 Scott, and so our second speaker this morning is Hans-Jo Kempain from Imperial, and he's going to be talking about singularities of Taylor-Einstein metrics and Kalabi algorithms. Thanks, so it's a great honor for me to speak here. Like Simon said yesterday, Kalabi's ideas have given us work for the last 60 years, and it can be a little bit more specific. They've certainly paid my bills for the last six years, so I want to thank you for that. Okay, so I want to give some sort of overview lecture about singularities of Einstein metrics. I mentioned some old results, some new results, and some speculation. So looking at Einstein metrics, let's remind in metrics with constant Ricci curvature. So you're saying that Ricci curvature of G is equal to lambda times G for lambda as the Einstein constant, and that's going to be normalized to be either plus one, zero, the Ricci flat case, or minus one. And so more specifically, I'm interested in singularities of Kala Einstein manifolds, and so the Kala case has this beautiful feature that you can sort of split up the problem into a complex part and a metric part, or an algebraic part and a metric part. So you can always look, so forget about the metric data and look at the underlying complex manifold, hopefully. Algebraic manifold, possibly algebraic variety, and then the idea is if you understand the algebraic geometry well enough, singularities of algebraic varieties in some form, then hopefully that's going to tell you something about singularities of the associated Kala Einstein metric. So it's going to be about the correspondence between algebraic singularities and metric singularities. And the talk is going to have two parts, hopefully. So in the first part I want to, so notation, so X is always going to be a Kala manifold. So complex manifold in the first place, or even algebraic manifold, of complex dimension N. You have a complex structure, tensor J, Riemannian metric G, that's Kala, so that J is parallel with respect to G in terms of the associated Kala form, omega. So that's that standing notation. So in the first part of the talk I want to talk about the sort of first case where not everything is known, so that's complex dimension two, the case of complex surfaces, in particular Einstein metrics on real four-dimensional manifolds. And there I'm going to talk about sort of, I'll show you a bestiary of phenomena that are known or are conjectured to appear in all three cases. So I mean the three cases are markedly different. So for all three Einstein constants, lambda equal plus, equal to plus one, zero or minus one. In the second part we're going to go up to the complex dimension at least three and I'm going to restrict to the case that the Einstein constant is zero, so Kala Ricci flat metrics. So in the first part I'm probably going to be mentioning some stuff that I did in my thesis and the second part is parts joint work with Ronan Conlon and Aaron Neighbor. Okay, so the first part is more of an introduction. So we'll just recall that from yesterday that we have a necessary condition at least in the compact case for the existence of Kala Einstein metrics with a necessary topological condition for KE. And so that's the relationship of the first-turn class of your manifold. So let's see one of x be either positive or lambda equals plus one, zero or negative. So in complex dimension two you can say very explicitly what the manifolds with positive first-turn class are. This is so-called del-Petzl surfaces. So what we've got here, so the blow-up of CP2 in k points in sufficiently general position were k's between zero and eight. And then you've got CP1 cross CP1, but that's kind of an easy case because it's obviously Kala Einstein, just a product of two-round metrics on the Riemann sphere. Then here up to, in the Ricci flat case, up to Feynard covers, we have Tori, so C2 mod gamma for some lattice of full-rank gamma, or otherwise famous K3 surface. And then for the C1 negative case, we just attach a label. I mean we would just call them surfaces of general type. They're just way too many, so we don't really know how many there are. So maybe two quick remarks. So the first one is that the subdivision is sort of reminiscent of what we know in the complex dimension one case, the Riemann surfaces. So for complex dimension one, the first on Riemann surface is the first-turn class. So it's just minus the Euler characteristic of your surface. Then you have the sphere, Tori, or surfaces of genus at least two. What's very different from the case N equals one is that there are many more complex surfaces than are exhausted by these three cases. So I mean you could take products like Cp1 across a Riemann surface, sigma g of genus at least two, something like that, because of mixed cases where the turn class doesn't have a definite sign and where there's a priori no hope of finding anything like a