 I think one doesn't need to necessarily see the people. Can I see the old page? Sorry for the technical issues. So this is the first lecture of the IGAP ICT seminar. So the seminar will be lecture by Lothar and Alina about moderate space of sheets. And this is the first talk by Lothar about the anthropological invariance of moderate space of sheets and back of it any much. Okay, thank you very much. I mean this, this is just an overview talk. So later there will be more lectures with more details. Therefore, I also have the slides, I mean later we will have lecture talks to explain this more slowly. Now, maybe this will be better to be much of a material, to be short time, but we can just try. So what are we talking about? So we want to look at, want to compute and study topological invariance of moderate spaces. So, so let S be a smooth, objective complex surface, H and M for line compass on the surface. We assume that the first 30 number of S is zero, and the geometric genius, which is bigger than zero, that means they exist in non-vanishing form of two falls for S. And so then we can look at the monolith space of stable sheets on S, the strength of C1, C2. So, so rank are H semi-stable sheets. So I have here written down the definition of what the semi-stable sheet means. So it means that the sub-sheets are not too large in the sense that they don't have too many sections, even though they're twisted by power and line-up. So there exists a so-called coarse-modelized space for this, for the G, so there's a space which permacryles all of them up to isomorphism and just some nice properties. So, this is the monolith space, and we put the MSH as one of two. It usually will be singular or very singular, but not in many forms. There's a number attached to it, which is the exact dimension, which is the dimension that it should have if it was a nice space, but when it's not nice, it would have a different dimension, but I will talk more about that later. Obviously, so the virtual dimension is this 2R C2 minus R minus 1, C1 square plus R square minus 1 to the whole multiple of S, where these C1, C2 are the term classes. Remember, I mean I bet you most of you know what they are, but there's some homology classes which tell us about how far away the boundary is from being driven. And, you know, here on the surface, you know, we can turn the set in the section of C1 square, the number if we evaluate it on the fundamental class, like here, and the form of the second term class can also be built as an integer. And so, then this is just a number. And so, let's first look at the simplest case, which is in some sense the case of rank one, although first doesn't look like it. This is the case of Hilbert's scheme of points. So, you know, SN, the Hilbert's scheme of zero dimensional substance of length and on S. So, this is a nice scheme, which parameterizes, you know, the zero dimensional sub scheme so what are these. And the other point of it would just be a set of n distinct points on S. But these points can come together in which case, we have a non-reduced scheme structure. And so, in principle, such a sub scheme will be given by an ideal sheet, I see, is a structure sheet of S. So, essentially, the whole model can be finished on the sub scheme, the quotient with the structure sheet of the scheme of the sub scheme, and the length and is the total dimension of this structure sheet as a vector space. Just, you know, it's an R team. Bring it to its final dimension and that's the number of points. Count it with my business. So, it is a random fact to the property that this Hilbert's scheme is smooth productive and has dimension to it. Now, what does it have to do with more life space of sheets, but take the more life space of rank one sheets with respect to any number on S. With the second chance to be as a number that is equal to the item of it for the Hilbert's team of secret points by just associating. So, the sub scheme is ideal. The ideal sheet will automatically stable, because there's no condition and the, the, the second chance to be equal to the next. This is the case and in this case you want to complete some logical invariance or the simplest topological design one to think of is a topological oiler. But in numbers, and it's all something which is very easy to complete for any kind of reasonable logical space. And so we want to continue this. And so the answer is something I was very long time ago. So we can make a generating for this oiler numbers. So we compute the order number of the Hilbert's team and points. And then this is given by this simple generating function. So we have a product. We see that the other number of the human being depends only on the order of S is also that this is very complex formula which gives them all once. And now we could ask ourselves, are there similar formulas in higher rank. No, maybe I mean that this nominator is essentially a model of form if the data function up to this. And so therefore, and so the. Now that the from physics now becomes a paper. Which is also very old, and which gives an explicit contextual formula for the generating function of the other numbers in the rank two case. In terms of other. I will say to work. For the forms but not very much. So let me try to first state this result. So in the talk, we assume that so I say was equal to semi stable, which is the