to get the one parameter subgroup, which generates this perfect, but it's an important point. Thank the speaker again.