Kela-Heinstein metric. So mixed cases or fiber cases. So that's quite different from the Riemann surface case. So that's a necessary topological condition and then we have the corresponding existence theorems that the necessary condition in complex dimension two is almost always sufficient for the existence of a Kela-Heinstein metric. So I'm going to make three cases one, two, three. Just continue here. So case three is in some way, in some sense the easiest case. So that's the Oberleer theorem. There exists a unique Kela-Heinstein metric on, general tribe is really a bit more general even than having the first turn class negative, but in the first turn class it's actually negative. Then there's just a unique Kela-Heinstein metric. So the second case is Yau's theorem that there exists a unique Ricci-Flat Kela metric in each Kela class. And then in case one, that's the hardest case that's been open for the longest time. So if that's been settled in work of Tian, Yau, there exists a unique Kela-Heinstein metric when K is at least three and it had been known for a long time. It's as much a Schema's theorem that was mentioned yesterday that there can't be a Kela-Heinstein metric when you blow up one or two points. So that's what's known about the existence of Kela-Heinstein metrics on complex surfaces. And now I'm going to turn to that question, what happens to the metrics if the underlying algebraic surfaces degenerate? If you have a sequence of algebraic surfaces that acquire singularities, then what do the associated Kela-Heinstein metrics do? How do they behave? All right, so that's the question. What happens if Kela-Heinstein metric when Xj degenerates? Okay, so maybe the basic picture to have in mind is what's going on in complex dimension one. So if you look at hyperbolic Riemann surfaces, then the standard picture out if you start with a genus G surface, you can degenerate that to something that has a node. So it's an ordinary double point from the point of view of algebraic geometry. And then what happens, what happens metrically in this picture is that well, here it kind of looks the same. It's a random hyperbolic Riemann surface. And then sort of in this limit, the constant curvature minus one metric sort of gets stretched out and there's sort of an infinite cusp forming here that separates the two pieces. These hyperbolic cusps really infinitely long. But the question is what's going on in complex dimension two? What can we say in complex dimension two, really? So I'll try to give you one example in each of the three cases, sort of positive Einstein constant, Ricci flat and negative Einstein constant. I'll try to indicate how the cases are different and what kind of phenomena you can see. So if you go to the case of positive Einstein constant, so Ricci is equal to G, then the first observation is Meier's theorem that tells you that the diameter of the manifold is uniformly bounded by the square root of three times pi. So all your Riemannian metrics sit inside some sort of fixed box of fixed diameter. That's some kind of compactness already. And then you sort of have the bane of Riemannian geometry that if you take limits of sequences of Riemannian manifolds, they can drop dimension. It's called collapsing. If the metric sits inside a box of fixed diameter, it could be something like this a priori, sort of shrinking one direction and converting to some lower dimensional space. We don't really want that. Luckily, we don't have that in the positive case. We can use volume comparison to see that that's impossible. So volume comparison with a positive lower Ricci bounds of inequality tells us that the volume of a ball of radius R divided by R to the fourth, so it's volume ratio. This goes down as the radius R increases. So I have a lower bound. Just take the radius R equal to the diameter of the manifold. So it's going to be like root three pi to the fourth. And then up here I have the volume of the manifold. But in the positive K-aligned stand case, it's just proportional to the square of the first churn class. So up to this normalizing factor here, this is basically C1 of X squared. So that's a topological invariant. So it's a topological lower bound on this volume ratio. And this means that geodesic balls of radius R look uniformly four-dimensional on all scales R. So this means that if I take a gram of Hausdorff limit, so I have a sequence of X i, J i that degenerates in some way and have the associated Kepler-Einstein matrix X i, G i. And then I want to take a gram of Hausdorff limit as the sequence goes to infinity. So I'm converging to some metric space, a priori X infinity, D infinity. And I'm definitely going to know that X infinity, D infinity is sort of a metric space of finite diameter and it's four-dimensional because of this non-collapsing notation so X infinity has finite, finite diameter and is four-dimensional. So