 to speak about this work. Yes, so I'm going to be speaking about joint work with Uri Durvan. And yes, it's about constructing extremal metrics on blow-ups. So a central theme in complex geometry is to search for canonical metrics, such as those of constant scalar curvature. And what I will do in this talk is discuss the construction of certain kind of canonical metrics. So these are these extremal metrics on the blow-up of a manifold in a point. And this problem really has a long history of study. So I will also survey some of the previous results before discussing the core idea, new ideas that we implemented that allowed us to go a bit further in the problem. And this is all joint work with Uri Durvan, as I said. Okay, so the beginning of the talk is going to be some kind of more background talk on canonical scalar metrics. And then I will gradually get into the problem that I'm discussing today. So throughout x is going to be a compact complex manifold. And so we have a holomorphic structure, and that can be encoded in an endomorphism of the tangent bundle, which satisfies that j squared is minus the identity. So this is really coming from multiplying by i infinitesimally. So we have this endomorphism called the almost complex structure. And what we want to do is that we want to put Riemannian metrics on this complex manifold that in some way are compatible with this complex structure that we have. So a Riemannian metric is said to be Hermitian if j is nice, if j is an isometry, so the length of a vector is the same as the length of j of a vector. So if you plug j into both slots of the Riemannian tensor, you get the same value as you did beforehand. And whenever you have a Hermitian metric, you can produce a two form from it by just putting j into one of the slots instead of both of them. And it's a quick computation to show that this then implies that this omega that you get from that is a two form. And the metric is called Kailer if this is a closed two form. So if this omega satisfies that d omega is equal to zero. So we're interested in Kailer metrics in this talk. And whenever you have a Kailer metric, you have this condition. So you then have an associated chromology class that you get in the second chromology of X called the Kailer class of the metric. So I will interchange saying of the metric or of the form omega because they determine one another. So this is the Kailer class of this data. And now if you have another Kailer form whose Kailer class is the same topological class, then this IDD bar lemma says that you can describe the difference of the two by a function. So there exists a function phi such that omega prime your new method Kailer form is the old one plus IDD bar phi. We use the notation omega subscript phi to mean this omega plus IDD bar phi. So that means that if we fix the Kailer class, we can describe all Kailer metrics in that class through functions. All right. And actually the set of functions such that this omega plus IDD bar phi is a Kailer form meaning it is a form coming from a Kailer metric. This is an open not empty subset of the smooth functions on X. So we really have loads of Kailer metrics in a given class once we have one. So yeah, the set parameterizes all Kailer metrics in that class and there's really an infinite dimensional set of Kailer metrics in the given class. So it's a very natural question to ask if we can find a canonical representative once we have fixed the class. So to answer that question, we have to really say what we mean by the question. So we want to ask ourselves what is really a good notion of a canonical metric. And I think in dimension one, there's a pretty good canonical choice and that's a metric of constant curvature. So the uniformization theorem gives a unique metric of constant curvature. And that gives you a good canonical choice in complex dimension one. In higher dimensions, however, you would then think that it should also involve some kind of curvature property, this notion of a canonical metric, but there are many curvature notions. So this could lead to many different notions of canonical metrics. So instead of surveying all the different kinds of notions you could come up with, I'll just jump straight to the ones that will be relevant for this talk. And both of the ones that I'm talking about today involve the scalar curvature. So from the Romanian metric, you get the curvature tensor. And if you average that a couple of times, you get a function on the manifold, the scalar curvature function. So one good notion of canonical metrics are those for which this function is a constant. So these are called constant scalar curvature metrics. We want to solve s omega phi equal to a constant. And this is actually a predetermined constant. It won't be so important for us what it is today. So predetermined from the data of the scalar class and the manifold x. Normally, when I talk about this sort of stuff, I would usually just focus on these metrics. But for what I'm saying today, it's kind of important to include a more general type of metric than constant scalar curvature ones. And these are the ones featuring in the title, so extremal scalar metrics. So these are due to Calabi in the 80s. And they can be defined as follows. So let's let d omega be the following operator. So d omega action functions. And it takes a function. And then you compute the