 We defined these Waffa-Whitney variants in the stable case when there were no strictly semi-stable sheaves on the local Kalabiya 3-fold, the canonical bundle of the surface, or strictly, equivalently, strictly semi-stable Higgs pairs on the surface. Okay, so in this, it's important to try and define invariance when there are strictly semi-stables. For instance, in Donaldson theory you can do that. So here, once you have strictly semi-stable sheaves, you have automorphisms. If you remember, right in the first lecture, we proved that stable sheaves essentially have no automorphisms, apart from multiples of the identity, but that's not true for semi-stable, and so you end up with moduli stacks instead of moduli schemes, and obviously nobody wants that. So we rigidify the stacks by considering instead pairs. So we consider this sheaf on the 3-fold. Remember, it's a torsion sheaf supported on a surface in the 3-fold, spectral surface, and we pick a section of a very positive twist of the sheaf so that there's no highercormology. Okay, so that's this twist by N. So we fix a very large N so that none of the sheaves in our moduli space have highercorm... excuse me, have highercormology when twisted by N. And then we have as usual the usual conditions, because we're doing SUR instead of UR, Vaffa-Witton theory, so centromass 0, and determinant trivial or fixed. Or you can phrase it in Higgs pair language. You now have triples, so you have your Higgs pair, E-phi, and then you have a section of E. That's the same data. Okay. And then there's a stability condition, and it's that E is semi-stable and then that any destabilizing sub-sheaf, actually, maybe I could use... can people online see the mouse? I mean, I know they can't see my pointer, but can they see the mouse? So maybe I should point to things like this. I don't know. Yeah, we'll see how this goes. Okay, so if you pick a sub-sheaf of E, which has the same Gizika slope, so it destabilizes, semi-destabilizes, then the section should not factor through it. Okay. So these turn out to be... This turns out to be the right moduli space to consider. It's as compact as you could expect, so it sort of maps to the moduli space of semi-stable sheaves curly E with it's a proper map. And then you can define invariants counting these pairs. Okay. So we have a moduli space. Once you fix some topological data, the moduli space of all pairs, this perp moduli space, that's the ones with, you know, center of mass zero and fixed determinant. And again, they emit obstruction theories. And the way they emit obstruction theories is by thinking of these pairs as two-term complexes in the derived category of your Kalabi out. So if you form these two-term complexes, which will be familiar if you know anything about stable pairs, then defamation theory of these two-term complexes is for the same reason, symmetric, virtual dimension zero, perfect. It has all these nice properties that we've seen before. Okay. And that turns out to be a perfectly good thing to do. So you can think of these p-spaces, these moduli spaces of pairs, as moduli spaces of complexes. In other words, if you have a, if you have pairs, you obviously get complexes, but conversely, if you have a complex like this, you can prove that any defamation of it is also a pair. So you really get, this gives you an obstruction theory not just for the moduli space of complexes i dot, but actually for the moduli space of pairs p. So I won't go into too many details. Again, use C-style localization to, because everything's non-compact, and you want to localize to this compact moduli space of pairs supported set theoretically on the surface inside x. And that is compact, the fixed locus is compact, so you can define invariance like this. And then the idea is that these obviously depend on n, but the idea is they depend on n in some universal way that is governed by certain underlying invariance, which we're going to call the Vaffa-Witton invariance. So we need a universal formula expressing these in terms of Vaffa-Witton invariance. And this is what it is. And it's not entirely clear, this is comprehensible unless you're familiar with Joyce or can save its Soebenman wall crossing formulae. But there's a, what you should do here, the way to understand this is to ignore the scary-looking formula and look at the second formula. So if the surface has any one forms or two forms on it, this is the formula. And the way you should understand this formula is that this is rather easy. Ignore the sign, there's always a sign in dt theory. This here, the holomorphic Euler characteristic of my sheaves twisted up by n is the topological Euler characteristic of the projective space of sections of my sheaf twisted by n, which is the fibre. If my, if my sheaf is stable, actually, I think I might say this next. So yeah, so the way to understand this is in the stable case, so in the case where all the underlying sheaves curly E are stable, then my moduli space fibres over that, moduli space of stable sheaves, and the fibre is a projective bundle. So it's just the space of all sections of my sheaf and that projective bundle has topological Euler characteristic given by this or signed virtual, signed Euler characteristic given by this coefficient here. So the way things work out, when you work out the obstruction theory and the virtual cycle and you work out what's going on, what you find is that the invariant associated to this projective bundle is the Euler characteristic of the fibre up to sign times by the invariant of the base. And so that gives you this universal formula here, in the middle of the slide, which says that the pair's invariant determines the buffer-witney invariant. And the buffer-witney invariant determines the pair's invariant. So they're, they're the equivalent information up to some universal formula. And in the general case where you have semi-stable sheaves, this is a conjecture. So the conjecture is that the, again, there's a universal