 So I've planned these lectures, two lectures today, to most of them were giving background to what I, so background to the technical level, to what I talked about rather un-technically on Friday. So at some point we should discuss about having maybe one further lecture, so then I can try to pull everything together. So to reassure you what I'm saying today doesn't immediately seem to tie up with what you're hoping to hear about. So this morning I'm going to talk about a paper with Song Sun, which I've heard about a year ago, it's on the archive, this is the number. And the point of this is to see that Gromov-Hausdorf limits have, under suitable hypotheses, algebraic structures. So this is the sort of slogan. So we're mainly interested in Keiler-Einstein manifolds, minor variants of that. But it's actually in a way, at least for me, natural to work more generally. And so we consider the following hypotheses. Let's consider compact complex manifolds, dimension n, fixed with a positive line bundle over it. With a curvature omega, it should be a Keiler form, 1x. So the key hypotheses are first we can let the volume be less than or equal to some number. The volume will be essentially an integer after you normalize because the Keiler class is an integral class, so the volume is bigger than some one, essentially. And we want to suppose the volume is less than some fixed number v. Let's suppose that the Ricci curvature of this metric satisfies a fixed bound. Well, we could take any number here. And finally, we want a non-collapsing condition of the kind which appeared in the talk of Haye in the meeting of the weekend. But for all proper balls, for r less than the diameter of the manifold, the volume of the ball is bigger than some positive constant times r to the 2n. So let's consider the collection of all for fixed v and c and n. Let's consider the collection of all such data. Let's call it say k, n, c, v, c. So we're really secretly interested in the Keiler-Einstein case when, obviously, this is satisfied with the supernormalization of things. This condition then follows, sorry, I should say in the primarily in the Keiler-Einstein case of positive Ricci curvature. So this volume non-collapsing condition then follows automatically from standard argument from Meyer's theorem, again, as appeared in Haye's talk. So if you have Keiler-Einstein with Ricci positive, then the non-collapsing is automatic for a suitable constant c, which you can calculate. This follows from the Bishop Romoff monasticity property that let's suppose in the case of positive Ricci curvature, say, then we take the volume of a ball about a fixed point of radius r and divide by r to the 2n, then this is decreasing. It's a function of r. So by Meyer's theorem, we have a fixed Ricci curvature, gives a bound on the diameter. So we have a fact that the volume is bounded below essentially by 1, as we said. So the volume of a ball of a fixed size is at least 1. So this Bishop Romoff condition says that small balls can't have very small volume. In fact it's the same to replace this non-collapsing condition under the other hypotheses by a bound on the diameter by the same kind of argument. But I brought this in because this volume monasticity condition for balls is one of the crucial things that appears again and again in the underlying the various arguments. So this is the hypothesis. So what we want to say is supposing we have any sequence of xi in this kind, then by very general facts there will be, if we were to take a subsequence, we should suppress, there will be a Gromov-Hausdorff limit, z, which as I wrote down the definition of the Gromov-Hausdorff distance in my talk on Friday, so let's write down again, this is a very general notion for arbitrary metric spaces essentially. So this thing is initially a metric space in the sense we have in the sense of dxy, distance between points. And in the short form of the statement expressing this fact would be that z is homeomorphic to a normal projected algebraic variety. It's called it say xinfinity. So I mean a complex projected algebraic variety. So this is a short form, easy to state, but it's not, there are better ways of making more precise statements which capture more of what's going on. One way I like to think is that in this sort of metric space of Iranian geometry game people are always talking about convergence of sequences whereas in algebraic geometry people like to talk about families over a base. So you can think of it that way if you think about, if you think about take a model set which is just a accountable set of accumulation point. So this is just a set 0, 1, it doesn't matter, it takes any sequence converging to 0. So we've got set s. So it follows from very easily from the definitions of Gromop-Hausstorff limit that we can take the disjoint union of all the xis and z and think of this as a, take that and map it down in the obvious way to this set s and endow that with a topology, in fact a metric. So we can think as we've got x goes to s where the fibers are xi and this limit z. So one slightly more precise statement would be to say that we can embed this whole picture in some projected space in such a way that z maps to this normal projected variety and the maps on all the xi are given by some fixed power of our line model L. Possibly after going to a subsequence. So xi embedded some fixed variety then? So you have