condition. One, so it means that the model of stable sheets is already compact. And, and we assume for simplicity for now, that there exists a smooth connective curve in the canonical in your system. So that means the occurs a smooth connective curve as a zero set for all of them. This will be two cases. For instance, for many others. So, and then we write this before the square is the first term that square. And we also get a lot of physical mind of the dimension of the modernized space. And so we have a dimension. And then we again, at this data function of normalize by taking away this. And we have another model of form, which is a better function that affects, which is just the sum over all integers. Thanks to the n square. And then we have this projector. This formula. I don't think that anyway, so which is. And then again, in generating function for the smaller numbers of the smaller spaces of the site differently. The order number of the one life space of rank two sheets. We'll be determined as follows. We write down this function. We want one over to a bar x square. And then this other term to a bar. x to the fourth square. And to the power to serve in the section. So these are two basic. So the logic of the. surface. And just in terms of this, one can. So one writes down the special and then one makes the position of extra the B, the expected dimension of this model. Okay, so it's a very nice formula. It's under these conditions for all. You know what the order is. Okay. So you want to first see in what sense this formula might be true. This is the two, and we want also to look at the finance of this formula, not in the sector of data. So it's kind of the other number of a finer. And we want to generalize it higher. So let me first make you another. So, first, I want to say how I want to perfect this one. So, as I said, this model life space is usually quite singular. And it's dimension might be different. As a matter of fact, it can only be larger not smaller than the virtual dimension, but so. And so, how can one describe the situation. The model space is singular, which means that the tangents dimension of the can be singular that means the mental and space to be larger than the dimension of the space. So at every point in the model space, we have a tangent space, which is given by the phase three. So, so, and your section for the natural abstract space would be the x to the same subject. And the virtual dimension actually turns out to be the difference of these two. The dimension of the tangent space, minus the dimension of the obstruction. It is a, maybe a non-trivial. I mean, it is the fact that this dimension is always constantly independent of that. But the individual dimension, the dimension of the tangent space, the dimension of the obstruction space. And actually, one can use this. So what is, there's something final is to this is to an issue map, which locally describes the model space near F. There is an analytic map to an issue map from the tension space from an open neighborhood of zero, the tension space at F to an open zero neighborhood of zero, your section space. Such that an analytic neighborhood. The quality neighborhood of F in the model space is isomorphic to the inverse image of zero. So in other words, you could say that locally. Indian ecology. And is given by the dimension of the obstruction space equations in effect in the in a fine space of dimension. So in particular, if your section space is zero, then the model space, not single dimension. So, now, however, the model space singular and it's very difficult to hear nice singular spaces. So there is. The notion of the perfect structure, which allows you to somehow that the model space even though it isn't. So, so what is to this case is that there's a context. So two vector bundles, one vector one to the next on M. So on the model space, such that for every point in the model space. The tension space is the kernel of this path and the obstruction space is the copper. So that means we have in one. You know, in one sense, or in one step, just this two vector months, whether the map between them, tell us everything about the tension and so structure. So, although, you know, the, so the tension of structures can jump in dimension all over the place but there is two vector months with no everything. And this information is enough to get some way. I think the first one that will actually things you can say, you find the virtual tension bundle, just as the difference between these two vector months and key theory. So, as a formal sample of vector bundles. The virtual dimension that I have said before can also be described as the rank of this, which is different of these two dimensions. Very interesting thing that one can find by that. No, in this situation, which was defined by is the virtual fundamental tasks. So this is a class in homology in the homology degree, which corresponds to twice the virtual dimension of the model space. It is like it was the fundamental class of a manifold of compact variety of a smooth compact variety of dimension. And it behaves also in many ways, like the fundamental tasks, like actual fundamental tasks. And so, and one can just to think of it, you can think of this. The distance of the perfect abstraction theory is some kind of virtual form of smoothness. So somehow, the modelized space is not smooth but