gradient with respect to the metric. And that's a section of the tangent bundle. And we're on a complex manifold. So we can take the one zero part of this. So there's an isomorphism smooth bundles from tx to the one zero part of the complexified tangent bundle. And this is then a section of the holomorphic tangent bundle. So this is a holomorphic tangent bundle. So it has a d bar operator. And you can apply d bar to that. This is second order in f. And the kernel consists precisely of the functions so that this section of the holomorphic tangent bundle is holomorphic. So meaning that this vector field that you get is a holomorphic vector field on x. And now extremal scalar metrics are those which land in the kernel of this. So here I should say if we change omega to omega phi, both the d omega and the scalar curvature s omega is going to change this d omega phi s omega phi equal to zero. So you want it to be this in the kernel of this operator for the new metric omega phi. So what this is saying is that the scalar curvature is a potential for a holomorphic vector field. If we take the one zero part of the gradient of the scalar curvature, then this defines a holomorphic vector field on x. This was suggested already by Calabi. Yes. So Calabi came up with this when he studied the energy functional associated to the scalar curvature, meaning the square of the L2 norm. So you do the integral of the scalar curvature squared and you look at the critical points of that. And if you're on a compact manifold, you get this equation out. So did he even produce examples of classes which doesn't admit constant scalar curvature metrics when he? Yes. So the first example of something like this that he produced, so he produced explicit examples of manifolds admitting these metrics, namely Hiddesburg surfaces. So on Hiddesburg surfaces that are not p1 times p1, you can't have constant scalar curvature metrics, but he produced extremal metrics in all scalar classes on those manifolds. So they admit extremal metrics, but not constant scalar curvature metrics. So he used some kind of ansatz to create these metrics because you have a lot of symmetry in that situation. And essentially solving the equation using that ansatz becomes an ODE rather than a PDE, and he was able to solve that PDE. But yeah, the study of this equation maybe comes from already maybe being motivated by the study in constant scalar curvature metrics. It's then natural to look at this functional, which is the square of the integrating the square of the scalar curvature. And critical points of that functional are precisely these these metrics. And I'll also mention another reason for studying these coming kind of from the linearization of the scalar curvature operator. Later, even if you start with constant scalar curvature metrics, it's kind of natural to allow extremal metrics in general, but I'll come to that in a second. Yes. So these are the metrics that I'm interested in today. And for a lot of the talk, you can just think of constant scalar curvature metrics, but many of the results only apply for extremal metrics. That means you can't just replace everything by constant scalar curvature metrics. So one has to work in this generality reading. But yeah, this is some finite dimensional space instead that you're trying to hit rather than hitting a constant. All right. But it's also kind of a moving target, because this d omega depends on omega. So yeah, but there are ways to describe that in a good way. Okay. So an important aspect of this theory is that the existence question is deeply related to algebraic geometry. So what we would like to answer is given a scalar manifold and a scalar class on it, does that class admit an extremal metric or not? And that question is very, and yeah, you can find examples where you admit it in all scalar classes. Some scalar classes are non-numbered scalar classes. So we would like to figure out when you do. So a central conjecture in the field is the Altium-Dolson conjecture. And this says that the existence of these metrics should be related to kind of algebra of geometric conditions of stability. And the predominant notion is called case stability. And there are variants of it, but I will be very rough about what this is today for those who are experts. I'm not maybe using exactly the right words, but what does these notions involve is that you have a certain class of degenerations, so that's called test configurations. To those, you associate a number called the Dolson-Tutaki invariant. And then you ask for that number, so case stability asked for that number to always have a particular sign. So that's what's involved in this algebra geometric notion. And the conjecture is then that a polarized projective manifold admits a C-C-K or extremal metric in the first turn cos of L, if and only if it's case stable in the C-C-K case, or relatively case stable in the extremal case. And there are variants of this when it's not, when you're not in the projective setting and so on, but maybe I'll just be happy with this now. And the bottom line is that it should be captured somehow algebra geometrically, even though the question itself is solving a PD for a metric. And this has had a huge history of study. Okay, so the problem that I'm considering today is a perturbation problem, meaning we're going to start with a metric that solves this equation