an algebraic family? You've got these isolated guys so they all sit in one variety then? Well that's what we're working around is that we don't initially have any. That's what you're trying to get to? We're trying to get to. So this is a kind of a way of trying to say that. So in other words we can say we can embed this thing in the child variety, parameterizing sub-variety, it's a projected space of another way of saying it. Or we could say the natural way you can define a sheaf just by just a very general way over this limit by saying a function first to find a pre-sheaf you would say you consider a little open set u, you consider functions which extend continuously to functions that are holomorphic on all the xi's so we know what that means. Then take the associated sheaf to that pre-sheaf and then the statement would be that that sheaf defines the structure of an algebraic variety on the limit set. Or another even more to a high technology way well let's say we'll be to it so let's go back. So let's to which I need to say something more about what's known about these Gromov-Hausstorff limits so initially this thing is just a metric space that has no other structure from the just from the definition. But a lot more is known and we would need to to use that what we're saying. So let's start off by saying that what we know by my work is a basic result of Anderson and also a general theory of Chieger and Colding which we'll be appealing to later says that in fact z is the union of two pieces the regular set r and the singular set s well this is this is an open dense and on this regular set well under the hypothesis because we only said bounded Ricci curvature what we get is a we get a Romanian metric but not quite a not quite necessary a smooth Romanian metric we get a c1 alpha Romanian metric but in fact say since we're actually interested in the Kader-Einstein case in reality so you can ignore that you could just think of this is a smooth Romanian metric on the regular set. So we have this small singular set which we know less about where this has got house this has got Hausdorff co-dimension real co-dimension at least four so we have although say z is initially just a very abstract object we know in fact it's got a dense open set where it's a standard Romanian manifold and then this singular set that well at least we know something the control of its dimension is small in the sense of Hausdorff dimension but but we should be careful that we don't at this stage we don't know anything else about this singular set really we don't we don't want to imagine that this is a a submanifold or anything this could be a bizarre kind of fractal object something at this stage. In any case just to just to pursue the further different forms of ways of stating this result what we get along with the so say we get a Romanian metric but we also get all the other structure that we started with on the the regular set so over the regular set we have a nice line holomorphic we get a complex structure and we have a holomorphic line bundle and we can talk about with the Hermitian metric and we can talk about bounded holomorphic sections of L over the regular set. The algebraic structure on the limit is unique because it's unique with all the things you need to define. The algebraic structure on the limit? Yes that's that's what I'm trying to convert the way of sort of writing down a sheaf gives a way of that's right so so another way of kind of saying things in a more fancy language is that for each k we can define arc so we say that arc k is bounded holomorphic sections of L over L to the k over R so these things obviously form a form a ring if you multiply two bounded sections you get a bounded section and using results of chi-li plus the other input we're going to put in so this is in fact finally generated so plus the input for what we're going to do from later implies that this is finally generated so we could just define this this this thing to be x infinity is just the prod of this does we can write down a formula but it is the point of the the difficulty is to prove that this is finally generated so it actually defines yeah it was graded by so there are various various ways you can make the true sophistication in which you can interpret this but okay so the crucial thing to do is actually and there needs to be done it's something which has been understood by by chan for many years when probably by many people but people chan has explained over many years why this is the why there's a crucial thing one needs to establish in order to have all that I have all this this nice compatibility between the metric geometry and the algebraic geometry so given again given k and a point x in so I just took x is just going to be a typical number of this considering we can define row k x is it's the maximum of the size of s of x where s is a section of l to the k and normalize so the l to norm is equal to 1 so maybe I something I didn't but I didn't say it rather the beginning which actually rather crucial to recall we consider this is a Hermitian holomorphic line bundle then we get a good that's how we get a compatible connection and a curvature form so that's what we're using here it's means the norm on the line bundle so this is this is to say that this is positive it's just to say there's a section that doesn't vanish at the given point but the crucial thing we want