formula, it's a bit more complicated, and the reason for that is that it expresses to leading order, it's just this formula that we kind of understand, and then to second order, it has something to do with sections which factor through one of these sub-sheaves which destabilise E, so of the same slope as E. And then to third order, the, when you get cubic terms here, where L is say 3, it's to do with sections which factor through two different sub-sheaves, and there's an inclusion exclusion principle going on. So you're not meant to absorb this formula, but this formula is somewhat comprehensible. Okay, so this is a conjecture that in the semi-stable case you can define buffer-witney invariants from this, from, by inverting this formula at the top here. So, if, if the conjecture is true, if the pairs invariant have this universal form for some coefficients, Vw, then you can recover them from the formula, it's an invertible formula, and that defines the Vw's, okay? And this is proved in many cases now, more or less, yeah, in many cases. Okay, so as I said, the lower order corrections count sections factoring through destabilising sub-sheaves, or pairs of destabilising sub-sheaves, or triples and so on. What do you have to prove to, to prove this conjecture? Well, the problem is it's a wall-crossing formula, and the wall-crossing formula that let's say Joyce proves uses barium functions and Euler characteristics. And so you need a version at the virtual cycle level, and he's proving one at the moment, and he will prove this formula. But you define the wall-crossing formula by the left-hand side, I mean. Yeah, but you've got to know that for all n, this is true, that's the problem, yeah. You know, these, the dependence on n needn't be a nice polynomial or whatever like this, it could be something much worse. Yeah, it's the dependence on n that this is about, the conjecture is about. Okay, so when stability is semi-stability, then this is proved, so we recover the old Waffa-Whitney invariance. When the degree of the canonical bundle is negative, then the whole moduli space is compact, and so for those of you who know about it, the, the, any invariance we define can also are equal to invariance defined by bare-end weighted topological Euler characteristics, and then we can use, we can use Joyce's wall-crossing formula, and that gives exactly this. Yes. I mean, I guess it's not a question, it's a comment, okay? It's a, Joyce's new wall-crossing formula does not cover this wall-crossing formula because of the localization, at least for now. Is that, oh, Cardi. Okay, okay, that's great. I mean, he knows better than me. Joyce told me he was gonna do it. Okay. I have faith. I believe this formula, and I believe in Joyce, so. Okay, when S is a K3 surface, we can prove this formula. So this is joint work with Davesh, and it's, I mean, it's kind of pointless that I gave this, that's too much information in grey there, but it, it follows lines that Georg was talking about yesterday, and when the surface is general type or has a holomorphic two-form, then computations I'm going to discuss later, and that Lota discussed by La Raca, prove the conjecture in prime ranks. We're starting again, let's assume we have these Vaffa-Witton invariants defined. I'm going to explain what the techniques are for computing them. So this is, you know, Gromov-Witton theory, it took 20 years before people could make serious computations or more. Vaffa-Witton theory, you know, we have, it took 20 odd years to make a definition, and then we had lots of tools from Gromov-Witton theory and DT theory, and we found some new tools, and then the progress has been incredibly rapid. I mean, these La Raca results are extraordinary. In some sense, you know, most of the theory that you're interested in has been computed already. So I want to explain some of the tools for computing these Vaffa-Witton invariants, and I'm going to explain them only in the stable case for simplicity, but more or less everything can be done in the semi-stable case using these pairs. And you just compute using these pairs, and then it turns out the answers satisfy that conjecture. So I'm going to talk mainly about, or entirely about calculations on the monopole locus, and then I'll talk a little bit about this estuality and how it relates to computations on the instanton locus. So everything comes down to these nested Hilbert schemes of points, and curves and points, but I'm going to start with points. So let's start with the rank two monopole locus. So it's a union of nested Hilbert schemes. So you have two sub-schemes of your surface S. So it corresponds to a sheaf which is supported on a thickening of the surface inside the canonical bundle. So you have your, here's your S, here's your KS. The sheaves are supported on, that's a good colour, on a thickening of this. So the doubling of this surface. And the sheaves are a C star invariant, and they're rank one on their support. So to leading order they're just the structure sheaf up to centering by line model. But more precisely they can have singularities. And so what you find is you get ideal sheaves. We've got some red. So inside there's the thin S, the reduced S, you'll have some, I can't see that, can you? So you have some sub-schemes of S, maybe there's a fat point, so curves or something. And then some of them will thicken into the two S directions. So maybe your sub-scheme looks like that, or maybe it's the whole, maybe it's all thickened. But the point is the sub-scheme on the reduced S always contains the sub-scheme on the first order deformation of S. And that's why you get a nested sub-scheme. But I mean the actual formal way of writing this down, I did last lecture, using in the Higgs language and using weight space decompositions. So the zeroth weight piece is the thing supported on the reduced S. The first, the weight one piece is the thing, or maybe weight minus one piece is the thing supported on the first order