remembers in the form of this obstruction theory that it wanted to smooth and one can use this intention to do things if it was. Okay. Now, in particular, we can use this to define the virtual version of the owner. So the classical index theorem, that if you have a smooth compact manifold or the variety there, and the Euler number is equal to the evaluation of the what turns us off. On on the fundamental problem. This is the result on time. And we just use this as a definition of the virtual owner. So we write down the same thing where we put the video whenever we feel like it. So the fundamental class is replaced by virtual fundamental class. So this is this thing. And now the projector that a kind of size form of this project that we formulate is that they were forbidden formula with mode. So instead of the actual order number of the modelized space, which one has no idea how to compute that which most likely will not be true. And one takes the virtual version. Now, let me, I want to briefly look at the slightly more general version of this to put it there. Anyway, so this is for the case when there's, we don't have it's not so necessary is more connected. So that is written in various are very long and very different from any force. And which many people have studied, but in either by geometry, they are very simple. So if the four manifold is a smooth project in by surface. Then, over the complex numbers then the associates with every class in the second homology in numbers, which is the first one is written in very because the class in the second homology, so we're going to class if this invade this number is not you. And it is the case that there will be only finally many eight for which this is not the case for either by taxes here. compute. For instance, if we are in the situation where before that the first pretty number as zero, you mentioned to you, you know, this bigger than zero and the canonical in the system, but it is more connected first, then there are only two different classes, meaning the zero. The class zero in the second for module for which they're written in one and the canonical class for which they're looking at several different inventors minus one. And then this was the reason for the assumption we made before because then the formula was simple in general the formula like this, you have the same. And if now we don't make this assumption we have to do so as to protect the surface with again the one here, which is bigger than zero, you have the same model for the data function, data function. And then we write down a very similar formula, where we have a now some overall several different classes. And if you basically have the same functions only we can see the formula very similar before, but it's more complicated. So this is this. Okay. So much for one number now I want to talk about the way we can get back. So the point is that I actually like to you. If you look at the formula of butter in Britain. It does not. It is not precisely the formula. They are more terms. But these terms are not supposed to correspond to the order number of model spaces or she's on a spot for something else for something else. And so, so there are these extra films in the water with the formula and the question is what do they need. Recently, some of the Thomas, you find. He's in there automatically, you know, via one life space. And so, maybe I. So first, it's again looks like some one last piece. So, you take again as the smooth project surface with the device. And the next pair is now pair of abortion tree she has. And it's so-called fixed field, which is a homomorphic from E to E cancer the canonical class. You put in an endomorphism of E, however, the values. And we want this to 30. And they again exists the modern life space for such pairs, you know, stable pairs where he has been trying to do. So the definition of stability, which allows us to construct a modern life space is very similar to the first one in position. She's the only the difference. So we, the difference is now we don't require this inequality that we have for stability for all subsheeps of F, but only for all subsheeps, which are invariant under P. So, where this mix maps F to F 10 pairs. So we have the effort on so it's therefore, and if you have a stable. So it's not necessary to that the corresponding bundle. Now, there's another completely different description of the same space, which I just briefly say we don't need very much. So there's an alternative description of this one. So here, namely as a modern life space of sheaves on a three-fold, instead of on the surface. So, we look at the total space of the canonical line bundle. Yes, so that means three four over the surface as where the fiber over every point are the fibers of the line. It's obviously non compact because the fibers. And so we look at the space. Now, you want to associate the thing we had before so he gave a sheet on the space. I give only a rough description because the fun does the property would. I mean, I think, and I'm a Thomas things like pages or so correct. So, therefore, let me just think of it. So, assume we have such an experience. So he is a chief, and they have been from the sense of yes. And now, if I have a, I want to see what this does. So if you take a point, fix the S, we can look at the corresponding actually restricted on the fire. So there's a