or is close to solving that equation and then we want to perturb it in a different setting into solving it. So we changed the situation slightly and then we want to perturb to actually solving it in the new situation as well. And in those kind of problems, the linearization of the equation is really key. So the Liknerwitz operator is the operator you get by taking this operator we had when we defined the extremal metrics and then using the formal adjoint of it to create a fourth order operator, d star d. So d is this operator I defined earlier. And this operator is deeply linked to the linearization of the scalar curvature. And moreover, the kernel is precisely those holomorphic potentials. So it's potentials for holomorphic vector fields on x. And yeah, the linearization of the scalar curvature operator is essentially this operator, it's up to a sine and the term of lower order. But the important thing to take from this is that the kernel of this operator here appears in the study of the extremal equation. And therefore I will be talking a lot about obstructions and so on coming from holomorphic vector field fields on x and on the blow up in what's going to come. And the reason I'm doing that is because it is linked to the linearization of the equation, which means it's very much linked to what we can solve at the how we can perturb at the infinitesimal level. So this is why obstructions that I'm talking about is going to be linked to the holomorphic vector fields on x. All right, so now I want to go a bit into the problem that I'm actually considering today. So I'm just going to call it the blow up problem. So first I'm going to say what the blow up is for those who don't know. So we start with a with really a complex manifold and we pick a point. Then one can define the blow up of x in that point. And it's a manifold which satisfies that you have changed nothing outside of p but you have replaced the point with sort of all the directions out of that point. So the preimage of the point via this map that you have from the blow up to the original manifold is just a copy of pn minus one called the exceptional divisor. So really the local model here is blowing up cn in the origin and this consists of the pair of points in pn minus one cross cn so that the cn component lies in the line determined by the point in pn minus one. So the cn component is a constant multiple of the pn minus one component. So if you project to the second factor there then this is a bi-holomorphism away from the origin. So if the second factor is known zero then the point in pn minus one is predetermined. So we have a unique point in this product in the blow up but if we look at the preimage of zero then it's all of pn minus one because that parameterizes the lines through the origin. So the origin goes through is in all of those lines. So what we have done is that we've replaced the origin with a copy of pn minus one that's the local model for the blow up and that's you can do that globally via gluing. So this is the manifold that we're interested in constructing extremal metrics on and we need one more piece of data so we need to also fix caler class on the blow up and for that we need to understand a bit the caler comb of the blow up so meaning the kind of caler classes you can get on the blow up. So first of all the second cohomology of the blow up changes by adding on the Poincaré dual to the device you've added. So we have this exceptional divisor in the this copy of pn minus one in the manifold and the second cohomology changes by the Poincaré dual to that new sub manifold new divisor. And if omega is a caler class on x then the pullback to the blow up is going to be on the boundary of the caler comb of the blow up. Every sub variety will have non-negative volume but the exceptional divisor will have zero volume for this class. So in order to actually get a caler class to move into the caler comb you need to give this exceptional divisor a bit of positive volume and you can do that by subtracting a small multiple of this Poincaré dual to the to the exceptional divisor. So these classes omega epsilon on the blow up they will be caler for all positive epsilon sufficiently small and it's in those classes we are going to be interested in constructing these extremal caler metrics. So yeah that's a caler class on the blow up. So the question then that we're interested in today is under what conditions does the blow up admit the cck or extremal metric in in these classes for all epsilon positive epsilon sufficiently small and here we're only going to be interested in when epsilon is really really small not necessarily all epsilon for which this is caler. So you can think of that as when the volume of the exceptional divisor is really really small so we are in some sense close to our original manifold we've only ordered on this little divisor with very little volume and in that way we can think of it as a bit of a perturbation from the original manifold that we had. So now this question has a rich history of study. There are several constructions constructing extremal metrics on the blow up in these classes and what these constructions so these are constructions of Editsport card Editsport card senior and second heedy and they start with an extremal metric before blowing up and by a gluing construction constructs new extremal metrics on the on the blow