thickening of S. Okay, so I'm going to start with the case where these sub-schemes are all points, no curves. So this is the simplest case. So we have this nested Hilbert scheme denoted SN0N1. Sub-scheme, so which way round is this? So Z0 is the big one, so the ideal sheaf is smaller, and it's got length N0, and Z1 has length N1. And it's singular in general, but it sits naturally in a smooth space, so the product of Hilbert schemes. N0 and N1, so the moduli space of Z0's is this one, and the moduli space of Z1's is this one. And we want to exploit the fact that these Hilbert schemes of points on surfaces are smooth. So what we want to do is see this as sort of the, going back to the first lecture, we want to see this as being an example of the fantasy model, a global, we want a global Kuranishi model where we have a vector bundle over this smooth ambient space with a section cutting out this nested Hilbert scheme. And that's not quite possible, but we have something very close. Instead we can describe it as a degeneracy locus, so the main idea is the following. So it's the locus of points where there's a non-trivial map from one ideal sheaf to the other. So I guess exercise check, you're happy that there's a map between these two ideal sheaves if and only if there's an embedding of the ideal sheaves, if and only if one sub-scheme is contained in the other. And in this particular case where I've only got points, a further thing which simplifies things is that when this HOM is non-zero, it's only one-dimensional. Okay, there's a unique HOM up to scale. And at such points X1 will then also jump as well as HOM. So what we've got is that we're seeing this nested Hilbert scheme as a locus where some HOMs and X1s jump. So we consider the family version as we move over the Hilbert scheme. I should have put a curly HOM and a pie in. Anyway, you consider the family version. Oh, okay, there it is. So there's a complex over this ambient space which computes those X groups at points. And that is the usual family way of doing it. You can't possibly, you know, you either know how to do it or you don't, but it's best done by yourself. Since we're simplifying things, we're going to assume some cosmology vanishing on the surface. You can get round this, but it makes things simpler if we assume it for now. And what that does is that ensures that this complex here only has HOMs and Xs. It doesn't have any higher Xs. No X2s or anything because they're dual, yeah, I'll stop there. Yeah, because they're dual to HOMs, maybe tense the canonical bundle. And yeah, so that's, I guess that's an exercise. So use serduality to check that this condition holds. All right, and then it is an exercise to see if you understand base change properly. This complex is quasi-isomorphic to a two-term complex of vector bundles. It's very important they're locally free over the ambient space. Okay, so this is the thing that when you restrict to a point, this complex of vector bundles is the complex whose core homology gives you the HOMs and the Xs at the previous slide. So the nested Hilbert scheme sits inside this smooth ambient space and it's a degeneracy locus. It's the locus where this complex, where the map in this complex, the differential D, drops rank. So D is generically injective. There's no HOMs generically. You know, when Z0 and Z1 aren't contained in each other, there's no HOMs. And then on this nested Hilbert scheme, suddenly you get HOMs. In fact, a unique one. And so that, what that means is D stops being injective and it suddenly drops rank by one. And it has a one-dimensional kernel and a one-higher dimensional co-kernel. Any questions about that? That's kind of the key to why all these computations can be done. They're not well defined. They're only defined up to quasi-isomorphism. So this is just one of these things where to compute this R-HOM, you're going to take a locally free resolution somewhere and then push down. And then you're going to trim that locally free resolution using the fact that there's no higher core homologies. And when you do that, you're going to end up with some abstract pair of vector bundles. So this is just a complex. Which are you happier with? Are you happy that this is a complex over... You're saying there is no HOM, but there is a non-trivial direct HOM. Right. And this is a complex. You can see that there are no HOMs. So my claim is D is injective at the generic point. Therefore, as a map of sheaves, it's injective everywhere. It certainly isn't injective as a map of vector bundles. And so what you're finding is that this nested Hilbert scheme is the locus where D drops rank and suddenly it's D is not injective as a map of vector bundles. But it is as a map of sheaves and therefore this HOM sheaf is zero everywhere. But the X1 sheaf jumps. Anyway, either any one are not really well-defined. They're just abstract resolution. But the complex is well-defined up to quasi-isomorphism. So we're interested in where this complex degenerates. You can give the degeneracy locus a scheme structure. This is the usual fitting, ideal stuff using minors. What you do is you say... And you can do this much more generally than I'm doing. I'm interested in where the rank drops from full rank to one less. You can do this for any rank drop. You can drop from rank K to rank K plus R or something. All these different loci all have descriptions like this. And everything I'm going to say applied in those situations. But I'm going to try and do it in this simpler situation. Suppose you're interested if you have a map of vector bundles and you're interested in the locus where the rank drops to what? I'm interested in where the rank drops from being a full rank. So that's the rank of E0, R0, to one less. So that happens, D stops being objective when the top exterior power of D which is a map from the line bundle given by the determinant of E0 to this vector bundle which is the R0 wedge of E1. So this is a non-trivial vector bundle. So