 And some sense it's based on collaboration with, I had earlier, I mean, at least my side with Iraku and Kudaiyoshioka, at least they are some similar. It's OK. So I work overseas. And I want to study the topology of some modular spaces of sheaves on algebraic surfaces. OK. So maybe to start a bit. So I just want to say what we understand by a modular space in algebraic geometry, roughly. So by this, I mean, essentially, this is an algebraic variety or a scheme. Is it big enough that M that parameterizes some objects, some interesting objects in algebraic geometry in a natural way? So naturally. So parameterizes means that the points of M, so this one could maybe just write M of C. This will just be the objects we are interested in. So for instance, the sheaves. And naturally, it's compatible with families. So for instance, if E over pi over T is some family of objects parameterized by some scheme T, so that means the fiber, for instance, the fiber of pi over the point T and T will be one of these objects. So then there is a natural map from T to M, which sends a point T and T to the fiber. And this is a morphism. So it's an algebraic map somehow given by polynomials. So this is very roughly the thing, depending on what objects one wants to study. For instance, for sheaves, it could be a formalized piece of sheaves. It could be sheaves up to current sheaves up to isomorphism, could be vector bundles, it could be curves, whatever. So the example which is important for us is the Hilbert schema points, because the whole story is built on this thing. So I noted Sn with square brackets, or Hilbert N. This is the Hilbert scheme of points, of N points on the surface. So a general point of S will correspond to just a set, P1 to PN, of N district points. And I should maybe say, in general, Sn parametrizes zero-dimensional sub-schemes of degree N on S. So as I said, generically, this is just N different points. But these points can come together, and then you have a multiple structure. So such a sub-scheme Z in Sn is given by an ideal sheave, Iz, among OS, or the holomorphic or regular functions. So these are the holomorphic functions, or regular functions, vanishing on the sub-scheme Z. And such that the quotient, so this is, if you want, just an ideal in the ring of all holomorphic functions, such that the quotient is a finite dimensional vector space and has finite support. The quotient is OZ, which is just all holomorphic functions divided by its ideal sheave, has finite support. So only over finitely many points, the fiber of this quotient is non-zero. And the degree of the sub-scheme, which is the dimension of, if you want, the sections of OZ. So just everything which is in the quotient is a finite dimensional quotient. It's just the dimension of this quotient, which I can write as the sum over all the points in the support of Z. So where this thing has support of the dimension locally at the point, so the local ring at the point times. So if I take the sum over this dimension of this thing, it's the sum over the dimension of the fibers of this thing over the points where it lives, and this should be n. And so if these are n distinct points, then this ideal sheave are just the functions which vanish to first order at these points. And then you just have one dimensional vector space at every point, and it adds up to n. Very simple. Let's look at some just from concrete list. Let's look at some very simple examples. So we have that the fiber scheme of zero points on S just consists of one point, which is the empty set. Then the fiber scheme of one point consists is just S, namely the set of all points P in S. And the fiber scheme of two points is a bit more subtle. So either you can have two distinct points, or you have one point plus some additional structure so that the quotient has dimension two. And it turns out this is the same as a point P plus T, a tangent vector to S at P, up to modification, so up to rescaling the tangent vector. And so it turns out that this thing is obtained as follows. We take S times S, we blow it up along the diagonal, and we divide by the action of the symmetric group, which permits these two factors, and the action lifts to the blow up. And the later ones will be a bit more complicated. So this thing is, as you can see in these small cases, is related to the symmetric power, which is a simpler thing, parameterizing endpoints. So this is just as n with round brackets or sum ns, which is just the end food product divided by the action of the symmetric group by permuting the factors. And this will parameterize just endpoints counted with multiplicities. So it's a formal linear combination, sum n, i, p, i, the p, i, i, and s. The n, i are distinct points. The n, i are some positive integers. And the sum of the n, i is equal to n. So that's very simple. And there is an obvious map, which forgets this scheme structure and just remembers the multiplicity, sending a scheme to its support with multiplicities, which is called the Hebert-Schau morphism, which is, turns to be an algebraic morphism, so given by polynomials. So p from the Hebert scheme of endpoints to the symmetric power, which, as I said, sends a subscheme to its support with multiplicities. So by which, I mean, the sum over all points where the subscheme lives, the multiplicity at that subscheme, which is the dimension of this local space, times the formal sum, times