map from ex ex. So this is basically an end of morphism of the ex so it has, I'm spaces or generalized I'm spaces and I'm going to generalize I'm great. So it will have some I'm spaces. X, the I, where the I'm values, as an app goes for the essence of S actually elements in the fiber of pairs. It has, I'm spaces ex, the I, for the I values, the I, yes. And so, so, our GP, he will associate she on the total space and on the final line, but the fiber, over the point X, the I, in the total space of this is just the corresponding I was obviously not very clear that this whole thing moves together to the sheet on X kind of just told you what the fibers of the sheet are, you can see, it will be something which is kind of finite map to as it lies in the, in the fibers of them. It lies in the kind of over S and there will be over every point and as there will be points where it's supported. And so in the end, you get that is the torsion sheath and who support as dimension to on this canonical house and in fact, and you can go back. And so that it means that these, the modernized phase of these things is isomorphic to modernized space or X pairs that I find before it's not it gives me a different description of it which looks quite different. You can also do it as some kind of. And, and so now it's not. Maybe I shouldn't have that the thing but you can see that there is C star action on on this modernized space to look at it in this way. I can see star X on the total space of yes, by just rescaling the fibers of yes, the back door. So yes, just if I multiply by T and C star, I just need, I just, if I net T and C star operating it, it just needs a multiplier. And then this action lifts an action on this one last day, but just pulling back the sheets by action. And so given an action of C star on the water. And then. And the point here is that. Anyway, so we have to remember what we want to do is we want to compute some want to find some of the victim events we want to associate some numbers to the topology of the space. Usually you can't do this if the space is not compact. And so we have to, and this modernized space NSH as the ones to certainly not compact, because we have parameterizes the CP together with fee, and we can always multiply fee by any complex number. And so it will not be compact. And, but we have this other action. So, and the fixed point will be compact. And it's actually, you can see the moment what it is. And just, I should say that this one last is also as an obstruction theory, perfect obstruction theory as before, which in addition is what is both symmetric, but which in particular means that the expected dimension of this one last days is zero. So in some sense, virtually, it's just the final set of points, even though it's. And so, now, just, now I want to define these invariance, although the space is not compact, and that's a little bit difficult, maybe, you will not be able to maybe send all the symbols here but certainly not everybody could understand all the symbols of the law, but those who understood them after now will not understand most of this. So anyway, if M is compact, many pole of virtual dimension zero, then I can integrate one of the virtual fundamental classes would call this the virtual number of points of this. If we were to take the virtual fundamental class, but the fundamental class in the extra dimensionals here would be the number of points. And so, if M has a C star action, which is, whatever, competitive with a certain theory there's a theorem which is called virtual localization, which allows you to replace the integration for the very thing on the virtual fundamental class by that on the fundamental class of the fixed point. So you have a much smaller space in which you can compute, which you can maybe understand. But you get the, you know, as a price, you have to integrate something crazy, which is the one over the equivalent of the virtual normal bundle of the fixed point. Whatever. So, now, the point here is, this was to get M is compact, and there's a C star action. And it's not compact, but it has a C star action. And they expect connection zero. So then why aren't we just there and say, okay, this theorem. I mean this would be a theorem. This theorem doesn't hold it doesn't make sense because one of the two sides is not defined. So this is this thing to define the, I don't know, precisely how this. And now, okay, so this is what they do now, unfortunately, it's a bit difficult to see what this is now on the next slide, which I could show you or not. I have defined what this thing is but it's a little bit complicated. I don't know whether you want to see it. What. Okay, so I can really keep the same. So this, as I said, Thomas have taken this as definition so the virtual number points is defined the interval over the virtual fundamental class of the fixed point locals of one over the C star. Now, what is this first to see what is normal. So we have the. The tangent bundle of and we can restrict it. Now, as this is fixed, it means that now, then, this bundle is a very big C star x on fire. And then, now, therefore, it's decomposes into hiding space for the actually where the eigenvalues are powers of tea, tea and C star x five. Now we have unfortunately number that the virtual tenant one is a difference of two vector bundles so we have this difference of such people