it's Hausdorff dimension four if you want. That's a good thing. Much more is true actually. So there's a problem I studied and there's actually an ingredient in some sense in the existence proof of Tian in the positive case that you have an a priori result of what the gram of Hausdorff limit could be like so if in the worst possible case... That's... I guess there's Mike Anderson, Bando Kazunakajima, Tian who proved that in the late 80s or early 90s that this gram of Hausdorff limit is actually in orbit fold with isolated singularities. And the singularities are the form C2 mod gamma where gamma is a finite subgroup of U2 acting freely on the unit, on the unit sphere. So what does it mean geometrically? So imagine that's your gram of Hausdorff limit this four-dimensional orbit fold. So the easiest case is when there's only one singularity when gamma is equal to Z2, the smallest possible group, to forming one orbit fold singularity looks like C2 mod Z2 approximately your space. So that's X infinity. And then here you have a member of the sequence sort of X sub i for very large i. You can imagine that there's some sort of little bubble forming. So this is X i, G i for i very large. So there's a little bubble of curvature forming. Symptomology that gets pinched off and disappears into the orbit fold singularity in the limit. And then you say, well, we want to understand what that looks like exactly. It's called the bubble. So what we do is we rescale here to make sort of the topology that's being pinched off have unit size and then what's going to happen is that all the rest of the manifold, if we sort of scale up, if we zoom in with a microscope, all the rest of the manifold is going to be pushed off to infinity and we're going to converge to some complete space here that's Ricci flat because before we rescaled the Ricci curvature uniformly bounded and what we're going to be converging to is then a complete Ricci flat. KLM metric in complex dimension two, that's a hyper-KLM metric. So it's actually KLM with respect to many different parallel complex structures. And that's that if you dilate out, you get a gravitational instanton. So this is actually one I constructed in my thesis. So it has the same basic structure at infinity. So it's a half ray topologically across a three-dimensional nil manifold. Some topology inside. And the volume growth of this thing is... So we had Eguchi Hansen with Euclidean volume growth. We had Taubnacht with pubic volume growth. So the volume growth here is r to the fourth-third. And the sectional curvature decays as slowly as it possibly could in dimension four for an Einstein manifold so precisely quadratically. There's scale invariant decay, so this means the metric size of the plug is roughly one. That's the only thing that fits, but of course you have to prove it. It's a complete type of KLM manifold with a sort of asymptotic behavior. It's the cross-section at h3 or h3 model like this. Yeah, sorry. Let's just say that this means Heisenberg group modulo lattice. Okay, moving on. What is the complex structure of this thing? If you put one here, you get an affine cubic, so if you want it's... What's that? Or you can hypercalabotate it and you get a rational elliptic surface. So complex dimension, at least three, in that case. So here's a simple algebraic singularity model. And we want to understand it from metric point of view. If you take x, let's kd star for a final manifold, d. So d is a final manifold. What this means is you take the total space of the canonical bundle of d and you collapse the zero-section to a point. What does that mean? That's sort of an algebraic cone if you want. So somehow the picture is this. You have d that has complex dimension n-1. x is n-dimensional. And you're taking the complex cone over it. So you have a circle bundle sitting over here. That's sort of the theta direction. And then you just sort of cone it off in the usual real sense. That's the r-direction. So you have r and theta. It's a complex cone. So that's sort of singularity. We'll say d is a final manifold or maybe an orbit fold. So perversely, the most natural examples actually have d in orbit fold rather than a manifold. Or orbit fold. So that's the type of complex singularity that you encounter quite often in limits of sort of Kalabiyao manifolds. Just algebraically if you take an algebraic limit. So what's sort of the metric model for this? Why should there be a metric model in the first place? So there's some sort of theorems in the French school of complex analysis about weak solutions to the Monjom-Pierre equation. Let's say if you have a singular Kalabiyao manifold with sort of singularities, as you break singularities like this, then there's sort of weak or singular Ricci-Flat matrix on these guys. You just can't say anything about what the metrics actually look like. It's sort of an abstract existence