the point, OK? And so for a surface, this Hebert scheme, this symmetric power will be singular, because you have some co-dimension in the fixed look, look closer if the action is not in the right co-dimension. So then it follows that, so this Hebert scheme, however, is non-singular and complex dimension to n. And it's actually a resolution of singularities of the symmetric power. So this is a very simple example of such a modernized space. And now you can also, in simple cases, if S is simply connected, I can also view this as a modernized space of rank 1 sheaves by looking at the corresponding ideal sheaves. So as an example of the modernized spaces of sheaves you want to consider. And so therefore, you can look at the topological invariance. And so there's, for instance, the simplest one one can think of is the topological Euler number. So this is, so for space, topological Euler number of x will just be the sum, the alternating sum of the depth, petty numbers, two times complex dimension of x of minus 1 to the i, the dimension of the i-th homology group, some integer. And the old result says that if I take the Euler number of the, if I want to look at the Euler number of this symmetric power, then it's particularly nice to write a generating function. And it's given by this attractive product formula. So we have a simple formula. Take the product of all integers, 1 minus t to the n. It happens to be that if you have it without the Euler number it's a generating function for partitions. And to take this to just the power of the Euler number of the surface, this gives you the generating function here. And it's also interesting for us that if I look at this factor here, so this product 1 minus t to the n, this is up to a trivial factor, t. If I now call tq to make it more nice, I mean, more familiar. So this is up to this trivial factor, the Dirichlet eta function, which is the 24th root of the discriminant model of form, and so it's a nice model of form. And so in particular, we also see that this Euler number, the generating function for the Euler numbers is expressed in terms of model of forms, in this case just by this one model of form. And so we want to, so the aim would be get a similar formula for modular spaces of sheaves on s, also a nice generating function in terms of model of forms. So most of you will notice, briefly review, model of spaces of sheaves on surfaces. So we take s, most projective surface. So that means it can be embedded into some projective space as a closed sub-variety over, always I work, always over c. H should be an ample line bundle. So that means there is an embedding, I mean, essentially it means I can have an embedding of s into projective space, so that this is the pullback of the hyperplane bundle. And we fix the rank r, which I assume positive, the first-journ class in the second homology, and c2 in the fourth homology, which one can always identify with integers if I assume that s is reducible. I expect you know what the churn classes are. So the churn class of a vector bundle are some invariance, ci of e associated to a vector bundle in the 2i homology of whatever the vector bundle lives on, which somehow tell us something about how far e is from being topologically trivial. For instance, if this would be the trivial vector bundle, then all churn classes would be 0. And so we want to study modular spaces of rank 2. For the moment, in general, rank r, torsion tree, coherent sheaves on s with these churn classes, c1 of e is equal to this given c1, c2 of e is equal to c2. I mean, I don't know, maybe some of you don't know what a sheaf is. So you can think of a sheaf as a vector bundle with some singularity. So there are some fibers where the dimension of the fiber is larger than the rank of the bundle. And the torsion tree sheaf would essentially mean it has only finitely on a surface. It's not quite true, but essentially it means it has only finitely many singularities and they are not too bad. So just to have some vague idea. OK, so there is no modular space of all these sheaves. You need to put some condition, which is stability condition, which somehow means that these sheaves don't have two big sub sheaves. Or if they were for vector bundles, it would mean that they don't have two big sub bundles or sub sheaves, where big means having many sections. So let me just remind you. So we have an E is a coherent sheaf on some steam x or some variety. Then we have the ith sheaf homology. So H0 is just the global sections. And the HI are somehow, HI are some kind of the obstructions, some obstructions to have in global sections. And then we have the holomorphic Euler characteristic of E. It's just the alternate, is again like the Euler characteristic before, the alternating sum of the sheaf homology group. So it's chi of xE, which is defined to be the sum i equals 0 to the dimension of x, and to which minus 1 to the i, the dimension as a vector space of HI of xE. And then now let's E be a coherent sheaf on our surface S. And H was ample, ample line bundle. Then we have the Hilbert polynomial. This will be just we take this polynomial in some number. M is just the holomorphic Euler characteristic of E. So of E tensor H to the m. This turns out, for instance, by Riemann-Roch or by whatever, to be a polynomial in M. And then I should maybe say for what follows, one doesn't really need to talk about this. If M is sufficiently