is to get a uniform positivity on this over all manifolds in our class so the essential theorem to say that exists some k that we can take k to be arbitrary large given any number we can take a number bigger than that such that and and b bigger than zero such that row k x is bigger than equals to b for all x in x and x in our class so you know though we're getting from for large enough k we get a kind of a universal lower bound over all things we're considering so when I when I said input from later but this is what I really meant that Cheely approves that if you know this then you get this final generation condition so most of the consequences of this in a sense fairly stable things people have known how to do for quite a while in some form the crucial thing is to prove this this lower bound which is kind of an effective form of saying that the sections generate the line you see we know what we know is that what we know is if we take a fixed x then indeed at any point indeed we can take a large enough k to make this positive but we need to do that in this kind of uniform way this is all making sense should I carry on so let's just dive in and talk about the proof of this which is really the heart of the matter and then come back to say something about how it implies the other things so the proof will combine two things one is this theory left from these people I just rubbed off this is mainly to the chieger and colding about the convergence theory of manifolds under Ricci curvature bands and the other is in complex analysis complex geometry it's called the the L2 or Hormander method for constructing homomorphic sections so we want to combine those as we put those things together and basically the arguments beyond that are this elementary nature so I want to review the L2 technique first so initially just in the simplest possible setting of how we how we were proved from first principles that the fact that I stated about we have a positive line bundle then indeed take a super high power we do get sections that didn't finish the point so it is now a fixed X it's a big X and a point in X and we want to construct a section out of the K with S of X on zero whether there's one one very basic thing to see it's obvious but but a very important thing to understand what is this what is this out of a geometers good quite familiar with saying you take a take big powers of a positive line bundle to do things what does that mean more in terms of the metric geometry that this corresponds to a scale factor if we change L to L to the K that corresponds to changing our our K-to-form omega to K omega so as we're scaling the scaling the metric in the sense of the distance between points by a factor the square root of K is sensibly so if you draw a picture we want to think about our point of X we want to think about we've expanded the manifold out by some large amount so now so we take a sort of a little ball about X now this expands out to some huge ball and in this expanded thing a unit ball goes down to a ball of radius one over root K about our point so our constructions are going to be as you'll see in a moment essentially localized on these little ball around our point which when we rescale that'll become this standard size ball in X so of course what we know about a Romanian manifolds we take a small ball and scale it up it becomes approximately flat that's the basic thing we know so let's for simplicity we're just reviewing the idea here let's suppose that in fact our metric actually was flat precisely flat in a small neighborhood here so that when we do this rescaling this is now isometric to a ball in Euclidean space in reality of course that won't be true but it'll deviate by a small amount so it'll just be a small extra error term which won't really cause any difficulty so we should so that's the first idea then is this scaling idea the second idea is then to focus on the flat limit supposing we just take CN with its standard let's say omega standard flat metric doesn't seem very interesting but let's think about we should think about a line bundle over CN with a metric whose curvature is omega and that's that's then comes that's a bit more surprising and since it's the same thing well what you then find is that this will be the trivial line bundle but we're going to change the metric so that the trivializing section does not now have norm one but will be Gaussian so we have a section it's called it says sigma nought but the norm of sigma nought is a function on CN will be e to the minus z squared over 2 this Gaussian thing that's just saying that all we're saying is that d bar d of mod z squared over 2 is the standard this is this is the caler potential for the standard metric that's all we're saying so this is this is just a different language for what and if you look and say Hormander's book would be expressed in terms of weight functions rather than took rather than taking holomorphic functions on CN and taking the standard L2 in a product which of course will not work very well because it would never be finite then we take the weighted function weighted by this Gaussian factor so talking about weight function is just the same as talking about the line bundle with a emission metric so now we can actually say it so we want to restrict to a big a big ball of radius r say r is some fixed some large number that we're going to fix very soon a way of