that section where that vanishes is where D has ranked less than R0 which is what I'm after. So that's where you get the scheme structure. You look at where this section of a vector bundle vanishes and that gives you a scheme structure on the degeneracy locus and then there's an exercise to check that that's really the same as the scheme structure on the nested Hilbert scheme. Of course you need to define the nested Hilbert scheme properly. I've only told you what it is point-wise. You need to define a functor, represent that functor and so on. But then what you find is that the scheme structure, so this is kind of a lengthy exercise but it's done in my paper with Gollampour. What you find is that the natural scheme structure on the nested Hilbert scheme is the one cut out by these minors of the matrix D. So what we've done is we've seen that this describes the nested Hilbert scheme as being cut out by a section of a vector bundle on a smooth ambient space but it's far too many equations. These minors have far too many relations between them so this is not the right virtual cycle that we want. It would almost certainly be zero and it certainly wouldn't be related to the one coming from Buffer-Witton theory. So we need to see a different way of cutting out this degeneracy locus Z by equations. And you can't do it in the ambient space we started in but you can do it in a bundle over it. And in this case that I'm interested in where we're dropping from sort of top rank to one less, it's a projected bundle. In the general case it's a grass mania and the general case works just as well but let's just do this vector bundle case. So recall the setup, we have this map of vector bundles and we're interested in where does it drop rank, which points does it have some kernel. So we work on the projected bundle over the ambient space with this composition. So at every point of the projected bundle a point of a projected bundle corresponds to a line in the vector bundle in the vector bundle fiber, that's O minus one. That's a line in this E0. And then we apply D on that line and we see if we get zero. So we see if the point of the vector bundle I'm at corresponds to a line in E0 that's in the kernel of D. So that's the way I'm going to try and find which I'm trying to find the point wise kernel of D by looking in E0 which lines in E0 are killed by D. So this is defined by D but literally it's the section of this pullback bundle on the projected bundle whose push forward is D. So all I'm saying is it's defined by D and it's equivalent to D, you can recover D from it. Exercise, I think you can do this exercise on many levels. I think the best thing is to just convince yourself that point wise this is cutting out a copy of the degeneracy locus upstairs. So you have the degeneracy locus downstairs but this is cutting out a natural section of it upstairs. So convince yourself that set theoretically that's correct and there's another exercise which is do it scheme theoretically which is a bit more difficult. So all we're doing is the degeneracy locus is the locus where D has a kernel, a one-dimensional kernel. It's a natural lift of that to the space of lines in E0. Why? Because the kernel is a line in E0. All right? Because it's one-dimensional. So that's set theoretically point wise how you do this exercise. Yeah? Since the rank drops to one, that's the reason you were able to do one-dimensional. So if it is straddle from two, then you get a hyper plane. If you're interested in that degeneracy locus then you take the grass manian of two planes in E0. Yeah, and everything goes through. So now we have our global Karunishi chart, our fantasy model. So we managed to see this degeneracy locus as being cut out of a smooth ambient space. It's this projected bundle by a section of a vector bundle. And what's a bit harder is you can show that the perfect obstruction theory this global Karunishi chart gives you is the Vaffa-Witton perfect obstruction theory. And that's the bit that's harder when S has some higher core homology of its structure sheaf. It's not true then. You have to remove some pieces from the obstruction theory and that turns out to be quite hard, but you can do it. So in particular, the Vaffa-Witton virtual cycle, once you push it forward to this ambient space is just the Euler class of that bundle that you've cut it out by. It's just the Euler class of this bundle here. And then you can push back down to the original ambient space and what you find is the push forward of the virtual cycle to the ambient space is given by this characteristic class. And this is the standard characteristic class called Tom Porsche's formula that governs degeneracy loci. So when your degeneracy loci are all transverse and have the correct co-dimension and so on, it's well known that they are Poincaré dual to this Tom Porsche's formula. And what all we're saying is that when they're not transverse and when they've got their incorrect dimension, their virtual cycle is still Poincaré dual to the Tom Porsche's formula, as you would expect. So that's how this virtual cycle game always goes. But that's enough. I mean, that's fantastic. This is one case where you can really topologically identify the virtual cycle and get characteristic class formula. So now if you want to integrate anything over this virtual cycle, well, you can do it so long as the thing you want to integrate extends to a co-hormology class on the ambient space. But all the ones that we need and that arise in Vaffa-Witton theory, like one over the Euler class of the virtual normal bundle, they all extend to this ambient space. So you get an answer which, importantly, doesn't depend on E1 and E0. It only depends on their difference. So it's invariant under changing that complex up to quasi-isomorphism. And you can extend it to any surface without vanishing. And you can extend it to nested Hilbert schemes as long as you like with more than one nesting. So we're almost at the point