large, then there will be no higher homology. So this thing is just the global sections. And then we find that, so then the definition is that definition. So a torsion-free sheaf E on S is called H semi-stable. If for subsheafs, F of E, we have that if we take the 0. So if we take the Hilbert polynomial of the subsheaf and divide by the rank of F, this is always smaller equal to the thing for E. So as a polynomial, which means if I put M into it, this will be true for all M sufficiently large. And as I said, if M is large enough, this is actually the global sections of E tensor or of M on F tensor or of M. So it also just means that subsheafs are not allowed to have too many sections as compared to the sections of F. So OK. And then there is the theorem that such immodellized space for these things exists. I am a Gisika. There exists. I should maybe have said, so if this inequality is always strict, when I assume F is non-zero and here we also obviously F is not equal to E, then it's called stable instead of semi-stable. So there exists whatever that means, course is a technical thing that we are not going to use. Modellized space M has hrc1c2 of h-stable, h-semi-stable torsion-free sheaves on S with given Schoen classes, so of rank R. And the first Schoen class given c1 and the second Schoen class given c2, which is a projective variety. So this is M, or not necessarily a variety, but it's projective scheme, msrc1c2 is projective. And inside this we find as an open subset in the risk etopology, the space which parameterizes all the stable sheaves, which I can work like this. And so this is then quasi-projective. OK. So now we want to study the topology of this thing. There's one problem, and there's one thing here. So this modelized space has something called expected dimension, called vd of M, where M stands for this. So assume maybe that the first petty number of S is equal to 0, so that is the dimension of the first homology. So this vd of M is given by some formula, 2rc2 minus r minus 1, c1 squared, so this is the intersection in the homology, gives us a number, plus r squared minus 1 chi of S, where this is the holomorphic order characteristic of the trivial bundle. So it has this expected dimension. So now first one could ask oneself, what's that? It's not actually true that this thing will have this dimension. It's just the dimension that we expect it to have. In gauge theory, you have a similar modelized space. You can look at the modelized space of ASD connections, which corresponds up to gauge to some open subset of this. And somehow you can describe in terms of some Fretholm operators, and then you would see you also have an expected dimension here, which is 0. But for us, just let's say, so vd of M is for the moment just the dimension that M would have if it was nice. Or let me, for instance, just one way to describe this is the Kuanishi picture. Namely, locally, in the unedited topology, this thing can be written as a 0 set of a holomorphic map from some, say, C to the M via this map to C to the K, where the virtual dimension of M is equal to this difference. So near different points of the modelized space, these numbers might jump, but the difference is always the same. And so thus, if this was a smooth map in the sense that it's a submersion, so the differential is surjective, then it would follow that this modelized space would be non-singular of dimension equal to this virtual dimension. So we'll come back to this in a moment, because this fact somehow will allow us to pretend later that the modelized space is smooth in a certain sense. So now let me talk about this rougher written formula, which is some generalization to these modelized spaces of this formula head for the Euler number of the Hilbert scheme. So we say, again, S is a projective algebraic surface. As before, H is ample on S. And we also assume, as we did a moment ago, that the first particular number of S is 0. And we also assume that so-called geometric genus of S is bigger than 0. So this is the dimension of the sections of the canonical line bundle. So the canonical line bundle, the sections of these are the holomorphic two forms. So that means S has non-zero holomorphic two forms, global non-zero holomorphic two forms. So an example would be S is the k3 surface, or an elliptic surface, or a surface of general type. Well, not always, but many surfaces of general type most. So we make a restriction. So we choose our H and C1 and C2 such that stability is the same as semi-stability. So first, we assume we put ourselves in the situation that our rank is 2 for now. And then I write the modelized space just like this. So it's understood the rank is 2. And then I want to say I make, I choose these such that this is equal to the stable ones. This will very often be the case. It's basically mostly a condition on C1. But we can also make, so there's coarse condition on C1, which will guarantee this. And the finer one, if the one for C1 doesn't hold, which will also guarantee it, but it's not always the case. And let's remember, in this rank 2 case, we have our virtual dimension is given by this formula, 4C2 minus C1 squared minus 3 times square of 4S. And you write KS squared is just, we evaluate, we take the first-journ class of this line bundle, take it square, and we integrate it over S, so we get a number. And then we will, so then I make another assumption which