 which has to do with attention to obstructions here. So I think that's what I said about that. Just say, you know, so if I have, say, that is in an embedding, get the idea sheet, and then then there is a standard exact sequence, I don't know, there was five, I swear. And then a perfect, and then you could say that. And so perfect abstraction theory, so it's a complex of extra bundles, very short context of two extra bundles, such that we have a whole, with a homomorphism of two taxes in the same, let's say minus one. So that means the rules, such that two percents on homology, this is on one side, nice one is on the other side. Subject, let me see what that's called, H zero, five is the map from the coconut tree and H one, where they said that the dual of the kernel of the fiber-wise would be the obstruction space or the canonical obstruction space, the dual of the coconut here is the space of that. And so that means the map would be an isomorphism on tension spaces, and so this is injected on the structure of the map, injectively into a cage for the dual of this, okay. So, and the virtual dimension of that is then just equal to the length, and then if one has such a context as a virtual dimension, homology corresponding to the virtual dimension, and one can define invariant, right degree. Now let's look how this looks for the monolite space sheaths. Simplicity, I would just say two words, not much. So, for simplicity, we have the monolite space, section five, we can first say that the monolite space is a virtual dimension, section five, we can first say, so what's the tangent space corresponding to the sheath E, and what one finds is that this may be x, one. I can't say a few words about it, not too much. What's the tangent space, the monolite space, this would be the families over spectre Fs on, what Fs on square, which specialize at the close points to E. So, the vector bundle, we have an E over epsilon times K, we have an E over K, and it's an extension that we just had. So this shows that the tangent space in some sense should be x, one, E, but here we have the zeros and the trace B in the matrices. This comes from the fact that from the monolite space sheaths to the car variety, you have the trace map, and so you have the map which maps the sheath to its determinant, and the tangent map to that is the trace map, these endomorphisms, and so the kernels means you are looking in some set of fiber over the car variety, over the, where you fix the return class as a line bundle. Now in our particular case, so we have assumed that Q over S is zero, so we have in some sense in this way, indirectly fix the determinant, and the point is that, but anyway, so one has to be a bit careful but what one gets is this trace. And then a bit more work which told that the construction space would be x2, and in some sense it's also what always happens over there. You have the tangent space is a certain homology group and the construction space is one step up. So trace B, I mean, what trace B means for homomorphism, say a vector of analysis clear, what precisely means, I mean, can extend it to sheaths and then it's in the right kind of that to do it for the x group. And you can globalize this, we have that the dual of the obstruction theory because the obstruction theory is always something which is potential, but the intuition always comes from the tangent. The dual of the obstruction theory is, you know, if I want to, I was with projection, so this is the R, you know, minus one is the potential, it's the x1. So this is the right factor of the homo sheet. So this globally, and you can, so this basically just goes globally right. And so one can only, so this is it. And so then with this, we do have all, and you know, you can by Rivan-Roch, you can compute what the holomorphic order characteristics so what the alternate sum of the x i is, and this will give you the x zero, the trace free home from e to z is zero, x2. And so this is just the virtual dimension, it's just minus key of pi of e, e, zero, and that is given by the formula p. And I mean, we also recall that virtual tangent found zero minus. As I said, the conjecture was that the evaporated formula holds for, so the form of the character at the time, virtual or unknown, which is integral over the, I wanted to formulate, as I mentioned, the symptoms of logical invariant and algebraic variety is the, but it's also the causes, so it's not all that much information. So one refinement that one can keep up with, obviously it could be the better numbers, but this doesn't work the way here, but another refinement that can keep us there. I want to know what this average is, but I'll call it the option numbers, for more, okay. And it is, so by perhaps theory, we'll know some things about this, first, we know that they're symmetric, this actually is a position of the homology, so we have the, I think the number of x, which is the dimension of the i homogeneity, is equal to the sum of all four t, such that equals to the dozen follows, all in a number of i-genus in certain parts of the sum. So we take the sum q is this homology, so take the automated sum, all of them also say sum over all p minus y to the p. Then we can see that for that, this is actually given by the section numbers, you can, by Riemann-Roch, you can compute it in terms of the section numbers. Okay, so this is all, and that, and so it is certain, instead of just one number, you know, it's not much less than I mentioned, x. And incidentally, I think, so I can show that this i-genus is certain, one's utility properties, so that you can compute it by cutting your right pieces and reassembling that, I mean, I should completely teach it, and in fact, it is the finest invariant, which both is in that sense utility, and is equal to the most invariant, is expressed in terms of general numbers. And so this is the one we want to use it, and by what I said, obviously we have a tight minus one, with y equal to one in this, a tight minus one of x, is by this formula equal to the one I said, so now we look at the virtual savagings, but basically just by the same thing, we have the virtual e-forms, and if I look at the e-forms, that's obviously the only reasonable way is what would mean by that, for such a difference, and it's completely that this is minus one, with a j lambda i sensor, if you're alternating power of something negative, symmetric power, if you wish this would be a definition, you know, if you try to think how it's actually not much to go, whatever, it's the only