 Thank you for this invitation to this very enjoyable and interesting meeting. And congratulations to Maxime on his coming birthday. This is a talk aimed at non-specialists and will mainly be about formal differential geomethric aspects of Keilor geometry, in particular the study of things related to Keilor Einstein metrics and constant scalar curvature Keilor metrics, and as we'll see certain related objects of differential metric interest. So let me remind you of the basic setup. We want to think about if we have an algebraic variety, the polarisation, the positive line bundle V, we want to, if we consider Hermitian metrics on this bundle whose curvature form is a positive form, the Keilor metric, so H will be our bundle metric, and this gives a curvature form, say, omega H, which we want to assume is positive, then we can, we would like to consider the problem of finding an H such that the scalar curvature of this metric is constant. This is the scalar curvature, which of course is the trace of the Ricci curvature, which our writers wrote as Ricci. We think of the Ricci curvature as a one-one form, just like the metric tensor. So this is the general problem of the existence of these things. Can we find such an H? And this is a fourth order partial differential equation for the metric H. H is locally represented by a single function. This is a fourth order partial differential equation function because we have to take two derivatives to get to the metric, and then the scalar curvature depends on the two derivatives. So a special case of this setup is when the first-journ class of our manifold is a multiple of the first-journ class of our line bundle, but we would just consider the case when it's a positive multiple there, suppose that they're equal, so we can actually suppose that the L is the dual of the canonical bundle of V. This is the Farno case as discussed in a different context in Paul Seidel's lecture. So then we could write down the scalar Einstein condition, which is that the rho is the Ricci form is equal to the metric form, so rho of omega H. But this can be written in a much simpler way. This thing which is on the face of it is a fourth order differential equation, just as here, can be reduced to a much more tractable looking second order equation. In the following way, if we just algebraically, if we have a metric on the dual of the canonical bundle, that determines a simple algebra at each point, a volume form on the manifold. H goes to, say, capital omega H, the volume form. That's to say, a way of saying it would be a metric on this line bundle gives a metric on the dual bundle, so you take a form alpha of length one with respect to that metric, and then the volume element is alpha wedge alpha bar. So this equation here is, in fact, equivalent to solving just the simpler equation, that the volume form omega H is equal to some constant multiple, which we can normalise to be one, to the standard volume form, sort of Romanian volume form given by our metric tensor. So this is a second order PDE. So, of course, clearly, if you solve the Einstein equation, the Ricci form is a multiple of the metric, when you take the trace, you get constant scalar curvature. A slightly more, a less obvious fact, is that the converse is true. If you're in this situation, just the topological conditions you're considering here are satisfied, then any metric with a constant scalar curvature would also satisfy this a priori stronger condition. So let me just remind you of how the proof of that goes. So in case two, what we always have is a scalar identity, which says if we take d bar star of the Ricci form, then that's equal to taking d of the trace of the Ricci form, which is just d of s. So, if you have constant scalar curvature, it's always equivalent to saying that the Ricci form is a harmonic form. So s equals constant is equivalent to rho is harmonic. Well, that's always true in whatever case. The point about case two is that in case two, the carmology class of rho, which represents the first-gen class of the manifold is equal to the carmology class of omega, so by uniqueness of harmonic forms, since the metric form is harmonic, we can deduce that rho is equal to omega. Once we know the harmonic form in the class of the calymetric itself. So actually, in this Tharno situation, in a sense you could study either equation and you know you're studying solutions are the same, but the framework for the equations is somewhat different. So, in this special Tharno situation, there's a functional, which is important in the theory, which is the ding functional, which introduces a slight variant of a normalised functional later. But the important part of it is just that we just take let's say d nought of our metric H. This is just the logarithm of the integral over V of this form omega H. So, it's a very simple thing. If we choose some reference metric H nought, and we write H is, say, e to the phi times H nought, then this is just, so everything, not a variable is this function of phi, then this thing is just the log of the integral of e to the minus phi, it works out, times omega. H nought. I think it's alright, I think it's, yeah. I should say that this will be a talk where random signs, probably random factors, so if you get the one right, you might as well get them all right, and that's a lot of work. But I think that's the right sign. Anyway, this is a very simple function. It's not just the integral of e to the minus phi. And this was introduced, I think, by ding in perhaps the 1980s or something round about then. But relatively recently, it was discovered by Benson, that this function is convex in a certain sense, in a sense that I will recall later on. And this is a very important point in the theory, for reasons I'll try to outline, but I really only have time in the talk to just talk about as the formal setup, without trying to go