positions. This would be the virtual normal one is a difference of some sort of eigenbundles of weight I so we're on the fiber. And then the virtual normal bundle is just the part where this way is no zero. So you just take it. So that's why we call it the moving part where the square team. Now, so then we know what this is. And then we have this. That's even more scary. Now, yeah, I can have just like homony. And so let's just take epsilon to be available. Some extra variable, which then eventually will cancel out. And then, if I have one of these. And then the order number of it. The order number is this expression to take the sum of k equal to the rank of the I, the I churn class, the page churn class of the I, thanks. I know that I was way. So we have this expression. So just see the table for that. And then the equivalent virtual equivalent. This portion of product to multiply this product. Okay, so this is the answer. I wonder, where does this, where does this live. These are formal classes available. But so somehow this will be in. This will be a rational function. And then one can integrate this over this, which means when integrates the formula just part of it over it, and what comes out with a rational function in epsilon, which are what turns out to be another. That's the answer. So, I mean, it's complicated, but you know, it's, in the end, I want to just show that it's not some, you know, it's actually something complete. And now, let's look at it more. We wanted to compute this is virtual number. It's by integration of a fixed point locals. So if you want to do that, we have to know something about this point. So let's describe the six point. Let's say, if I isn't an element in the six point on the small life space. So one can always assume that our sheet is activated. That's C star X on it. And if it's a command you have as before a way to competition you can write the direct sample sheets, where the way of the action on the eye is I for each guy. I've kind of said how the action can pass by restraining the fibers on. Yes, on this total space. This leads to the fact that the weight of the action of the, on the canonical line model is one. So, if you have an accurate map from the eye. He has this can be invariant to weight zero, which means the fixed point of the action, if and only if the difference. No, of course, here, it acts on one side. The J as one, you know, to be invariant means it sends everything to itself so it sends T to the I to the I. So this only works with J. So, if one puts this together, you find that the fixed points can be described as follows. You can write E as a direct sum of such spaces where all of them occur from zero, one and so on. And if I respect our fee, so once I've summoned the eye, it will always map to be I minus one. It doesn't have any other sum. No, the fee is from the sum to the whole sentence of yes, but the restriction to one of the I always have to the to the next one, which comes from second. And the last one, or first one, there's nowhere to go. This is zero. And, and so this follows from this description. And in addition, so we wanted to see to be stable. And the pair to be stable that it's there that the pair is stable, in particular, that the kernel of the map can only have one summit. And so that means the one summit is zero. So all the other steps. And so therefore we see that the ranks must decrease. And, and obviously the sum of all the ranks is half to the ranks from a position of up. And so we get the decomposition of points of course into unions of connected components. And so this, this is the description of this. And so if I think I want to single out two of them. One is what something is called the horizontal component is where the partition just consists of one number of the number R. So that means in this case that he is equal to zero. So P is the zero map. So P is the zero map. Then the X pair is just the same as just a sheet in the modelized place of cheese and the sheet has to be stable. So we see this six point component for the two partitions just the modelized space of station. And then you could go all the way to the opposite end. And have the partition does the 1111 times. So now I want to, you know, it shouldn't be. So if one doesn't know something, one puts it into generating function and looks as if one would know something. So we make a generating function of all these numbers that we don't know. And, you know, there's, you know, we have to give you a factor because you want to make a similarity to the virtual dimension. We have normalized. There's this sign for something you can see a moment. And then the number one is more like. So we have a generating function of all of those qualities. So. I would have just said, I can write this. I can write it as a sum of all the contribution. This part of the fixed one, because the invariant is complete is on the fixed point. This John union of these components. So the total thing is just the sum over. It's one goes over these. One goes at five partitions. So in particular, you can look at the one corresponding to the. I've seen that the model space. This is just the model space stable cheese. And then it's not difficult to prove that everything matches up nicely with it. With the abstraction theory and so on, so that have to sign this virtual number of points of this. And so we get in