definition, which we need to analyze and make a statement. And now again, one could show, and so now if I imagine is to put i minus one, m, see we have the virtual structure sheet, and we get set for the sum above the p, where the statement was that the virtual looks more like other times of some vector bundles on m, is just inside, so I have not made the statement. And now again, we can show that this formula actually holds, so it shows the difference, especially I write the one that we keep i minus one, this is equal to the virtual one I can see. This is the whole point of this virtual stuff, is that one wants to be able to work as if everything was smooth, but meaning in particular that one wants that the formulas that one had the same, that one has more interactive varieties, where we know in some of the cases one has to put here, and it's the same formula. And so this particular formula puts this one as a formula, which is the same as the standard gets put in a normal formula, so the virtual whole of the product stays vector bundled, is equal by the normal formula of the integral over the virtual fundamental part of the transfer of the virtual central, and the standard normal formula is the same without the wheels, which one would want to know what the multiple effect is of the vector bundle on the fact that right here, it's given by the integral over the fundamental part of the transfer of the virtual central. I expect you know what these things are, but as it is so fast, I can also use a vector bundle, as we had 10 classes from the other time, that can formally write them as a, so write formally, just write this as product, if it's one to one, on some sense, might be common classes somewhere, maybe not necessary on our variety, such that the turn classes are the geometric symmetric functions in the XI, and so basically the idea is whenever we write an expression in the XI, which is symmetric in the XI, and so, and then in terms of this, you can write the turn character, some symmetric expression in the XI, it's what we can express in terms of turn classes, and therefore we make sense of it, okay? And obviously, in the virtual setting, the key group, so different of two vector bundles, obviously the turn character of the difference of two vector bundles is the turn character of the XI, the turn character of the top genes of the difference between as a product, say for the portion of XI, and this makes sense, because if you look at this, of course you start to write it in this sense, in this formulation, which can formulate the fine version of the XI piece, so bigger than 0 to 0, and again, for this piece, soon XI is built into a non-device, then we have the formula for, in terms of model of forms, for now we have an extra variable, so we somehow have to upgrade our model of forms to something which has more variables, in this case, that's what we call a set of functions, so we write them in the standard set of functions, set of X, Y, which is the sum over all the integers, and X to the n squared Y to the n, and then it's the only such, and then we have to make again this model of a set of functions, and then C, S is the two times, eight of R, they can write down this big thing, I mean everybody can write down some complicated policies, which will be complicated, not yet, but it would, but then the thing is that this is the generated function, this model like this, so this is the, so okay, so this sets up a big time, or obviously the virtual one, one, all these very, all the last places, you want to do, you have one, that's one close to the question. So this is, very fine, so now I will later talk about the, how we want to try to check such a project, but for the moment, I want to, want to introduce also the R for written, modernized space, actual R for written invariance, the point is that these, all the numbers of the modernized space are only part of the numbers that are put by R for written, and they want to know what the meaning of the other numbers are on it. So the thing which describes all these numbers is this R for written modernized space, and I showed by one, introduced. So as I said, to give up with the formula, the R number, and so this was only, in the essence, Chanakai and Thomas gave the definition of R for written invariance, which was supposed to be the one step, and this is in terms of this, of the modernized space, which is the modernized space for X, X is zero, equal to zero, E, the portion of the machine S, and then, is an element of P where it's in the current line number, so that means P is a map, because the whole number is equal to E, to E, center, in the current line number, and I assume that the trace of the sum of the zero is also equal to six K. So, one to consider, one last basis for this, so we have to say what it means, is stable or semi-stable, and then, the stability condition is the same as for machine E, but we have to restrict, so this is something about sub-sheets, but we have to restrict the sub-sheets, which are invariant under P, so that comes, so we'll take the whole map with all our characteristics, the line number, divided by the rank of S, same thing. Now, normally, so if it was just for all sub-sheets, now it's there, we take all sub-sheets, which are invariant under the X-feet position, which just means that when the feed actually is contained, restriction, one can show that this is the monolid space of six Ks, a semi-stable, obviously, means that you have two sub-sheets equal to right half, and the third half is equal to that bunch of nodes, so that is, you know, this feed, the one possibility is that this home office is the zero, then the condition that P of F is contained, contained with F tens of Ks is the empty condition, so it means here all sub-sheets are allowed, that means if P is equal to zero, then the pair EP is stable, if and only if P is at the sheet, and here it follows that the monolid space is the number square of six Ks, so we have a monolid space that's contained. There's one thing, which it is obvious, if one thinks of it at the moment, that before we were very much insisting that our monolid space should be compact, because we want to integrate something over it, you know, you have the virtual fundamental class, we only have its compact and so on, and so now these monolid spaces are never compact, so it takes a non-zero T is an element of C star, and it is a point in the monolid space, so you can see you have kind of got all these, when you have some point, it contains the whole line, and so it cannot be