up so these give sufficient conditions for the blow up to admit these extremal metrics. There are also necessary conditions by taking the algebra geometric point of view due to stopper originally and then stopper and second heedy in the extremal case and they provide necessary conditions so in particular what they show is that if you start with something that is strictly unstable so in some sense far away from admitting an extremal metric then that will also be the case for the blow up. And yeah I should also mention that there are versions of this for blowing up higher dimensional subvarieties by Sayed Ali and second heedy and also some work about Jimoto but today we are focusing on just blowing up a point and the strongest result in the point case is due to second heedy who completely settled a YTD correspondence in the CCK case when the dimension is at least two so saying that it's exactly captured by an algebra geometric criterion and saying what that algebra geometric criterion is in this case. So I'm going to go a bit deeper into these constructions just to say exactly what was known before a result and then I will state what the main result is. So the first case is when the automorphism group of the original manifold is discrete so the assumptions by Eritz-Pochard are that X admits the CCK metric in this class omega and that the automorphism group in this is discrete and in that case any point will do in the construction meaning if you pick X in that way and pick any point then for all sufficiently small epsilon the blow-up admits a CCK metric in these classes making the volume of the exceptional device are really small these classes omega. So there are no restrictions on the point and it will do but this changes when the automorphism group is non-trivial so when the connected component of the automorphism group is non-zero. So the simplest case is when you pick the point to be fixed by a maximal torus in the reduced automorphism group of X in that case there are still no obstructions provided you allow extremal metric. So if you start with a CCK metric it might be that you have to allow for an extremal metric but that's all you need to do so you will always be able to produce an extremal metric starting either from a CCK or extremal metric and the key really to understanding why there are there's no obstruction in this case is that all of kind of this action of the of the maximal torus lifts in this case so the all the relevant holomorphic vector fields will lift and the extremal equation is then kind of unobstructed at the linearized level in this case. So yeah just to summarize what what this construction is so both by Ritz-Pokart Singer and Sekihidi using kind of slightly different constructions suppose X admits an extremal metric in a given class and pick a point fixed under the action of a maximal torus in the reduced automorphism group of X then for all sufficiently small epsilon the blow up admits an extremal metric in these classes that we consider. So there the construction still works and we can always construct an extremal metric in these classes. So what happens if if this is not the case so if the point is not fixed by a maximal torus then the the the real issue comes in so that's where you actually see an obstruction to solving the equation and it might be that you can't. So the core reason for this can really be understood from the linear theory so the reduced automorphism group of the blow up can be seen as a subgroup of the reduced automorphism group of X and it's really the subgroup generated by the vector fields that vanish at the blown up point and remember I said that the linearization of the scalar curvature is very related to sorry to holomorphic vector fields the kernel of linearization is generated by the potentials for these kind of vector fields. So what we're seeing here is that if we are blowing up a point which is not fixed by this maximal torus then there's going to be a discrepancy between these two things which means also that there's a discrepancy between the mapping properties of the linearized operator before and after we blow up and that's really the core reason on the analytic level why sort of the construction is obstructed in the case when you blow up point not fixed by a maximal torus. So yeah just to say this slightly differently again if you pick a point which is not fixed by a maximal torus then there are holomorphic vector fields on X that won't lift to the blow up so these two automorphism groups won't agree and there will therefore be a discrepancy between the mapping properties of the linearized operator before and after blowing up and you really want to see the blow up situation as a perturbation of the of the situation before blowing up but now the mapping properties don't match so that's why you get on the analytic side some issues. I suppose that the, sorry if you don't mind me, I suppose that the presence of holomorphic vector fields makes extremal metric a more flexible, they allow for more flexibility in these metrics right? Yes I mean you allow more flexibility, that's true so if you did sort of the CECK problem then you would have even more sort of obstructions to solving the thing. You can think about doing something like changing the Keiler class instead of blowing up which is an easier problem like Ribbrym Simanka and there if you start with a CECK metrics