where you're computing all of these Vaffa-Witton monopole loci with corresponding sort of young diagram 111111111. But we haven't dealt with curves yet. We've only dealt with points. OK. And a perfectly doable exercise is to work out there's two different perfect obstruction theories on the ordinary Hilbert scheme SN. You can see the ordinary Hilbert scheme SN as SNN. So that's where all your blues look like this, where they're identical on the reduced S and on the first order deformation of S. So that's one way of going from SN to an nested Hilbert scheme is just to thicken it up. And the other way is to concentrate all of the points just on the zero section. And then have none at all on the first order deformation. So you just stick them in here. Why is this a different color? OK. You can't see that. So both of these inherit virtual cycles from this construction. One of them is just the ordinary fundamental class of the Hilbert scheme. And the other one is given by a Chair and Class formula of this topological bundle on the Hilbert scheme. And you have to work out which is which. Yeah, sorry. So the nested Hilbert scheme N0 to Nk for k greater than 2. Yeah. So is it true that this is kind of like the previous case, this nested thing is also direct product of this all this stuff? I didn't understand that. Say that again. This A in the arrow, SN0 product of SN1. The ambient space, yes. Yes, so this N0, the ambient space is SN times a point. In the nested case also this is true. This kind of product structure. No, it's embedded in a product. It's not itself a product. It's cut out of the product by this degeneracy locus construction. OK. And then more generally, again we do it in rank 2, you have to deal with curves and points. And so up to centering with the line bundle what you end up with is these weird nested Hilbert schemes. So here you pick a curve class beta and some numbers of points N1 and N2 and then you pick this nesting so that the blue guys on the reduced S, the zero section can include curves. There's this curve D in class beta. But on the first order deformation of S there should only be points. This turns out to be the relevant thing. So your nesting now consists of a curve and maybe some points and then the next one is just some points lying on that curve or lying on the points. So your first subscheme is one-dimensional and your second one is zero-dimensional and your second one lies in your first one. Or your first one lies in your zeroth one or whatever. I've surely said it wrong. And when there are no points we just call this S beta and it is just the Hilbert scheme of pure curves, Cartier divisors in class beta. And for now because I'm assuming the vanishing of the Jacobian of S and the higher coromology of S it's just a projective space, just a linear system. So it's also smooth. When you drop this simplifying assumption it's suddenly no longer smooth and that causes problems and we come to that in a minute. But for now the linear system is just smooth. Now what we do is we map this nested Hilbert scheme again to the product of the two punctual Hilbert schemes. So at that point we're sort of ignoring the curve but that will come in the fibres. So this will no longer be an embedding. But what we're going to do as we see its image again is a degeneracy locus. So it's the locus where there's a harm from one ideal sheaf to the other. As you see from this point wise description of the nested Hilbert scheme. So again its image is the points where there's a harm from one ideal sheaf to the other and that gives you your embedding. Any harm then gives you an embedding of one ideal sheaf in the other and puts you in this nested Hilbert scheme. But where things are different is now that the harms needn't be one-dimensional anymore when they're non-zero. They can be higher dimensional and what that means is you have fibres of this map. So this map here from my nested Hilbert scheme to this smooth ambient space is not an embedding anymore but it can have fibres. But it turns out it's still part of a general framework. So of degeneracy loci where the rank drops by more than one now. So the rank now drops from being full so from your complex, your map of vector bundles being injective that's the case where there's no harms to suddenly the map of vector bundles having a big kernel that's where you get these harms and so the dimension of this harm is the dimension of the jump in rank of your map D. We have a question. Should N01 be N1 and 2 or there is some non-zero shift? Oh yeah, sorry. Yes, I will correct that on the web. Thank you. I could have stared at my slides for an hour. I would never have seen that. How do people see that? That's incredible. Do people actually look at the numbers? It's just a blur to me. So this is part of a general theory where the ranks drop by more than one and this theory is called virtual resolution of degeneracy loci and again it's a much more general theory but I just do it in this case. So what is it? So again, we managed to write this complex here that's the family version of the X groups I was dealing with before. This is the complex that's jumping around. It's cohomology groups are jumping and we're interested in the jumping locus. We write it as a two-term complex effect of bundles and then we're interested in the locus where that D changes rank. So generically it's injective but then over some locus it's rank drops by more than one, maybe. So again we work on the projective bundle of E0. Again we're interested in the lines in E0 which are in the kernel of D and so they're precisely the zeroes they're cut out as the zeroes of this composition here. So you take a line in E0, that's this. That's tautologically, this is the tautological line. It sits inside E0 on your projective bundle and you apply D to it. This is really Q star of D I guess and you see whether you end up at zero in E1 and if you do you're being cut out by this D tilde and you really correspond to a