expressing what we're doing is that we can take a map chi from b of r to x and when we pull back when we take the kth power of our line bundle pull it back by this map is a holomorphic map but also this map will be an isometry of the rescaled metric and we can lift it so that we also identify the line bundles over the same thing so we can think of we can think of studying our line bundle l to the k over this ball is just the same as studying our standard line bundle over some people's ball in CN and we have this so we can transplant sigma 0 to a section of l to the k over this ball the obvious way using this isomorphism so regard sigma 0 as a local section so remember I'm going to start writing down norms of things and so on so it's really better I think to work with the rescaled norm so we had we would always work with the rescaled metric on x so this this thing really is an isometric embedding what we're doing so certainly this is a this is a local homomorphic section which doesn't vanish at the point we started with the origin what we want to do is roughly speaking to extend this to a homomorphic section over the whole of x which of course we can't do precisely but the wonderful thing about these l2 tech technique is it shows that you can do that you can essentially can do that with a very small adjustment you can do that what we do is we get it we take a cut-off function with a standard kind so this is a picture of our Gaussian function we take our we have our large number R which is the moment we're out to vary we take some cut-off function of any kind of standard kind so we can consider beta r sigma naught we extend by zero over x so this this this is a function vanishing outside the arbol as indicated but of course it's not homomorphic because we've got this cut-off function so what we're going to then define is say s is pi of beta r of sigma naught where this is l2 projection to the homomorphic sections this is certainly something we can write down that remember this is just a homomorphic session is to find dimensional vector space is all we're working in over over this compact manifold we can certainly make this l2 projection and certainly what we get is a homomorphic section but of course just from the symbols on the board we can't say anything about it it could be zero from just from the symbols on the board so the crucial thing is to say that actually this projection doesn't do very much but this thing is almost the same as the things as the thing we started with what we're really saying is essentially this Gaussian is compactly supported so because it's not exactly if it were compactly supported we've just been home straight away because it's not exactly compactly supported but we can get around this because the very rapid decay we can get around that so how does that go the standard kind of hodge theory gives you a formula for what this l2 projection is and so s is going to be beta r sigma naught minus tor where tor is d bar star of delta d bar minus one of let's call this thing sigma one this is this is the standard assuming to justify as I was a moment why this operate this Laplace operator is invertible if it is this is the hodge theory formula for that l2 projection so this is what we're considering the d bar complex on sections of our line bundle and this is this is the Laplacian of the usual kind d bar d bar star plus d bar star d bar on the zero one forms so we want to know this is that we want we want to know first of all this is well defined I is this or this this thing has got no kernel the class has got no kernel but also we want to actually a bound on the operator norm of this thing that's what that's a crucial thing we want but why do we get that we get that from a voctner feistenbach formula which which runs in the form delta d bar is some other well a certain other p star p for a certain other first or differential operator p plus a term from the Ricci plus one so here what do I mean so point about this is just this is going to be positive that's all we need to do is we don't need to know the details of what this is by Ricci really I'm working here with the rescaled metrics so this is really one over K times the Ricci curvature we started with but the Ricci curve if we start with this bounded so this is so this is definitely bounded and this is even smaller so and then the one is really the curvature of this is the curvature of the line after we've rescaled the curvature of the line bundle is the caler form that's the one that comes and this is essentially to just the there's just the same formula as in the kadara vanishing theorem just it's just the same thing the fact that you get this you get the so as far as Ricci over K but we should this original Ricci so what we're seeing is that whatever whatever Ricci was when K is recently large it'll be killed dominated by this one okay so what what that shows is that the operator norm of this the L2 operator norm is less than a quick by making we can suppose is that by making cases in large is less than one half or two it always seems like it's a completely elementary thing but it seems like a miracle to me you don't reconstruct this inverse we get the existence of this inverse with the definite control we get a way of solving the plus equation with definite control but we don't actually only solving we just write down this formula for the for the this sort of inequality