is not necessary. It just makes the formulation of the result simpler of the statement. Namely, let's assume for simplicity, we assume there exists an irreducible curve in the linear system KS. So that means there exists curve C, which is irreducible. For instance, if you want a non-singular connected curve, which is a zero set of a section, S is a section of the canonical line bundle, and C is irreducible. So in this case, we get the following baffled written formula, which is one of the statements. So we write down, again, this eta of x with a bar is what I had before. You take the eta function times x to the minus 1 over 24. So this is the product 1 minus x to the n. And we also look at the, so this is a modular form up to this trivial factor. We also look at the standard theta function, theta of x, which is just the sum over all the integers x to the n squared. So these are two examples of modular forms. And then, so if you write it out, you know this is 1 plus 2x plus 2x to the 4, and so on. And so I just write, now, CS of x is the following expression. It takes 8 times 1 over twice this eta bar of x to the 12. This I take to the pi of 4s. And I multiply this by 2 times eta of x, so eta bar of x to the 4 squared. I didn't leave enough room, so maybe I write here. So 2 times eta of x to the 4 squared divided by this theta of x we had. And this I take to the power ks squared. So I write down this crazy power series. And then the Waffelwitten formula is that the Euler number of this modelized space, so rank 2 she's, which is 1 plus c1, c2, under suitable assumption should just be the coefficient of x to the virtual dimension of m of this power series. So I have to say a few things to that. So the first thing is that as also Pavel mentioned in his talk, there's a more general Waffelwitten invariance with more terms. And the complete Waffelwitten formula talks about also that. This is only a part of the formula coming from this modelized space of she's. So let me just write this down for once, because it will also be mentioned by other people. So the actual Waffelwitten invariant, so which is what is used in the paper by Waffelwitten, was defined mathematically by Tanaka and Thomas. And then for this, we consider instead a modelized, consider invariance of modelized spaces of x she's. So we have n s h c1 c2, which is, so the parametrize is she's with these rank 2 with these turn classes. And phi, the x field. So this is e is a rank 2 coherent torsion to the sheave on s. And phi with these turn classes. And phi is a homomorphism from e to e tensor, the canonical bundle. And I think it's also assumed that the trace of phi should be 0. So one looks at this modelized space of these things. And there is a modelized space for this. There's a stability condition, which says basically the previous condition for sub she's should hold for all sub she's, which are invariant under phi. And this has a c star action by rescaling this x field phi, namely lambda in c star. If I multiply it to a fair e phi, this should be just e. And then rescaling this field. And so the full rougher written invariance would be, say I could call this rougher written invariant of c1, c2 would be, we'll see later what that means. It's in some sense the Euler number, but in some virtual sense of this. I look at this modelized space, but I look at its c star invariant part. So we look at the fixed points under the c star action on this space and compute some kind of Euler number of it. And one should note this fixed point set has several components. So one or some of these components just correspond to the fact that phi is 0. I mean, so are components where phi is equal to 0. And so these components, the union of these components, is just our old modelized space. And then there are also other components. So this is sometimes called the instant on branch and other components, which would be called the monopole branch. So what we are computing here is only the instant on branch of it, which is the modelized space of sheeps. And I think there will be several people in the second week who talk about rougher written invariance. At least I know that Laraca will talk about rougher written invariance. Maybe also Atanchesh Mani. I don't know whether somebody else. So therefore you will hear more about it. But I concentrate on the instant on branch, so just on the modelized space of sheeps. So we have here this, you already mentioned here this virtual Euler number. I didn't say what that is. Maybe in this case it was. So I have to say what I mean by virtual topological invariance of such spaces. And this has to do also with the stuff I said here about the expected dimension. So how much? So virtual topological. Yeah, I should maybe say that. So here I have this rougher written formula, but I'm not actually saying that I claim it to be true. I will try to explain in what sense it's supposed to be true. Virtual topological invariance. So as I said, this modelized space will be usually quite singular. It will often have many components, and these can all have different dimensions. And so not of expected dimension. But it has this expected dimension dd of m. So this means that, so what one could say it is that it is what one could call virtually smooth of dimension dd of m. So and something about it behaves as if it was smooth of this dimension. So this will mean that one will be able to define some