degenerate point plus an element of HOM, plus a line in the kernel. So plus an element of the kernel of D or the zerothcormology of D plus an element of the HOMs from IZ0 minus beta to IZ1. So what this section cuts out now is not the locus downstairs which was the image of the degeneracy locus. I beg your pardon. So remember downstairs you had this degeneracy locus but we had this nested Hilbert scheme mapped to that onto that you had fibres and we're now seeing those fibres because we're working upstairs we're cutting out the whole nested Hilbert scheme upstairs. So again the exercise is to convince yourself of this point-wise. Scheme theoretically it's a little harder but it's fine, I mean it's perfectly doable. So again what you should think of is what is this nested Hilbert scheme here? It's two ideal sheaves plus a HOM between them up to scale and that choice of HOM is what's encoded in this p of E0. Because what we're cutting out by this D tilde is the lines in E0 which are annihilated by D. So they're really lines in the kernel of D and they're lines in HOMs from one ideal sheaf to the other. So up to scale, up to scaling they are just HOMs from one ideal sheaf to another which is precisely what the nested Hilbert scheme comprises. Okay so that's just pointing out that you can do this in much more generality using maps which you can look at some locus where you don't have to take maps which are generically injective. Okay you can do this in much more generality. And so again you've got a global coronetian chart cutting out this nested Hilbert scheme and it turns out to induce the perfect obstruction theory that Vaffa-Witton gives, that Vaffa-Witton theory gives. So it says that when you push forward the virtual cycle to the smooth ambient space you get the Euler class at the bundle that cuts it out. But that's not the thing you really want to do because E0 is not a derived invariant. Remember E0 going to E1 makes perfect sense up to quasi-isomorphism but E0 and E1 themselves aren't really invariant things. They involve some choices. You don't want to use them. We can't just push down. Last time we just pushed down to the product of Hilbert schemes but we don't want to do that now because that loses information because there's fibres. It's no longer an embedding. So that would be fine if you were integrating coromology classes which were pulled back from that map but there are coromology classes like one over the Euler class of the virtual normal bundle which are not pulled back from that map. So you don't want to just push down because that loses information. So instead what you do is you observe that there's a natural embedding. So essentially you start using the linear system. So there's a natural embedding of each fibre inside the linear system. In other words, whenever you have an embedding of a zero-dimensional subscheme of a one-dimensional subscheme you can just remember the pure one-dimensional subscheme. You can throw away all the points. So this gives you a natural embedding here. As you mean here, rank drops by one. No, it drops by more than one. Yeah. It drops by the dimension of that home space. So it drops by... The amount it drops by is varying all over the place. Depending on... This home space is jumping around in dimension. So is it not kind of... Why do linear system comes? It can be much more than... There are many other sections for each one. I made an assumption early on to simplify things that the surface had no one forms or two forms. And that means that this linear system is just a projective space. We'll come back to the general case in a bit. So what we can do is instead of embedding these fibres in P of E zero, which is kind of a silly thing to do, we embed them naturally in this linear system. And what you can do is prove a theorem that you can transfer across the virtual cycle. Yeah, I'm not going to do it. It's a bit tricky, but you can do it. And so what you can find is that there's a Tom Porsche's formula in this ambient space over here. What you're effectively doing, what you're trying to do is replace E zero by some kind of canonical E zero. And in some sense, this trivial vector bundle given by the sections of your line bundle or divisor, all the divisors linearly equivalent to your... in this linear system beta, that gives you a replacement of E zero. And that doesn't quite work, but that's how you should think of it. So there's a Tom Porsche's formula here, and there it is. I mean, it really doesn't matter, the form of it. It just says that, again, the virtual cycle, when you push it forward to this smooth ambient space here, it can be computed by a Turing-class formula of tautological things that people are experts of computing with. And then the nasty case, the last case, which I won't do too much about, is when you take arbitrary surfaces with higher core homology, because now your linear system fibers over a Jacobian and the fibers keep jumping in dimensions. And so this S beta is no longer smooth. So this S beta now, which is the Hilbert scheme of all pure curves in class beta, this is now singular. But it turns out you can work over a singular space. You can still prove something. What's the best way of describing this? Yeah, maybe I won't go into this. I mean, I'm sure I've lost you by now anyway. It turns out that you can still get a Tom Porsche's formula, and this is what it is, but it's a Tom Porsche's formula in a singular space now. And again, you can extend this to arbitrary, so now really to arbitrary nested Hilbert schemes. The entire, this means you can compute the entire monopole locus in Waffa-Witton theory for these young diagrams L partitions, which are of the form one, one, one, one, one, one, one, one. And they're the really important ones. So in this blue line, you want to be in multiply with the virtual class on this S beta or... Yeah, that was the thing I didn't really want to get into. So you, because I've forgotten. Yeah, I'm a bit nervous about