into more the analytical aspects of why these things are used. But let's see what's, one reason why this is, that this Benson convexity is such a good thing, is that because this is such a simple functional, it can be defined in situations of very low regularity. We don't need this to be a smooth, a long way from being smooth in order to find this. One virtue of the ding functional, compared to other things we're going to mention, is that it can be defined in situations where we want to consider metrics of low regularity, or, more generally, where the space we're working on is not actually a manifold, but some kind of singular space. The theory developed by Benson and other people working in that kind of pluripotential theory direction extends all these ideas to singular spaces, and that's technically very important. But it's rather crucial, for all that extension, depends rather crucially being upon in this sort of second order framework, as opposed to this more general fourth order framework. So what this, the question that this talk aims to answer is how to fit all this, these developments into a certain standard framework, which I'll explain. We want to fit this. So that's what I'm going to try to explain, how this can be done. So I should say that actually there will be minimal new content, in a sense, in what we're going to do. What we're going to do would be look at critically, say we're writing down Benson's proof in a different notation. But, nevertheless, I think it's a point of view. This framework, to me at least, is a useful point of view, which unifies what one's doing with various other situations. And also points, as we perhaps be able to indicate at the end of the talk, towards constructions one might not have thought of otherwise, related to Ricci Fleur and similar things. I should also say this is an old manuscript, unpublished manuscript of Graham Siegel, going back to the 1980s, I think, which I managed to not throw away of all these years, which is somewhat related to this. This isn't the context of the days when Quillan did a calculation of this kind of Riemann surfaces, complex dimension one. Graham wrote this slicker version. But I don't really have a way of explaining how it's, well, I'm going to talk about it directly related to construction. So I'm not, I say this is really an expository talk. I'm not trying to say that I'm doing anything really exciting and novel, generally speaking. But I hope there's the point of view, may give a way of, especially for non-specialists, to kind of understand some of these things. So what I'm going to do next is to try to explain what I'm, indicate what I mean by this standard framework. So what I mean here is we want to consider the standard framework of respect to a group acting on a caler manifold preserving all the structure, both the complex structure and the symplectic structure. So g at z omega, so this is a caler manifold by holomorphic isometries. So we should think of, we should think of, for the moment, think of g as a compact lee group. In a moment, we'll be extending the idea as to have an infant dimensional analogy. But think of this as a, omega is the symplectic, well, a caler form on a space. Before you define a form of a caler form. Ah, yeah, that's right. Maybe it's a bad notation. So this, forget all about, this is starting, this is a, forget all about that. This is just saying we have a caler manifold and a group action. No, we want to. So the, the, the, the, um, the relevant object under a call is that, under favorable cases, I'll assume there is a, there's a moment map, an equivariant map from z to the dual of the lee algebra of g. Which is just, this is just the generalisation of the concept of a Hamiltonian, which should, if you had, since g was a circle, say, then we would know that the action would be generated. It would be generated by a Hamiltonian function. This is just the, the generalisation. It's to say, just to fix our notation. At a point in z, we have the infinitesimal action, which I might as well say rho sub z. The derivative of the moment map, the moment map is characterised by saying its derivative corresponds to the adjoint of rho. When we identify the tangent bundle with the cotangent bundle using the symplectic form with a suitable sign convention. Then the, the basic principle in this setup is that on the one hand we can consider the, we can consider extending this action to the complexified group. So gc acts holomorphically on z. And we can compare the, the complex quotient. It's to say, looking at the, the set of orbits of the complex group in z. Think of it as a compact group. If it's act, if this is a fine dimensional manifold and it's acting by our symmetries, a compact manifold, it better be a compact leak group. We will just have to verify that we have the structure piece by piece. But for the moment, think of this as a compact leak group. So the, the basic thing is we will compare the complex quotient, which as a set, this thing with the symplectic quotient, which is given by looking at the, the zeros of the moment map, which is a g invariant set, and dividing by g. So the, the, the general idea is that these things will not be precisely the same, but if we restrict to a set of stable points in our manifold to find that we, if we leave out some sort of exceptional or bad points in z, then these things will match up. That's to say, inside the orbit of a stable point, we'll be able to solve the equation mu equals zero uniquely up to the action of g. So I don't, I don't want to take the time to go into saying too much about reviewing too much of the detail about that. But just to say, in this situation, there's always a certain convex function, which one is relevant. So fix an orbit, fix a gc orbit, say gc times some z naught in z. Then we can define a function unique up to a constant defined by its derivative that if