but you have holomorphic vector fields then there are kind of directions you could go where you would admit the extremal metrics that are not CECK so it's not true that in all directions you could you could construct the CECK metric and in that situation it's precisely captured by the classical to talk invariant whether or not the metrics you construct are CECK so it's kind of the same philosophy there's yeah if you want to solve the constant scalar curvature metric you would like your linearized operator to have kernel or co-kernel the constants but if there are holomorphic vector fields then the co-kernel is bigger than that so at the linearized level you can't solve the equation you're trying to solve but you can solve this equation which is determined by precisely the co-kernel you're having and that's the extremal equation and the philosophy is a bit the same here there's going to be a discrepancy between what we see before and after blowing up so we can solve everything modular the things that are on x but that's not really what you want to solve on the blow-up and that's why there's there's an issue somehow but maybe i will have time to say a few more words about that later as well yeah so so this breaking or whatever of the of the ultramorphism group this discrepancy between the ultramorphism group is really the core analytic reason for for why you see some kind of obstruction maybe very quickly so the goal is to obtain some kind of geometric understanding of this obstruction so when do we actually have a solution despite the fact that it in general is obstructed so from the Jartian-Dolmeson conjecture you could guess at what the obstruction should be it should be given by case stability and there's kind of a natural candidate for what should constitute the obstruction so i mentioned this algebra geometric work of supper where he produced test configurations on the blow-up from test configurations before blowing up and the the leading order term in this numerical invariant that we had the Dolmeson to talk in variant on the blow-up is the one you had before blowing up and if you blow up a point that's not fixed certain product test configurations will no longer be product so this is maybe for experts but things that were allowed to have zero to talk invariant before are no longer allowed to have zero Dolmeson to talk invariant but the leading order term is still zero so the now the sub leading order term will matter and it's these kind of test configurations that you should test but that's maybe maybe for those who know a bit more about case stability but the algebra algebraic geometry is telling you what should constitute the obstruction to solving the equation but the the goal is then to kind of do this analytically to bridge the gap between those things and in sake he this approach he his approach allowed him to allow him to relate the conditions on on the point to case stability in this way so in the in the cck case when excess dimension at least three he showed that the blow-up and it's a cck metric if and only if the matter for this k-polis table so if you start with an extremal metric then what's kind of remaining in full generality is the dimension two case and the non-cck extremal case and moreover he gave a finite dimensional gite type condition that captures precisely what is needed to check k-polis the building this case so it's you don't need the full full definition of capability there's really like these concrete test configurations that you need to test on and they are the ones I talked about in the previous slide so the main result of my work with Rui is an extension of this to the non-cck case and dimension two and also to certain manifolds that themselves initially don't admit an extremal metric and that's an interesting case especially for examples so now I will state state this and some consequences and then I will talk a bit about some aspects of the proof so the main result is a y2d type correspondence for the blow-up of certain k-manifolds so either we work in the setting of those previous constructions of a Ritz-Bakard singer or Sekihidi that is when X admits an extremal metric in omega or we work on a small deformation of such a manifold so X will be very close to having an extremal metric but won't actually have an extremal metric itself so collectively we'll call those manifolds analytically relatively case-emistable it's kind of a horrible thing to say out loud but it implies the relative case-emistability of X omega and so it's a kind of analytic way of having this because you can you can see your manifold as a smooth deformation of something smooth that admits an extremal metric whereas in general you would expect some kind of singular thing anyway so the main result is that in those two cases so if X omega is analytically relatively case-emistable and you pick a point on X and you consider the classes that we've been considering then the existence of an extremal metric in these classes whether the volume of the exceptional divisor is really small is precisely captured by this algebra geometric criterion relative case stability and moreover this case stability criterion is really an explicit finite dimensional sort of condition in this case so it's precisely those type of test configurations that I was mentioning before so before talking about some aspects of the proof of this I just want