that now. You should do, right? I'm just wondering if it's embedded in the right-hand side in some clever way. Yeah, I've confused myself now. I think you're right. I think it's this capped with the virtual cycle of S beta. I will check and update my slides. Yeah, sorry about that. Okay. Okay, I think I said that already. So what this means is that up to the point that Georg mentioned, which is this virtual cycle of S beta, it means you stand a chance of computing all of the one, one, one, one components of the Faffer-Witton theory. So that corresponds in the language of sheaves on the Kalabiyar three-fold, the local Kalabiyar three-fold. This is all sheaves of rank one on their thickened support. So they have support R times S, where R is the rank, and they're rank one on that support. Okay. So that's what you're able to compute. Modulo this virtual cycle of S beta problem. But the beauty of that is the virtual cycle of S beta is very well understood because of cyber-Witton theory. So it turns out that if you're just dealing with curves in your surface, that's more or less the algebra geometric version of cyber-Witton theory. And that's been studied by other methods for a long time. And so we actually understand the S beta part in some sense, or at least we can call it something so then we can pretend we understand it. So we can reduce the whole of Faffer-Witton theory to computations on nested Hilbert schemes of points or just products of punctual Hilbert schemes of points. They're things we understand. Times by cyber-Witton invariance. Okay. So now I'm more sort of just survey what this gives you and Lota talked about this as well. So this reduces monopole contributions to Faffer-Witton invariance to computations, topological computations on products of punctual Hilbert schemes and then the products of these curve spaces, these spaces of curves, Hilbert schemes of pure curves. And Laraca carries these out when you have a, when your surface has a holomorphic two form using something called co-sectional localization which I'm going to discuss a bit tomorrow. And the integrals over these nasty spaces of pure curves turn out to be cyber-Witton invariance so you can actually handle them or you can call them something. Actually, yeah, you can really handle them. We know what they are. So this is cyber-Witton invariance that you want, you want cyber-Witton? Yeah, yeah, that's right. And then the integrals over the punctual Hilbert schemes, so Lota mentioned this, he does this via the Elling's Root Gertscher Lane procedure from 20 odd years ago where they show how to turn integrals over Hilbert schemes to integrals over the corresponding products of surfaces and therefore you can integrate. And then what you find is the result is sort of a co-borderism invariant in some sense and it only depends on topological data. And once you've done that, incredibly, even though he's dealing with general type surfaces, he says, well now I can compute this topological data on any surface I like, so I'll do it on a toric surface which is definitely not general type. So essentially you end up doing these nested Hilbert scheme computations on toric surfaces even though for toric surfaces you have a vanishing theorem which says they don't contribute to the Vaffa-Whitney invariance. So you end up, to compute the Vaffa-Whitney invariance of general type surfaces, you do computations on toric surfaces which are not relevant to the Vaffa-Whitney invariance of toric surfaces. They're relevant to the Vaffa-Whitney invariance of general type surfaces. And so from computations on toric surfaces and K3 surfaces where we can do derived equivalences and change these things into sheaves and stable pairs, and we can use Mukai's symplectic structure and all the results saying that modulized spaces of sheaves on K3 surfaces are really nicely behaved and they're essentially just Hilbert's schemes of points. You really have a lot of control on K3 surfaces. And so eventually you can compute these things in low degrees, and you can prove general structure theorems that they only depend in certain ways on topological numbers. They have product formulae low to discuss this a bit. So what you end up showing is the Vaffa-Whitney invariance only depend on this topological data. So why don't you have a dependence on C1 times better or something like that? C1 times beta, you might do why? I haven't written that down, have I? Yeah, you might do. I'll check that. The virtual dimension of beta i must be zero. Ah, yeah, there's an issue there, that's right. Sorry, say that again. The virtual dimension of beta i is related to these numbers so that you can construct. Yes, that's exactly right. Somehow, the ones which matter, those numbers are zero. I remember that now. Thank you. Okay, so here's the simplest result of his that you can state. I hope I get it right. S is minimal general type. So minimal means that the Cyberg-Whitney invariance don't have any contributions from minus one curves and they're just the canonical class. That's really important. Otherwise, the Cyberg-Whitney contribution becomes a bit more difficult to state, but you can still state it. But in this simple case, what you find is that the monopole contributions to rank two Vaffa-Whitney invariance determinant equal to the canonical class can be written in this form. So the generating series can be written in this form. And these universal functions that you have, these A and B, which don't depend on the surface, are completely determined by K3 calculations. And, well, sorry, one of them is determined by K3 calculations, so we know what it is. It's a modular form, a certain type. And the Bs can be predicted by modularity and instant calculations of Gertrude and Kohl. And he checks these predictions in low degree. So he can do these computations explicitly in low degree and use a computer and check that he gets the right answers. Okay, so how are we doing this time? Okay, so next time I'll start