I, let's use the forward notation, if I, if I make a, if I make a infinitesimal change in z given by applying an element like that. Obvious, hope that notation makes sense. Then, then the change in f will be given by the pairing between delta g or i times delta g and mu of z. So essentially we're saying that we rotate this moment map by a factor of i, and then that gives the derivative of a function on gc. But actually this is g invariant, so it induces a function f from the symmetric space gc over g to r. And this always has the property that's convex from the, and the critical, so this is a convex function, which you can write down the critical points, which are just going to be the minima because of the convexity. So the minima correspond to solutions of mu equals zero. So that's to say the problem of understanding this identification, that's to say that's the problem of solving the equation mu equals zero in a gc orbit. Once we fix our orbit, it's the question of understanding whether this convex function has a minima or not. Now this is convex in the sense of the geometry of this symmetric space. It's convex along g at e6 in respect to the standard structure. So for example if g was su2, gc would be sl2c, this thing would be identified with hyperbolic three space, and this would just mean a function on the hyperbolic three space, which is convex along g at e6 in the ordinary sense. Okay, so maybe that's all, I could try to explain a bit more about this down the picture, particularly the relation with stability, but I think that will mean I don't have time to get to the main points, so I'll just leave it at that. What we would like to identify, we'd like to fix these ideas into a situation where we have a group hatching and a moment map, and such that the equation we're trying to solve is the equation that the moment map is equal to zero. Let's leave it at that. One sort of understands conceptually how that is related to notions of stability and so forth. So what we want to do is to put, so how can these questions be put into this framework? So the first question, evolving this constant-scaler curvature, this is something which I've been well-earned for many years, I'll get talks about that for many years, but not so much the second question, so that will be what is new in this talk. But let me nevertheless review what one could say in the first question first. What we want to recall is a bit of standard facts, some standard facts about the homogeneous space, called M, which is the, so let's suppose we have a two-end dimensional real vector space, called U, with a symplectic form. So let's let M be the set of complex structures, just a millennial sense, on U compatible with Omega. So this is the homogeneous space, SPNR, over UN, because the symplectic group, easy to see that transitively, on these complex structure was by definition, the unitary group is the stabiliser of the standard one. So all we want to say for the moment is that this homogeneous space has a standard, has an SPNR invariant scalar structure, the structure of a complex scalar manifold. So in a few minutes we'll come back to review more specifically how to write this down, but let's leave it at that. So to say for example, if N equals one, this would be SL2R over U1, which is, this is just the hyperbolic plane, so you think this is, in any of the models of that, either the disk or the upper half space model, the hyperbolic plane. So we're certainly familiar with the fact that this has got a natural scalar structure. But let's say we'll come back to review more specific formulae in a moment. Let's explain how it fits into our story. Now we'll go to a, let's take a, let's take a, we take a compact symplectic manifold, a fixed symplectic form, and also with a line bundle, which we'll still call L, so x symplectic, so this is omega, compact symplectic. This is line bundle with connection, with curvature minus i omega. So let's suppose we have that. Then at each point of x, we have a copy of our, of our space M, given by taking U to be the cotangent bundle of, at our points. So we get a, we get a space Mx, compatible complex structures on, say, T star, x. And we get a bundle, underlined M over x with fibre M. So that sections of this bundle are just compatible, almost complex structures. Then the, the scalar structure on the fibres of this bundle define, at least in a formal sense, a scalar structure on the input dimensional space of sections. That's to say, if we have a section, a tangent vector is given by a vertical vector field. If we want to multiply by i, we just use the complex structure on the fibres to do that. If we have two vector fields and we want to define the symplectic pairing, we use the structure on the fibres to define a pairing at each point, integrate over the manifold with respect to the standard volume form. So let's not write all that down. So that's, that's going to be our space zed in this, in this situation. So of course that's why I didn't want to try to fit any precise statements because now we have input dimensional spaces. We're just trying to develop the formal picture at the moment. What is the group? So g, this is essentially the symplectic diffeomorphism group, except we can, we'll keep the line bundle in the picture. So g is the group of automorphisms of L as a line bundle with connection but not necessarily covering the identity map of the base. So such things will preserve the curvature, so in particular they'll give symplectomorphisms. So this is just an extension of the symplectic diffeomorphism group by the circle by giving the way you lift the action to L. But it means that the Lie algebra of g is the whole space of functions because we haven't thrown away the, we don't have to divide by the constants. Write it in that way. So g match to x omega and the Lie algebra of g is the functions on x. So that's good. We're in this