so okay yeah so we'll take K bigger we said Ricci was less than one so K is going to be bigger than two or some some number yeah but in general it'll be sufficiently yeah cable is certainly large yeah but I'm not supposed to know exactly how big no everything's rescaled here I always want to risk all the form I can write down for the rescaled metric that's the easiest way but it is invertible for any K I would say more than one Ricci was bounded by one yes well the one was arbitrary I mean I one was I could have put 10 the one was really arbitrary the point is that once K is bigger than some definite number this thing is satisfies a definite so this means we get a if you take the L2 norm of tall squared is what is that this is D bar star delta to the minus one D bar of sigma one D bar star delta to the minus one D bar of sigma one we can take come we can take we can take this across but in here then it cancels with the delta to the minus one there so this is less than two times the norm of D bar sigma one squared but this thing this thing we can make very small because what is what is the D bar of what is sigma one remember this is just beta R times sigma not D bar of sigma one is D bar of beta R times sigma not to the size of it is this the derivative of beta R only happens out here so radius say roughly R over two or something where the size of sigma naught is e to the minus r squared so this is essentially e to the minus r squared or something like that so by making our large but not very large we can make this a tiny number so what we're saying is that we make the the adjustment in L two that we have to make is it can be make it a tiny number it's by then making our quite a moderately large number are so why did why did so that now we have our redrawing our picture that's a schematic picture of our Gaussian section now we know that we've slightly adjusted we now got it all over the manifold we got some sort of slightly adjusted thing actually it's a homomorphic section at this point we only know that the difference is small in L2 but if we restrict her we want to we want to get a pointwise bound at the point we started we want to know the difference is small pointwise but it's in fact less than one so that our section is still non-zero but if we restrict her some fixed-size interior ball then tour is holomorphic because because we're not seeing the cutoff function we restrict that thing so we by standard elliptic estimates the L2 norm over this thing controls the the pointwise norm that the C0 norm so if we if we again if we make our we should need to choose our R sufficiently large to fit in with the the constant and that elliptic estimate but then we can suppose we can get the norm of tour at our original point we started with is less than one-tenth say any number we choose adjusting our parameters yep so so the section does indeed have size essentially one at this point okay so this is really for time this is this is all the review of what is basically very well known to experts okay so I can I can see that I'm not going to I'll be carrying on with this first part in the second session this afternoon but that's fine so let's begin on putting let's begin on doing what we really want to do which is obtaining this uniform bound over all our class of manifolds so let's now review the Chico-Colding theory some I'm not expert in this but so review some small parts of it that we want to refer to the crucial thing is the existence of tangent cones that's to say for the Gromov-Hausdorff limits of the kind that we're considering so if we take a point P in Z then we can rescale metric and we've also rescaled the metric by a by a sequence of factors by sequence let's say ri tends to infinity but we can after rescaling we can restrict to sets of fixed size centered at z so it means very well so this is a schematic picture of our P in Z we can we can take a something of we can say unit size and expand this out size and the same very general compateness theorem says that when you do that after taking a subsequence you get convergence on bounded sets if we fix any number after rescaling we converge on balls of a fixed size centered at our base point P and the crucial so so the crucial point is that the limit is a metric cone so this is a this is this is a deep technical theorem I think it's sort of it's not hard to see roughly why they thought to be true from the volume of the volume of B of R decreasing it follows from this but this thing is a this thing is a volume cone there was the vol if we take this limit then the vol of B of R is equals a constant times R to the 2n and at least in starting with straightforward situations when you examine the proof of the bishop Romoff you see it actually if this is constant then the thing has to be a cone that the the inequality comes precisely from the deviation of the metric from being a cone when the bishop Romoff inequality so this the fact that you get these metric cones coming out it's not it's not too hard to see that it's plausible but the precise proof is some much harder because we want to take it as a as a a known fact but we should be aware that these this these tangent cones are not are not in general known to be unique so we're not we're not saying whatever sequence you take you get the same limit you could you could in principle take different sequences of rescaling's and get different limits