invariance for m, which behave like for a non-singular variety of dimension m. So technically, what I mean by this virtually smooth is that m has one perfect obstruction theory. So I will briefly define this. So maybe not in complete generality, but only what I need. So let m be a scheme which has an embedding into a smooth scheme. So that just avoids the most terrible problem. So in particular, so m is embedded in some x. So for instance, m could be projective, or quasi-projective. Then it's embedded into projective space. And so it's embedded into smooth scheme. And I take e to be the ideal sheaf of m and x. So these are all regular functions on x, which vanish on m. So a perfect obstruction theory on m is a complex, a very short complex, just the dot of two vector bundles of vector bundles on m. With the morphism of complexes, so we have this e minus 1 goes with d2 e0. And we have here this morphism. See, here we take the ideal model of the ideal squared, so basically the normal sheaf. So m in x have derivative to the cotangent bundle, or the cotangent sheaf of x, or in this case, cotangent bundle because one singular, restricted to m. Such that two things hold. First, if I take phi as a map from the co-kernel of d, so this divided by the image of n minus 1 to the other co-kernel of d, is an isomorphism. And secondly, so on this co-mology level, it's an isomorphism, and at this level, it is at least subjective. So phi from the kernel of phi of d to the kernel of d is subjective. So this is the condition. So roughly speaking, this somehow is, so you have a, here one can somehow see the cotangent bundle of x somehow is obtained here, and here one has the normal bundle. If one puts it together, one has somehow the, this means that this vector bundle here, this morphism of vector bundles, will capture both the information about the tangent spaces of our m, I mean, cotangent or tangent by reality, and which means how you can infinitesimally move points, and the obstruction spaces, which means how you can, what prevents you from having an infinitesimal deformation to become an actual curve. And so this is both covered just by these global vector bundles, and so instead of having all these local things, you have it all captured in one thing. And so we denote the virtual dimension of m to be the rank of this complex of vector bundles, by which one means this is an even dimension, this is an odd dimension, so a rank, so this is the rank of E0 minus the rank of E minus 1. So this is the expected dimension, or virtual dimension, of m. And then there's this theorem. So say, Berend van Tecky, Lee Tjan, which says, if you have this, you have some nice structure on m. So let m be a scheme with the one perfect obstacle theory. Then we have first m has a virtual function. So this means we have a class m where, which lies in the homology group, which the fundamental class of a complex variety of dimension vd would have. So this is in the second 2 vd of m, homology of m, my assumption like this, which, I mean, behaves in many ways, was the fundamental class of a smooth variety. And so as alpha in the homology of m, we can class. So we can define the virtual intersection number. Just we evaluate on this virtual fundamental class this class alpha. And it also, we will use later, also that m has a virtual structure sheaf. So we can consider the regular holomorphic functions on m, something else, some other sheaf or holomorphic function on m taking into account this virtual. So this is m here. So this, however, is not directly a sheaf. It's actually an element in the Groten-D group of coherence. Don't know whether you just finish. I give you the definition of that. So if x is a variety, so k groups, x is a variety, we can look either at k upper 0 of x. This is the Groten-D group of vector bundles. So these are formal linear combinations of vector bundles. So where n is some positive integer. And the ai are integers and the ei are vector bundles. And this is more some equivalence relation. Namely, we have that if we have an exact sequence, 0 goes to e, goes to f, goes to g, goes to 0 of sheafs, we clear that f is equivalent to e plus g. And k0 of x, the Groten-D group of sheafs, is the same with vector bundles, replaced by sheafs, replaced by coherent sheafs. So for instance, note that if so, we can define Schoen classes of elements in the Groten-D group of vector bundles just by applying formally the Whitney product formula, c, from some ai, ei will be, so the total Schoen class will just be the product c of ei to the power ai. And it's compatible with this relation. And in particular, m also has a virtual tangent bundle. So this is just tm here. This is e0 minus e1 in the Groten-D group of vector bundles. On m, where e lower i is the dual of e minus i. So we had in the definition of that. And then finally, we can at least define to close it up what is the virtual Euler number of m, namely, so assume that m is compact, so the virtual Euler number. So then I take the integral over the virtual fundamental class of what should be the top Schoen class, so cvd of m of the virtual tangent bundle. So this is the virtual Euler number. And this is an analogy to the standard fact, maybe sometimes called Hopf index theorem, that the Euler number of a smooth variety is given by the integral over the top Schoen class of the tangent bundle. OK, maybe I stop here. Thank you very much.