sort of standard picture, at least if we're willing to allow infant dimensional spaces. What you find indeed is that the moment map for, so this maps into, do you think this is space of volume forms? So the mu, if we identify, if we use the L2 in a product to identify the functions with its dual, and loose sense, then the moment map mu is given by the mu of j is equal to the scalar curvature of j. But we can, so let's write this, I'm going to say mu naught. We can always change the moment map by adding an element of the centre of our group. So we take the normalised moment map of mu of j, the scalar curvature of j minus the average value of the scalar curvature, which is just a topological invariant. So the problem of solving mu of j is equal to zero, that's just saying the scalar curvature is equal to its average value, that's just saying the scalar curvature is constant. So mu minus one of zero is equal to constant scalar curvature metrics. This somewhat ignores some technicalities. But if we want to go back to the problem we were originally interested in, we want to look at a subset of this. So let's say z, the gamma of m, set of all, almost. So we have a subset say z into the integrable, almost complex. So in fact for the problem we started with, we're only really interested in this subset given by the integrable, almost complex structures. So indeed we can look at finding the constant scalar curvature metrics as fitting into this framework. So I want to push on now to get to the main point. So the point is that how do we see, in this framework we don't see anything special about the scalar ion situation, we don't see the ding functional or anything like that. This is all. So in the Farno case we can study a different scalar metric on the space of integrable, almost complex structures. So let's go back to review in more detail about familiar fine dimensional homogeneous space, m. We can think of this as an open set in the set of Lagrangian subspaces of lambda n of our spectraspace u, complexified the, sorry, did we take Lagrangian subspaces in u, tends to c. If we have a complex structure then we can take the, we have the zero, we have the zero one and one zero parts of the complexification and these are Lagrangian, what we'd fix on say the one zero space is a Lagrangian subspace. So as such it can be identified by the plukar embedding in the projectivisation of the middle exterior power. So we can consider a bundle, say m hat over m, which is just going to see star bundle, which is just going to be the corresponding subset of these forms, just to the decomposable form. So this is just consistent to the forms that are equivalent to the standard model to the decomposable form to dz1, to dzn times a constant. So on this middle exterior power, we have a natural emission form, some normalisation that comes in depending upon the dimension. So let's define alpha bracket beta, so this is cn alpha wedge beta bar, where cn is chosen such that if I take alpha, let's call this thing alpha naught, so that alpha naught is equal to the standard volume form, that's cn. Normalisation with respect to dimension. So this then becomes a emission form on this exterior power. So the first thing I have, I have an alpha in m hat. So I look at the tangent space of m hat at the corresponding point. That contains alpha, because this is a cone, this thing contains alpha, so it contains c times alpha. But the basic fact we want to recall is that if we take this emission form, which is indefinite on the big space, if I restrict it to the tangent space of m hat, it becomes negative definite on the orthogonal complement of alpha naught. So, in fact, it has signature 1, what are the dimensions? So the basic fact is that alpha is negative definite on the orthogonal complement. That's to say, if we have any beta in saying that this thing, then if I remove the component along alpha, which is some formula like this, then that thing is negative, and zero if and only if beta is a multiple of alpha. So this gives one way, one standard way, of defining the invariant metric on m. We have this indefinite metric on m hat, but it's negative definite orthogonal to the fibres. So with a suitable normalisation, normalised by dividing by an extra power of the length of alpha, this induces a metric on m, which will be negative definite with our conventions. So, a metric. Now we can make a similar construction in this infant dimensional situation. So now let's suppose that c1 of x, the first shown class of any almost complex structure is equal to c1 of l. At each point in x, we're going to have, let's write it something like this, but x in, we're going to do the corresponding thing, but tensored with our line bundle, l at x. Because this is a projective, we don't care if we tensor our vector space with a one-dimensional space, our space of forms, to find it as above. So we get a bundle, with fibre, this slightly extended space. And this is precisely the condition that there is a global section of this bundle. So it implies the existence of it. So such things are going to be, such things are going to be alpha, forms alpha, and these are n-forms on x, with values in this line bundle, which at each point lie the side of these special decomposable forms. And just topologically, there are such things, but the invocability condition fits in very simply in this way. So it's a lemma one. So if we let, say, gamma j hat, what's it like on that thing? Z int, I call it. So z hat int. This is the set of alpha sections of this bigger space, where if we can take the exterior derivative on the n-forms to find coupling to the connection on L, this thing is a C-style bundle over our previous space. That's to say, if we have an integrable, almost complex structure, which is a section of the space underline M, we can lift it up to a solution of, to a section of a form, values in L, in this bigger