theorem on the level of Kepler potentials that tells you nothing actually, but sort of the asymptotic behavior of the metric. So you want to make a good guess for that and try to understand what that could be. There's sort of an easy case where you can make a good guess, at least from the point of view of making a good guess it's easy. That's when D is actually Kepler-Einstein. And then that's construction that I think is implicit in one of Kalabiyao's papers about the Ricci-Flat matrix on total spaces for canonical bundles. So if this complex cross-section of the cone D is Kepler-Einstein, you can lift that to an Einstein metric on the circle bundle. That's what you would call a Sasaki-Einstein metric. And then the cone over that is going to be a Ricci-Flat cone, actually a Ricci-Flat Kepler cone. So if D is Kepler-Einstein, you can sort of lift that metric and cone it off to get a Ricci-Flat cone structure. So there's your Ricci-Flat metric. That's a perfectly good guess for what the model metric should be. So Kepler-Einstein metric on D gives rise to a Ricci-Flat Kepler cone metric on X. Okay? So there's your guess. So now you might want to ask a harder question. Does that globalize? Next question. Suppose you have a compact Kalabiyao X hat. Some Kalabiyao variety in the algebraic sense equipped with one of these weak singular Ricci-Flat metrics that we know is from sort of Pluripoi potential theory. And let's assume that it has an algebraic singularity that's bihologomorphic to X. Just algebraically. Is it then true that the Ricci-Flat metric that's known to exist on the compact guy actually has to converge to that canonical model that you wrote down there? So do the singularities of the metric have to be conical? So is it true that the Ricci-Flat converges to GX always? I think that's that seems to be completely beyond the reach of current technology to say anything much about that problem. So whether you can actually have singular Kalabiyao metrics with isolated conical singularities where you sort of guess the right cone model in advance. There's sort of a theorem approved with Ronin Conlon. You could phrase that as saying that infinitesimally the problem at the linearized level the problem is unobstructed. So if you're trying to solve the Kalabi problem construct a Ricci-Flat metric using a continuity method sort of on a background space that has these conical singularities then you actually do have openness in the continuity method. So if the linear part is okay so you can solve the continuity method for a short while and sort of preserve the conical geometry. It doesn't sort of jump immediately to something else. The hard part is usually this closeness where if you can go all the way from 0 to 1 the continuity method and that I don't know if it's true. But so even the linear case the openness is a sort of non-trivial theorem that's kind of gap theorem if you want. So the statement is if X is a Kepler cone so in a metric sense really a cone with a smooth cross section whose cone metric is Kepler with non-negative Ricci curvature and if you have a harmonic function U on X harmonic so it's a linear theorem so we're passing U is equal to 0 and you assume it's sub quadratic so it's a little o of r squared then it actually has to be plurihomonic some kind of Leovall theorem or cancellation theorem that if you restrict the growth of a harmonic function on your space to be less than quadratic it's actually automatically plurihomonic it's a real part of a holomorphic function and that tells you in some sense the obstruction to openness in the continuity method on those spaces isn't there closeness it's a it's a different story as r goes to infinity so there's this also true on asymptotically conical Kepler manifolds with non-negative Ricci curvature so here's your cone now imagine you have an asymptotically conical space that's asymptotic to that cone at infinity you assume the space is Kepler with non-negative Ricci curvature as well if you have a sub quadratic harmonic function on a space like that it still has has got to be the real part of a holomorphic function and somehow this is false on TELB nut so TELB nut is Ricci flat Kepler but you can make a harmonic function of linear growth on TELB nut that's not actually plurihomonic so somehow this maximum volume growth condition enters into the picture and sort of tells you something about the behavior of these harmonic functions so TELB nut is not asymptotically conical it's cubic volume growth but that's sort of the easy case where we could easily guess the right local model but it's sort of probably still way hard to prove that the model is realized globally in these X hat situations so look at a different case where we can't even guess the right local model a priori so hard case is when D is not when D is not Kepler Einstein so then we have this singularity X is a star that's an