to mention some consequence so there's this conjecture of Donaldson's which says that on any scalar manifold it might not admit an extremal metric or anything but even if it doesn't you can find a collection of points on it so that if you blow up the manifold in those points then that blow up admits an extremal metric that's a general conjecture due to Donaldson and we show that for analytically semi-stable manifolds you can always find a point where in his blog we we get an extremal metric so the kind of criterion you need in the obstruction can be satisfied for some choice of point so yeah we show that if you are analytically relatively case semi-stable then you can find a point so that the blow up admits an extremal metric in these classes so in this for these very special manifolds there there is a good point that which one can use in this conjecture of Donaldson's this is far from giving any insight to how to solve this problem in general but it solves it in a very special case and moreover the fact that we can find that point allows us to construct many new examples of extremal manifolds so this relies on recent work by by really many researchers that is Tejita and Chevsov and and really very many researchers who have worked on explicitly verifying case stability for final three faults and in their work one finds many families admitting strictly case semi-stable manifolds degenerating to a smooth k-polis stable central fiber so a killer Einstein central fiber and for those examples we can apply our construction so as a byproduct sort of of their work we get many examples yielding extremal metrics on the on the board and often the semi-stable guys won't actually have any automorphisms so often these metrics are actually cck metrics okay so that's what i wanted to say about the main result and some of the consequences and then i'll use the remaining time to talk a little bit about some of the aspects of the proof but maybe are there any questions now about about the statement or or anything so when you mentioned the main result that manifold is deformation of deformation which has an extremal metric so what do you mean by deformation there like so it's really you can you can think of a smooth manifold right where you have a let's say one parameter family of almost complex structures okay where you're where parameterized by c or disc in c or whatever where over zero you have an extremal manifold and then for all non-serial values you have the same bihologomorphism class and it's this this these non-serial values that you're interested in constructing an extremal metric on so they are they won't have it themselves but they are infinitesimally close to to admitting one okay kind of infinitely close to the to the central fiber right thank you any other questions all right i'll i'll go on then so i'll explain the core new idea so the core new idea is that we want to work with a fixed symplectic manifold in the gluing process and instead let the complex structure vary so in the beginning i talked about this point of view where you use functions to change the cali metric by changing the symplectic form there's another way of using functions to change the metric by changing the almost complex structure instead and we take that point of view so when you blow up a complex manifold the complex structure will depend on the point you choose so i mean the example is blowing up three points in a line on p2 versus blowing up three points in general position those complex structures will be not be bihologomorphic but as as a smooth manifold it's always the same and in the symplectic category as well so as a smooth manifold the blow up is just connected sum of m with cpn with the opposite orientation and so we're going to think of it in in this way that we're always having the same smooth manifold we're always going to pick the same point on that and we're always going to take the same symplectic form but we're instead going to let the almost complex structure vary so that's what we want to put ourselves in the position of initially and that will allow us to see everything as a perturbation of the situation when you blow up a fixed point of this torus action and that's the unobstructed case we want to see everything that we're doing as a perturbation of the unobstructed case maybe i'll be a bit quick about this part but what we do is that we first construct a symplectomorphism that take this fixed point to a nearby point q instead so what we're really interested now is we should think that we want to blow up q and we're going to make q become p by applying some sort of ephiomorphism and one can do that using mozer strict so then we can pull back the almost complex structure that we had to create a new almost complex structure jq and blowing up m with this new almost complex structure in p is the same as blowing up the old almost complex structure in q so the point is that we have applied some we've constructed some sort of symplectomorphism that allows us to to view blowing up in q as blowing up in p just with a different almost complex structure and that's that's the thing we want to want to do we want to always work with blowing up the same point p no matter what and then let the q variance that jq variance that with yeah and since this is a simple ephiomorphism then the two caler manifolds we have before and after pulling back this