space, solving this equation, the coupled exterior derivative is 0, and that choice of lift is unique up to an obvious constant multiple. In fact, this just essentially encodes one of the standard forms of the integrability conditions. Time is running short, so let me not say anything else about it. More, more, relevant is lemma two. But let's let now, supposing I have alpha in alpha in this, so I have such a form, I can consider the tangent space at alpha of this space. So this is the tangent space to z int at alpha. So it consists of, so this consists of forms beta, such that they're closed in the sense, and also at each, so the line satisfy the appropriate algebraic condition at each point. So at each point they lie inside the tangent space of the corresponding five-dimensional object. So we can redefine, we'll define a omission form on such thing, given by just integrating the construction we made pointwise. It's a beta one, beta two. This thing is a volume form, so we can integrate it over x. So lemma two, which is one of the crucial statement is that it's the same statement really as we said in the Findemental situation. So lemma two says that this thing is negative definite on the orthogonal complement of alpha in this tangent space. So we have this omission form on the space of bundle-varied forms which has got no particular completely indefinite. The crucial point that when we restrict to these particular forms which are both closed in the sense and also because it has algebraic condition then orthogonal to alpha we have this form becomes negative definite. So given lemma two then we, in just the same way as we've discussed before this metric on this, the total space of this C-style bundle induces a metric on our space of integrable almost complex structures. Just in just the same way as we said before. So from lemma two we get an induced of our cosine convention negative definite because we can multiply it by minus one, positive scalar structure of this space zint. So the point of the which is all I have time is to say the main point and then stop. The point of the talk is that if we use this metric on this form this is something we can only do in the farno world but if we use this metric then we fit into this standard framework in a way which is adapted precisely to the scalar Einstein structure and for example the ding functional convexity becomes an example of the general convexity we have in this sort of framework. And I said that's a bit of a cheat to say it that way because the proof of lemma two is really a disguised form of Vance's calculation proving the convexity. But I think this is an efficient way of understanding what's going on. Unfortunately I was hoping to do the proof of lemma two which is not very long but I don't quite have time to do it. Okay so let's just see what we do have time to do is to see why how the moment map comes about. But what is the action? So what is the action of the Lie algebra of our group C infinity of X on well first of all let's take all the n forms with value in L. What you do when you have a Hamiltonian what that enables you to do is to give a lift of the the simplex morphism on the base to the line bundle but you need to multiply in the line bundle by the Hamiltonian. So what happens is that you get the ordinary formula so let's call it say R H of R alpha of H this will be the ordinary formula we'd have involving the the lead derivatives so we take the Hamiltonian vector field corresponding to H contract with D of some of any alpha plus the other thing D of V H contracted with alpha but the way the line bundle comes in is now we put in plus I H times alpha but let's so in this situation see we just got an action on a vector space with a fixed Hermitian form so it's very easy to understand the geometry that the moment map is just given by find the right page of my notes from the moment map so this is a map from our space so it takes alpha to the element of the dual of the le algebra so it takes H to just the pairing between this say the infinitesimal action R alpha of H with 8 with alpha so this is just the integral of but now let's specialise to our situation where our form alpha is in the situation we're considering that means that D alpha is 0 so we lose that term so we get D of V H contracted with alpha plus I H alpha this thing but now we can integrate by parts to write the term involving the this exterior derivative of this but because D alpha is 0 that goes out as well so in the end we just get the very simple thing factor of I which somehow should have been taken away so in the end we just get H alpha using the facts we know about using the fact that alpha satisfies this condition that exterior derivative is equal to 0 so that's to say the moment map so mu naught of alpha is indeed just given by the the volume form determined by alpha which unravels into being the correct thing to define with respect to the ding functional and so forth so this is quite different you see from the moment map in the other situation which was given by the scalar curvature which is a much more complicated thing involving derivatives using this different metric we get a much significantly different conceptual picture to work with right but that's I have to stop in fact the fantasies years ago almost in the same direction yeah you know from a theorem that with respect to the morphism group of CP2 it's a more topical improvement should become a sort of analog of fracture conjecture that are a different morphism group but yes we will look for rotation group and then one can generally ask what a classifying space of simply a morphism groups and if I got not confused with science I thought more about varieties of surfaces, complex surfaces of general type it simply added sense and the conjecture is to be more the space of complex structures on surfaces should be related should be should be related