with his vanishing theorem. So Georg talked about a version of this. And I'll talk about it in more detail. So this is what tells you that... So what we've done today is explain how to compute these Vaffa-Whitney invariance in this... corresponding to this maximal thickness of S with rank one over the top. So these profiles, one, one, one, one, one. And now we need a vanishing theorem so that the others don't contribute. And that's what I'll talk about next time. Any questions, comments? I have a question. With an SCK-Ber scheme, you're working with N01, but just one class beta. But don't you want also to nest two classes in homology beta, like theta one and theta two? It's because I was doing three things, right? I had a Z0, a Z1, and the structure sheaf. But what I can do is if you want to put in another curve, you can put that into the structure sheaf. So ideal sheaves are not really relevant. What's relevant is sheaves with a map between them. And it just turns out... So you have your Z01 and your Z11, and that's really what's relevant. And then what it turns out is an ideal sheaf is on a surface, is always a line bundle, tensor and ideal sheaf of points. So that gives you the map to the third thing, the map to the structure sheaf. So you're absolutely right. You might have IZ1 twisted by some minus beta prime. But we wouldn't then put that in the structure sheaf. The point is you just put it in its double dual. So that thing happens to embed in OS minus beta prime. And now you can just tensor everything. This is just a line bundle. Now you can tensor everything through by that line bundle and get rid of this. So the point is that this map is not really relevant. It's not what's coming up in Buffer-Witton theory. It's just the double dual map. It's just the fact that we're exploiting the fact that these abstract sheaves, by embedding them in their double duals, you can see as ideal sheaves. So that's what this last map is about. And that's what soaks up the last divisor that you're asking about. Yeah, it takes some getting used to. We have two questions in the Q&A. The first one is, since the surface S could be non-compact, where is the algebraic underpart of the boundary conditions of the field content? Since it can be non-compact. Yeah, it can't. For me, all my surfaces are compact. It's the Kalabia 3-fold that's non-compact. The canonical bundle of the surface is non-compact, this guy. But this is, for me, this is always compact. I appreciate for physicists that may not be the case, but we can't handle those cases. The second question is a general question. Are the singularities of the nest did by skin so relevant here? For example, does this wall machinery make any prediction or require any results about that? I doubt it. I would say I would guess the singularities can be pretty bad, but I don't know that. And this perfect obstruction theory does not really constrain them much at all. A symmetric perfect obstruction theory does constrain singularities, but they don't have this. They're the fixed loci. I think the answer is just no. I have a question as well. You mentioned that the Waffen-Wiffen in the original paper calculated some of these modular functions. Another method. Are those only coming from this instanton branch or do they also include things in the monopole branch? Mostly they computed on the instanton branch and then they did one computation on a general type surface where they didn't say what they were computing, but they said whatever it was, or maybe they just worked in... Anyway, they used a holomorphic two-form on the surface to localize the whole theory, whatever that means, to a canonical curve on the surface. And then on that canonical curve, again, they didn't say how the theory was defined. They didn't talk about monopole loci or instanton loci. They didn't use the equations at all. They were using the field theory or something. So they weren't just working on shell with solutions of an equation. They were really using some quantum field theory or something. And they were saying we can localize to this canonical curve and then they said cosmic string and then they said, and therefore the answer is this. And I asked Witton about this and he said, oh, that's two quantum u-geometers are never going to see that. Not without doing something drastic. However, the co-section localization I keep hinting at and I'll talk about next time precisely localizes the geometry to a canonical curve. And it really was the co-section localization was invented by Keeman-Lee precisely as a crude approximation to Witton's perturbation that Witton used it at the time to compute Donaldson theory, Waffa-Witton theory, many things. So yeah, we do have a crude geometric approximation to their argument. We'll see that it localizes to the canonical curve. But then, yeah, we have to work much harder. We can't just say cosmic string, unfortunately. So then we really have to do computations. The people who do say cosmic string, they can also get, they can also somehow calculate these. They can in rank two, they can more or less write everything down. But then in higher ranks, I'm not an expert here, but Martin Kuhl tells me that in higher ranks they could also write things down, but they were wrong. And so I don't know whether that means they just made a mistake or whether cosmic string is not powerful enough or I don't know the status of that. That's the question. Yes, go ahead, Eugene. So in case of Ness's Hebertino points, is it possible to define this abstraction theory which is kind of local on S so that like all these points, you can see they're like in sum of fine shares on S and then you want some abstraction theory which is also in that fine chart. Yeah, I think what we do is of that form. Let me just think a second. Precisely because we remove the cormology of the structure sheaf of S, I think what we do is of that form, but I can't promise it, but I could look at our paper. So I think the answer is yes.