almost complex structure is the same so we're kind of starting with the same sort of data but we're just having a different viewpoint so now we can take the point of view that we are blowing up a fixed symplectic manifold in one given point and what is changing is the almost complex structure not the point so we can encode that in a map q goes to jq from some ball in the vector space to to the space of almost complex structures compatible with our symplectic form and I mean once you have this point of view you can even let the complex structure vary in that in that way that I was just discussing so we can even allow the kind of bihologomorphism class to change a bit as well and that's what allows us tackle this sort of strictly semi-stable case okay okay so so we now have this way we're changing the almost complex structure instead of the point encoded in some kind of map and it parameterizes the the domain of the map parameterizes all nearby points to a fixed point by this action of the maximal torus and all nearby complex structure in this kind of current issue model and by work of Sekihidi and Brunner we can we can do this in a way so that we ensure that the scalar curvature of the metric we get lands in the space of holomorphic potentials for when q is equal to p so this is this is really only important here for this second situation where we're allowing the bihologomorphism type change that we can't necessarily solve the extremal equation but we can solve that the scalar curvature lands in the space of holomorphic potentials for the central fiber so there's some sort of finite dimensional obstruction space that we can have the scalar curvature in and that's a model for what we're going to do on the blow up as well so that's actually the next step so the next step in in the approach is that we divide the problem up into two more manageable bits so if we directly try to solve the extremal equation then we know we're going to run into problems so instead what we do is that we solve an easier equation first on the blow up and then we analyze when we have actually solved the equation we wanted which is the extremal equation after that so yeah we first solve a more general equation on the blow up than the extremal equation and then we analyze when this more general equation actually solves the extremal equation and this is sort of a this is a strategy that is really useful in these obstructed perturbation problems that you kind of solve everything that the linearization that the linearization is allowing you to solve that's what we're going to solve in the first step and then that's going to be easier than solving the equation you care about then you can try to analyze when you actually solve the equation you care about afterwards so from the analytic point of view this is a bit easier you've made the well the first step is easier anyway but you're not solving the equation you care about so you have to do something more but so I know this kind of strategy of dividing the problem up into two bits from ideas going back to at least Donaldson but I'm sure it goes further back than that as well but that's where I know it from so the first step in our in the approach is to obtain a lift of this map we had before so before we had a map from some ball in a vector space parameterizing nearby points and nearby almost complex structures to the space of almost complex structures on M and one could do that in a way so that the scalar curvature landed in this finite dimensional space and what we want to do is that we want to elevate this to the blow up so we want to find for every q and sufficiently small epsilon an almost complex structure j q epsilon on the blow up so that the scalar curvature lands in h bar epsilon where these are the lifts of these guys for the central fiber meaning for when q is equal to p and so this is I should say this is the kernel of the linearized operator on the central fiber when q is equal to p so in q is equal to p that's the unobstructed case corresponding to blowing up an extremal manifold at a fixed point and we're we're going to solve for everything being in the kernel of that operator which is what the linearization allows you to do which is not the actual equation we want to solve but it's an easier equation to solve so yeah omega epsilon is the symplectic form defined using using the earlier words that I mentioned so the advantage of trying to do this of working with sort of thinking of the symplectic manifold is as fixed is that now vector fields that don't correspond to holomorphic potentials on the on the fibers q not equal to p now now they actually have a natural lift so we can just use the one for when q is equal to p yeah so that there is a lifting procedure where you just use the one for the central fiber and the space of these lifted functions is precisely what you're seeing as the co-kernel of the linearized operator to leading order so that's why solving you know prescribing the scalar curvature modulo these things is possible at the linearized level and but I should say that not all of these will be holomorphic potentials on the blow up they're just some Hamiltonians but they don't all produce holomorphic vectors okay so just to reiterate what the the goal is in case I lost you a bit we want to construct an almost complex structure on the blow up so that the scalar curvature lies in this