algebraic singularity and we don't have an obvious metric model I should give you sort of hands on the example for that the example is you take D so that this has complex dimension N so this has complex dimension N minus 1 so D is actually an orbifold in this case that's Cp N minus 1 the orbifold multiplicity is 1 minus 1 over K along a divisor and the divisor is a hyperquadrex QN minus 2 it's a hyperquadrex and then this space X KD star if I do this complex cone construction I get a very simple singularity I get an ak minus 1 singularity so X is just the ak minus 1 singularity in Cn plus 1 so it's a hyperstrict for singularity that's the locus of all points in Cn plus 1 where C0 to the K plus C1 squared plus plus plus plus Cn squared is equal to 0 so in particular K K equals 2 gives you the quadric for the associated stencil cone metric which in complex dimension 2 is the same as a Gucci Hansen so when K is equal to 2 then on the quadric cone we do have a Calabi-Yau cone metric so we do have a good local metric model so the difficult question posits itself but how about other values of K so that's maybe the most natural kind of singularity that you could possibly look at so it actually follows that's known from work of Russ and Thomas also from work of physicists Gauntlet, Metrelli, Sparks and Yau who study Sasaki Einstein Manifolds so using stability type obstructions you can show that when K is greater than 2 times n minus 1 over n minus 2 then this orbit fold there so with this cone angle along the hyperquadric there's no K lines in that metric so with algebraic singularity and ak singularity that occurs very naturally as a singularity of compact Calabi-Yau but there should have no reasonable guess for a Calabi-Yau cone metric in a certain sense there can't be one so then what's the right local metric model for this on the singularity so I'm going to make a table here tic-tac-toe table let's say K and we have n so K could be 2, 3, 4 5 in this dimension 3 up 4, 5 maybe so K equals 2 again is the quadric cone where we have a stencil cone metric so this is perfectly well understood in any dimension this is okay now here you're obstructed in some sense so what's that region so here we don't know what's going on so for the little more word you can sort of extend this argument to include the equality case so there's some boundary cases that are slightly more difficult but you're still sort of ruled out and one mystery case down here there have been papers claiming that there can't be a Calabi-Yau cone metric in this case here it's really the only admissible cases of the quadric cones that were already understood before so then here's the next theorem that I'm writing up with our neighbor so for K strictly greater than 2 times n-1 over n-2 there exists a Calabi-Yau metric so reach if not counter on a neighborhood of the origin on the ak-1 singularity so it's a local metric and the metric has an isolated singularity at the origin but if you zoom out and if you pass to the tangent cone you see something that has a plane of singularities from an isolated singularity at the origin so metric singularity at the origin but whose tangent cone has a plane of singularities let's draw a picture so picture like is this let's say this box here is ak-1 that's singular exactly at the origin so again it's z0 to the K plus c1 squared plus cn squared equal to 0 and now you project that thing on to sort of the special coordinate so this is the z0 axis and you see that the singularity is fibred by quadric surfaces so if z0 is equal to 0 you have a singular quadric cone so the central fibre that passes through the actual singular point and here you've got smooth quadrics so these are all tangent bundles of the sphere so you've got these vanishing cycles here these topological spheres and the metric picture is that the radius of sort of when K is sufficiently large when this power is sufficiently large and this is exactly what sufficiently large means so it's actually sharp it's right at the boundary where these spheres shrink so fast as you move in towards the origin that if you take the tangent cone so if you sort of dilate out by larger and larger factors and pass through a gram of Halstorff limit the cycles get pinched off and the tangent cone that you see is sort of a plane, a flat plane across the singular quadric cone one dimension less but that's true on the level of the Ricci flat matrix that's how the Ricci flat matrix behave an isolated singularity is sort of the tangent cone so it leaves the borderline the borderline case when you have equality here so it's these two guys so we're not writing that up just yet but the only thing that can be true in those cases is that you again have a local Ricci flat metric for the isolated singularity and for the tangent cone which is not simple so whose cross section is not smooth but it doesn't split off a