particular. Part of the additional function corresponding. So. So the solution. And this you know where they should just to the generating function for the oil and other things. So, in particular, we see, in what sense. Partition function contains the general function of the virtual model. Now, I want to briefly talk about what the learning properties for this. And also, you know, I mentioned more than once before, so you should at least have to these ideas. So, obviously, I do not like to say something. That's very much so the model form of with K is by definition a polymorphic function on the complex, which satisfies some nice transformation properties under the action of transformation. So, make it. And it is sent to. I was sent to. And then. So one. So, this is basically generated by two elements one is T. And the distance formation. And actually, this invariance property will then just say that f of tau must be put out as one. Periodic. And then the other generator is as we sense how minus one. There's also another property. So, if you have this thing that is. So, if you have a periodic period one, it means it hasn't been in development. F of power can be written as a sum of all n in that a and two to the n, the number. But we need to require that only. Um, non-zero coefficients are positive. Okay. Well, the same way. And we can also look more general but forms for some. So, which then. So, often, instead of. So, and so this is just this and now the thing is that in some, the claim is that in some sense, this partition function will behave like a model of form. So, that's, so, that predicts the behavior of this generating function. It's a bit so we can write a more complicated version of the same so we take this generating. Now, here we take the sum over all the system of representatives of the second homology of s that. Second homology. This is the sum like this, this before the fact that this is the so-called language function. And then, then the conjecture. This is our form of. Number five. Is that under this. Element of the. These things are related to the function. And so why is that interesting. And one thing is that. But last year, we are interested. It's not obvious, but I just say now, if one looks at how this creates for the contributions for the different points local, it will exchange the contribution of the one for the partition and for the partition 11111. There is another result by Thomas, which says that if I was a prime number, then this will be zero. We have either. In fact, that is not just the particular number of points of the. It's more like the point before. So, therefore, there are only two terms in this case. And they are exchanged by this transformation. And so, if you believe in this expert conjecture, you know, one of them. So, obviously, you know, we have done some competition. We checked that we can do this. And we also use it to make some. We say a few words. What we actually do. Okay, I think. So, we computed up to some high order in Q. This thing, which is one of the chiefs. So I have to hide the fact I mentioned one of the computers in general for an S. When they're actually three, and he completed this other one. When the rank is at five, so therefore, on the other side of the time generator, so I could know when we. When we look at this kind of the first. We can identify some of the forms and so we find formulas for these generating which is the high as rain grows get more complicated. I have a lot of time which I didn't tend to have I could show you the form. But I won't. And so now let me be seen say how we want to do this. So, in both cases, no one is supposed to either compute on the smaller spaces or she's all on the six point low side. In the. Modern space. And in both cases one wants to reduce the computation. So computation. So let me first just say a few words how this will go for this thing. So if we look at this and see star one one one. This will parameterize the zero plus and four plus the arm. Where the eye or chiefs of length one. And so, therefore, if I have it. I have a portion fishing for one like here. It must be of the form. The sheet of a zero dimensional. So, therefore, these are usually some of them. So you can somehow see that you would expect that you can discuss the whole situation in terms of products of So, and one can do that. And so one can, in the end, if I want to compute this integral we have to do, we can do this. As in some intersection number of products. So I don't want to do this. Let's just to some extent. In the case for the other part. She's, she's on on s. It's less obvious how one would go to a point, but we use a formula by what you do. Which was proven. So, so, so to be able to roughly state it. Right. So, maybe I should show this, you know, one pattern. Anyway, but so we look at the one life space you want to compute the section numbers on this one life space. In terms of the section number of points. So on s times one last case you can assume you have a unit sheet. And that means that she on s. One life space that if I restrict it to s and the point corresponding to a sheet, it is that she. And then we can use this universal sheet. And then we can use the many common classes on the one life. They take classes of the chief. You might slide it down by the full bake of any. And we put forward to one life. And they say both. And now, what