actually that's not going to matter in what we in what we do luckily and probably a post-apostura after we've proved that these things are these algebraic structures it may well be possible to prove that the tangent cones are unique but that's another thing but also we have some structure theory for these for these tangent cones just like the structure theory we had we describe before in the ordinary limits so we have some singular set say sigma y in y and outside the singular set essentially we have a smooth romanian metric really we should say c1 alpha but actually it is I think it is even smooth in this because we've we're scaling Ricci down to zero so we have a we have a so this is this has got this is closed house dwarf co-dimension bigger equal to four and we have a say a smooth romanian metric outside so we're saying if we look at it we look in at a very small scale in our Gromov house of limit scale it up what we're going to see is essentially this this this cone this metric cone it has some singular set it's a small singular set that's just it's just symmetrical with this singular set is this but outside that we have a genuine metric cone in the usual sense so away from the singular set we can do we have a kind of a standard and straightforward differential metric picture we have a we have a line bundle at the trivial line bundle we can define over because the cone of y minus the cone on sigma y a mission line bundle with a section a whole more fix section with the same Gaussian decay where this just notation for the distance to the vertex in the cone this this this line bundle has a mission home of it line bundle has curvature the caler the caler form there's a complex structure here the curvature is the caler form defining the metric on the cone so again we want to really work with the bit we know where we have this nice differential geometry so we're going to we're going to do a bit more complicated cutting off this time with three parameters one we're going to take a large radius say R I'm going to take a small radius say Delta and but then we should be then we've got this singular set so we'll take a neighborhood of the singular set say the eta neighborhood and remove that so let's let's call this say u equals u of Delta Eta that's that's this is this is now a smooth this is now got a smooth Romanian metric a complex structure this is cone structure that's what's called a kind of what's called a sotarchy Einstein structure is what it will be called and we have this we have this home of it line bundle at trivial home of it line bundle with this Gaussian decaying section so so in a few moments I'm going to stop but maybe it's nice to ask any questions about what I've it's anything I've said not something I can usefully say now to terrify so what you're doing now is you're sort of trying to understand what the sections it's not going to be quite what I'm doing now is I'm it'll come in the next sets in the next paragraphs it will be clear why I'm doing it but I'm doing is I'm I'm getting a nice region in our in our cone in which we can do differential geometry and we have this homomorphic section smoothness is one alpha yes alpha is positive you said which house move this this delta this is good oh well see one alpha well it's alpha well I ignore alpha I mean in fact in fact when we done this rescaling I think we can get see infinity because in fact the metric here will be reach you flat so you have curvature yeah but you are working only with linear operators you are working only with linear no we're we're we're we're just we're just saying we wanted to fix nice region where we stay away from the singularities so we're doing in this cone we have this we have this this metric cone but the the base of the cone itself has got this singular set so we're fixing this nice region depending upon these three parameters yeah your choice of you said that this will give you a sasaki structure is precisely an Einstein manifold on a cone so away if we remove the singular set just we've got to correspond to the second form or something I mean I'm just wanting to well the sasaki form will be essentially D of this something like that that'll be the sasaki one that'll be a sasaki okay well let's let's stop does that make sense to stop now and then I'll start again to explain why why we're doing this but let's just say I'll say it now and I'll say it again supposing we have now some cutoff we have a cutoff function of r as before so beta r we're going to have a cutoff function depending on delta something vanishes in this thing and because equal to one outside distance to delta and then we want to cut off function on why which is equal to one outside this eater neighborhood oh sorry support in this eater neighborhood and equal to one outside a slightly larger neighbor so we have our section sigma naught and we can consider this thing just just saying just saying multiplying by a compactly supported function we can do that so if we can embed this you in a essentially isometric and holomorphic fashion inside our x then we can do just the same thing as before we can regard this sigma one as a section defined over x not quite holomorphic because these cutoff functions and then use the same L2 projection to construct just to do just what we wanted to do before and that's what we're going to do but let's let's stop now and take that up again in the next