compact. Now, in order to have some variance, so we still want, we eventually want to have some kind of virtual fundamental class, although it will be more complicated, and so for that we need an abstraction here, so I can say some words about that, so Tanaka Thomas, so assume, this is not let's say by monolid space, that it's not necessarily universal sheet, but let's just assume that it is universal sheet, so if it is a universal monolid space, so on S times 10, I can first globally say what the dual abstraction theory is, as I said the dual is what the abstraction theory corresponds to, what happens on the 10th level, which is the only one, you need to understand, whereas technically being able to work with it, you have to work with the abstraction theory, because the whole range of things are easier to work with than the 10th, even though the equation is always about the 10th, so this is the dual. So the abstraction theory, I can write down this thing, this is just that here, ahom, e, e, e, so this looks very similar, so it's the ahom, p, the red signature action, so it's the same, so we go over here, match the same thing, the map here, is just the map, which on the second factor applies p, here it's the identity, so to speak, on the e, so, okay, so explicitly, this means the following, we have so, if I look at this, we have so the 10th, so it's e, and inside we get something like x1, so the mapping corner is to take the sum of these two things, so here we have each one of the first, each zero, second, and the structure, so fiber-wise, the abstraction theory is given by this, the map like this, so much of notice that x2, e, e, 0 is dual, and this is the structure, and now here one has the, if I look at it, by side reality, this is the same as sum, e, s, dual of that, and by side reality, this is the same, to therefore see that the engine is dual to those structures. What? Let's go, I don't know what to do about it, but okay, is that better? So the 10th is dual to the abstraction, they just change their roles, so, and that's what it means, the set pool of structure, and so that, obviously again we can represent this by the complex of two vector models, which one needs for this, it follows the dimension of n, which is in this case the range, this will be 0, so we have an abstraction here, of dimension 0, so therefore, we should expect, which of the matrix are squashed, we should give them, now there is some, give us this section number, something which is not, we have decided to do the nonsense, but now one instead, unfortunately, we just did the solutions, I should say, there are actually two solutions, and there's actually a solution which seems to be the nicer one, which has the disadvantage that's wrong, and then there's a solution that we will use, rather than a kind of Thomas. So, one of the things is, if you have this set of abstraction theories, there's a way how you can, how you can kind of forget, two things in a different way, in terms of the so-called theorem function, so you kind of compute some kind of Euler number of space, where you weight the pieces according to some abstractive function, which you construct out of the abstraction theory, you can always do that, so you could just integrate over n, the sphere function, you could get some number, and you can do that, and you get some nice numbers, which are a little bit too simple, and you still have nothing to do with the predictions by the physicists, they are supposed to be the wrong numbers. Now, instead one can do something crazy, so instead we apply localization, so then so apply to n, and the fixed localization is localization form, localization is localization form, so if we have something which is compact, and it has a c-style, then there is a formula by which one can express intersection numbers on the fixed space in terms of intersection numbers of something else on the fixed load. Now, this thing is not compact, but the fixed load is compact, we just use the same formula as if our n was compact and declare this to be the integral over n, so we just formally apply this localization formula and say that's the answer. Usually, if you have a theorem which is true under some assumptions and then you decide to apply it when this assumption is don't hold, you will only get constants, for some reason. Now, I don't know how do we do with time, because I just used my time, now I would have to explain about the localization problem, so it can do next time or anyone can also just as a question. So now, I now have prepared to first review the localization formula and then do the virtual localization and then use it as a definition and it goes all very slow. Is that fine or do you want it with less detail? I mean, maybe go as I did, but anyway, that's the next slide. Okay. Are there any questions? I have a question, I'm not sure if this works in the hybrid format or not. Ah, oh. Okay, so brief question, maybe it's not so important, but do you allow for strict semi-stability in this whole discussion, so you allow your modular space to contain strictly semi-stable sheeps and in this case, Higgs pairs. Also when there are strictly semi-stable sheeps, you have to be more careful with the definition, it's done in terms of something which is called choice song pairs, you can define everything and compute everything in all the formulas that are non-restricted. Okay. All right, so basically it's you do, so in, yeah, it's okay in your discussion, you don't make any restrictions. Just. Whenever I explain something, I will, you know, you always assume stable is equal to semi-stable or whatever, but in fact, they are non-restricted. Okay, thanks. Thank you. I have one more question. Yeah, okay. So if we started with one of our curves, there is, I think there is an open embedding from a collision model of the models of curves through the big modern space of Higgs pairs. Hi, hi, hi. So are there any similar questions? Okay, I mean, here, you know, I, so that's actually something that the, you know, one could discuss whether to call these things Higgs pairs that I have is the correct, you know, that's how Tanaka almost called it. No, on model life spaces of curves, you have the thing is here, we just take this thing, tensor ks. So in order to have, so another thing would be if you take this thing, tensor, all alternating, tensor omega s, all alternating power, so then you get some kind of things. So then I think the potential bundle will lie there, but like this, it doesn't, you know, go back again. Somebody does this.