you took this formula gives us any polynomial. Any problem in these classes. What you do formula will express it. If I integrate this. Then this number. This is then stress. In terms of the section numbers. So now that is the school. So we have this. We will see to work on s times. Now we have two different kinds of sheets on on the one is, we take the idea she. So we take a sheet on this product. That said, if I restrict this to the surface times the points for one. It's just the idea she of the I. The one or two. The other one. Is it. So this is the sheet on the product. And this is the one. So. Which is where we strict it. So, which, you know, on. So. The structure sheet of the iPhone. So this could be two natural. And then. There's a terrible formula which I don't write, but. Something like this. And then the claim is the following exists. Some. Is. It's not so important. So you have a. Such a long one. And in a variable s. Where we face now. And so also. Where now. We take, instead of the right on the same thing as the. When we look at this one. We have these. We have, we have the universal sheet. We just replace. The universal sheet. By the style of these. Right now on the same formula. This gives us some formal tasks. On the product of liver schemes. And we have. So this. The one for an S was proficient. And in the trend. So something very complicated. I've said we can. Generating function like this. But anyway. This is a bit faster. So. We have written down this. Now we can. First of all. We integrate this thing. Over. This product. So as it was. For. We will get just. In S. We call it like this. And then. So we have this fashion. And it's not important. What the formula is, but what we do is. What we wanted to compute. It's obtained. By summing over all ways. Right. And. The first chance. We had some. In the second homology. Is that a good name. The first time for coefficient of S. Of this thing. You define. So it's a complicated thing. And it's much more complicated. But so we have a terrible formula. But. The advantage is we have. Before we had the. Simple formula on the terrible space. Now we have a terrible formula on a simple space. And so. That's. Kind of. That's how it is. So that we just. Finish. So, so we have. These things. That we wanted to. So. So. You don't know what this is. If you don't know what something is. Generating function. Not always. So now. You make a generating function for these things. We don't know. But if you knew them. We would know the answer to our question. And so instead of looking at one of them. And so. This. Is. That. In some way. This thing is simple. So. The way it depends. On these classes. A1. A2. And the surface. Is just by. Some. Simple numbers. The intersection. A1. A1. A2. So on. And. And the way. And. And the way it depends on it is just that you have seven universal. Power series. And you just. The whole generating function is a product. These are the corresponding power to the way. The geometry goes. The formula is just in this way. I got the power. And so this is a real life. But this is actually. And. It's a very long time ago. With. It's a very long time ago. And. Improve this for several. And. And the. Modification of the. I would show that. And then. And then. Given that one's basically. Done. So. So if we. So. Therefore. You need to know. And so. We want to know them. How do we. Can we know them. In order to know them. We only need to look at seven examples of the surface. And two line. So. If I take seven triples of the surface. And the line. Such that. The couple of these numbers. Is linearly independent. So. So. We need to know these seven universal powers. And so we want to know them. How do we. Can we know them. In order to know them. We only need to look at seven examples of the surface. Then. If I know. This. For them. I can. And so. So I only need to find those. And I can do this very simple. I can take. For instance, my purpose does to be. One. And take either one. Or one. Of the. And so. If I do that. Now. We have. S is a smooth. For the surface. The surface. With an action. Of C star. And then. Then we have an action on the surface. The action lists for an action on the. Point. But. Pulling back. The subspeed. And. It turns out. Still. In fact. You can. I can easily see. That the thing. I can easily see. I can easily see. I can easily see. I can easily see. I can easily see. I can easily see. I can easily see. I can easily see. That the fixed points are. I can easily see. That the fixed points are. Carmatized by. Partitions. Partitions. And. And. If you look at the things you want to compute. These are. You see what. You can use them. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. A. I think that's why I want to stay online. I don't know whether... I think I don't know whether anyone's still there. Okay, thank you. Thank you for the effort. Anyway, I promise that the actual lectures are a little bit slower. Certainly, I have to, there are so many... I went over so many... Yeah, yeah, well, but not to close. I compute the first... Just so many... The first, whatever... In some cases, for this stuff, maybe the first 30 coefficients, but then you can check. And also, in higher range, you get many coefficients, and then you can read off what the monochromes are. Let's go a little bit... I have to get my... Yeah, if there are no questions...