line so if it's more complicated it has singularities but it's just not the product of a plane and a simple cone so the way this is proved actually is so if you guess a good local metric model you make an ansatz for what the metric could possibly look like and then you compute sort of its Ricci potential and you see that the Ricci potential goes to zero so it's approximately Ricci flat in a suitable sense and then you try to perturb it into a genuine Ricci flat metric sort of maintaining the geometry of the derivative problem so you want to solve the Monjampere equation the linearization of the Monjampere is the Laplacian so what you need is good estimates for the inverse of the Laplacian on a space with roughly this kind of geometry you need charter estimates for the Laplacian on a space with this kind of geometry and we do that using blow-up arguments sort of Leon Simon style blow-up arguments you need Liouville theorems for that and the possible blow-ups that you can get and that's sort of the flavor of the theorem that if you don't go too fast actually have to be special real parts of holomorphic functions for instance, things like that if you're ready to have five minutes to say something about the mystery case so what's going on there so again the proof of this this theorem is based on writing down guessing the right ansatz, a right approximate local model but the formula for that local model that we have works for every k that makes sense for every k so in particular for the cases sort of below below the threshold but in those cases the ansatz works at infinity, it tells you something at infinity and not at the singularity so if when k is strictly less than 2 times n-1 over n-2 you can say that our ansatz works at infinity instead of at the origin so you get following geometric picture so again k is strictly less than 2n-1 over n-2 basically only leaves two cases sort of the vertical the column here, we have the stencil cone that's perfectly understood and sort of this one mystery case so let's look at the case first that we pretend we understand completely, so this one so what's the picture there k equal to 2 and n greater or equal than 3 arbitrary so here's this box picture of the A1 singularity I'm saying this, this ansatz that we made suddenly works at infinity in this case so you get an approximately Ricci flat metric at infinity that looks like c cross sort of an n-1 dimensional stencil cone but at the origin we're sure in the good case, at the origin we have the n-dimensional stencil cone metric so we can put that there put the origin further down so here we've got sort of the n-dimensional stencil cone so if you have faith in the solution theory of the complex monogram pair equation what this suggests to you is that there exists sort of a Ricci flat interpolation metric a Ricci flat metric on the total space that interpolates between an n-dimensional stencil cone at the origin and c times an n-1 dimensional stencil cone at infinity so it should be possible to interpolate in a Ricci flat way in between here so here's one obvious sanity check you can make compute the densities so the monotonicity formula for Ricci curvature tells you that the density ratio the volume ratio volume of a ball of radius rd divided by r to the real dimension goes down as the radius increases so you should compute the density of the n-stansel cone and the density of this cone and check that they satisfy the right inequality otherwise this is hopeless so if you compute the density ratio so that's the same thing that Brian White was talking about yesterday it's 2 times 1 minus 1 over n to the n here it's naturally 2 times 1 minus 1 to the n minus 1 that's a popular calculus exercise to show that goes up this way so when the dimension is 3 this is half, the density is 1 half and it keeps increasing and when I let n go to infinity it goes to 2 over e so that seems to make some good sense you know what does it tell us about the mystery case this is k equals 3 n equals 3 so schematically we have the same picture so this is now the a2 singularity in complex dimension 3 we have this ansatz metric this sort of fibrous ansatz metric that's sort of suddenly approximately reach a flat at infinity so here we have sort of c cross the two-dimensional stansel cone but we have nothing to put at the origin we have these claims that there is no nice simple cone structure on the singularity if this works out so nicely then how can this be true can the universe really be so cruel that this space doesn't exist but this one seems to exist you can do even more so the density at infinity here is 1 half and you can show that if this cone if the simple cone existed the density would have to be fader equal to 125 over 243 which is just barely bigger than 1 half so it still doesn't violate the monotonicity it's barely bigger than 1 half and so I was in China last summer