finite dimensional space corresponding to the kernel of the linearized operator when q is equal to p and that might not be an extremal metric but after solving that equation that's when we analyze when we have solved it and relate it to to this algebra geometric criteria case stability so to do the to solve this equation what we do is that we perform exactly the same construction as on the central fiber on the on the non-zero fibers and to actually solve the extremal equation we need this data curvature to land in some sort of subspace of this of this space but the point is that it's easier to solve s omega epsilon j epsilon q in h bar epsilon than it is to solve the actual extremal equation and so to do that we start with our initial j q that we construct almost complex or to j q that we constructed before blowing up we then apply most restrict a couple of times again to to obtain an a simplectomorphism to this m omega epsilon that we always want to use and if we then pull back the scalar structure we get an initial approximate solution to to this equation and really what we have is that the the structure that we have is a perturbation of the one we have on the central fiber and now we we do the construction of sekihidi in in in this whole family and we can then view it as a perturbation of the setup on the central fiber and we can solve this more general equation than than the extremal equation so at this point we have not necessarily solved the the extremal equation and that's what we want to understand now and the reason that this is this equation here is easier to solve than the one we're actually interested in solving is that we're solving kind of what the linearized theory is allowing us to solve okay so that kind of completes the outline of the of the first step in the approach so the second step in the approach is relating the construction to case stability and understanding when we have actually solved the equation that we wanted so um I only have like five minutes left so I'll be or four so I'll be a bit brief about this step but what we have is that we have um parameterized all nearby points to q and nearby complex structure by a ball in the vector space we have solved this more general equation than the extremal equation and now we want to understand when this actually lies in some sort of subspace um floating around in this setup is some kind of uh uh action of a torus on this ball so what we actually want is not just to solve this equation we actually want to find when there is some other point in the same orbit as q that solves this equation and the key to achieving this is that we can actually view this as looking for a zero or a critical point of a certain moment map and so a key proposition is that the image of the map that we've constructed to the space of almost complex structures on the blow up um that image is a symplectic sub manifold and now the scalar curvature uh is a is a moment map for the action of the Hamiltonian symplectomorphism group on the space of almost complex structure by Fajiki and Donelson and if we now project to the space the obstruction space h epsilon then this is a moment map for the restriction to this uh torus that we have of that action and then since we have a symplectic sub manifold um this also then holds on the image of this embedding so the option is that we have managed to put ourselves in a situation where solving the cck or extremal equation becomes a sort of finite dimensional moment problem and that's what allows us to relate things to g it or to relatives so uh just to briefly outline the the remaining steps of the proof so the functions in in the space age or age epsilon induce test configurations for the blow up this was for known through work of zikihidi and now there don't seem to talk in variant is precisely the value that you get out of the moment map that i just discussed so the upshot is that the value of the moment map can be realized as the algebra geometric quantity that we wish to relate the construction to and if you analyze the Hamiltonian functions um this um produces another point in the same orbit as the as the point q we started with uh where the Hamiltonians orthogonal to those that are holomorphic potentials vanish if we assume uh case stability this is a technique due to Durban uh in a in a different work so assuming relative case stability we get that the scalar curvature lies in the space of holomorphic potentials at the complex structure over this q prime point and that is saying that we produce the extremal metric under the assumption of relative case stability so it lies in this finite dimensional subspace um of this um construction space which is what we wanted um so the final thing i want to say is just that um we can also obtain explicit expansions that give the exact criterion for stability in this case this is really an explicit expansion of holomorphic potentials or almost into tachy invariance and this uses really similar ideas to the analogous statements in the work of um of Sekihidi um but yeah the the key is really to land ourselves in this uh in this point where we can uh in the situation where we can think of it as a sort of finite dimensional more zero moment map type problem okay um so yeah sorry that the end was a bit rushed but that's what i want to say so thank you for the attention thank you Lars thank you and so i'm gonna