sharing an office with Song Sun and I mentioned this problem to him and he was just finishing up this paper with Chi Li about Kale Einstein metrics with conical singularities so sure we can show that the cone exists so question is that cp n minus 1 with an orbital singularity of cone angle beta along the hyper quadric so beta doesn't just have to be like 1 over an integer could be an arbitrary real number so this orbital here according to Li and Song is Kale Einstein again using stability type arguments if and only if beta so it's always a final orbital for any value of the cone angle beta but it's Kale Einstein precisely for of course you can go all the way up to 1 because if beta is 1 then you don't have any orbital points for being a studi metric on cpn minus 1 so you want the lower bound and the lower bound is I hope I get this right so it's 1 over 1 over this critical constant that's appearing in this in this story so it's a saying that in particular for the mystery case so these claims that were in the literature that this cone doesn't exist they were actually wrong and the cone has to exist and sort of the picture is consistent again so you would suspect that there actually exists sort of a Ricci flat space interpolating between these two cones again to show the existence of these spaces that's sort of solving the monjamper equation on a manifold with isolated singularities where the curvature blows up and that seems to be something that sort of they have to exist they just have to but that's something that's sort of beyond reach of what we can currently do okay I'll stop there thanks any questions? that's not known that's sort of that's a sub question that's an easier step to the question I wrote down that's not known I mean you would think if you can make a good guess then you should be able to prove that that guess is always realized but you know we don't even know whether you know different compact collabials with exactly the same local singularity types have to have the same local singularity type that's not known but the other singularity types have to have the same local metric models of the singularities can you give us an idea where the 1.5 over 2.3 I mean it's it's a density ratio of the cone so that's the same thing as the volume of the cross section of the cone divided by the volume of the unit sphere we know the volume of the unit sphere now the cross section of the cone that's a circle bundle over this Kepler-Einstein orbifold so you want to compute the volume of the circle bundle over this Kepler-Einstein orbifold the volume of the Kepler-Einstein orbifold itself that's a degree computation you probably know that better than I do how you stick in the beta in the right place to make the degree computation come out right and then sort of you have to lift that to the circle bundle to the metric on the circle bundle and that's sort of the the collabial that's because the length of the I mean it's basically Kepler-Einstein orbifold times the length of the circle because you have a Riemannian submersion so you only have to know the length of the circle and that's something like the index of the final orbifold divided by the dimension so you have to compute the index the divisibility of the canonical class that's a collabial that's a degree computation plus collabial that's the statement of the theorem you say that you can always find some collabial metric even in these X cases where there's an X right so what the X means is that so you have 8k-1 topologically that's a cone topologically this or even holomorphically this is really sort of some compact cross section and then cross for the real line and you put a what you don't put a metric on there just topologically it's a cone so what these X cases mean is that there is no collabial cone metric on these guys so there's no sort of what metric of the form the R squared plus R squared G so L is the length L is the cross section which would be collabial whereas in the good cases there is so in the bad cases I mean it's topologically a cone so you would hope that it's actually medically a cone as well so in most cases it can't be nice cone metrically and what it is instead is the sort of more complicated thing where the cross section isn't smooth but actually singular we have sort of the cone sure we do so we make an ansatz for the metric and then we find that the metric is the ansatz is approximately rich flat but it's not quite rich flat we don't make it rich flat by bare hands we don't write down an explicit solution but so we have to solve the modern pair by perturbation so you can use the implicit function theorem to solve the modern pair so it comes down to inverting the Laplacian and then all the analytic work is solved in getting good enough estimates for the Laplacian on a space like this okay well if there are no other questions let's thank you and I'll be right back