 We had S was a projective algebraic surface over the complex numbers and we were considering the modular space of each stable rank 2 coherent sheets on S with Schoen classes C1 and C2 coherent torsion fissures with these Schoen classes and we also I also had introduced the Hilbert scheme of points Sn equal to Hilbert S which parameterizes two dimensional sub schemes and I had ended by talking a little bit about obstruction theory and virtual invariance so for the moment I just want to record that this means we have a complex so just 2 E minus 1 to E0 complex of vector bundles which somehow capture information both about the tangent space of say M and the obstructions to lifting tangent vectors to actual curves so to deforming points in the modular space so and if I so I have the virtual tangent bundle which would be E0 minus E1 where EI is equal to the dual E upper I and we had wanted to compute something like the Euler number of these modular spaces and this was supposed to be replaced by the virtual Euler number which would be the evaluation so if one has such an obstruction theory with these properties it leads to the existence of a virtual fundamental class which will lie in the homology corresponding to the virtual dimension of the modular space and so we can integrate any homology class over this homology class particularly we can take the top trend class of the virtual tangent bundle and we'll get some number which is the virtual Euler and now we want to see how this a little slightly more explicitly how this plays out for the modular space of sheaves so I don't need it but for simplicity I will assume that we have a universal sheave on the modular space so let E over S times the modular space the universal sheave so that would be sheave so that any family of such sheaves can be obtained by pullback from this by a map to this modular space but you know for us it's maybe enough to say that if we take the universal sheave and restrict it to S times this thing parameterizes the sheaves so every point in this would correspond to a sheave so if we restrict it to S times the sheave parameterized this will be just the sheave parameterized for E and M okay so this is this universal sheave and then I would just write down so what the dual of the obstructions theory is can be written as an element in the derived category so we have so we look at the projection from S times the modular space so the modular space and then the dual of the obstruction theory is R the lower star of R home E E 0 shifted by 1 in the derived category of M so this means so this is we have so an element here would be a basically a complex of vector bundles on M and this one means we have shifted it by one to exchange have all the signs the opposite if we look at the alternating sum and here what this basically means is that the tangent so this means trace free so the tangent to the modular space at the point corresponding to E will be X 1 E E trace free and the obstructions will be X 2 E E trace free and this is basically what is encoded in this and and so here I've said there's an element in the derived category but one can resolve this so we can write this so can represent this as a complex of vector bundles on M which would be E 0 goes to E 1 which is just a dual of this so it corresponds to the tangent bundle so the virtual tangent bundle will be indeed easier minus E 1 in the K group of vector bundles on the modular space so this is and so this gives us the following by this what I told you about this general thing about virtual smooth varieties and absorption theories this gives us a certain package of things namely so this gives us so we have first that M has a virtual fundamental class in the correct dimension and then we can integrate we have the virtual tangent bundle as we have seen so the virtual Euler number of M is the integral over the virtual fundamental class of this thing so the top churn class so let me write C V D of M of the virtual tangent bundle and we also have as I said a virtual as I said the other time a virtual structure C so M has a virtual structure sheath so this is O M here which is just an element in the Grotendie group of sheaves of coherent sheaves on the modular space and this allows us for any vector bundle so our complex of vector bundles on M we can compute we can define the virtual holomorphic Euler characteristic so this would be chi here of M V this is just the holomorphic Euler characteristic the usual one so the alternating sum of the sheave homologies over M of V tensor the virtual structure sheath and one nice property that this has which one is to use for these things is that we have the analog of the usual Rho theorem virtual Rho namely that the virtual homomorphic Euler characteristic will be equal to the adoration on the virtual fundamental class of the churn character of V times the taut genus of the virtual tangent I mean I expect you are familiar with these so churn so these are certain expressions in the churn classes of these things which are kind of standard so one can compute this by this formula in terms of homology of just churn classes of the tangent the virtual tangent bundle and and V this is the analog as the of the usual Rho if one just gets rid of the two ways here one gets the usual Rho I mean the three ways these as long as you cancel this this and this we have the usual Kirchhoch Rho form okay now in this context our conjecture is that this Waffa Witten formula holds for these virtual invariants so the for the virtual Euler number so I should say I should remind you that our assumption for this was that the first pattern number of s was supposed to be zero and we were assuming that the geometric genus of s was bigger than zero so that means there there are holomorphic two forms I mean there are global holomorphic two forms on s and I will also assume that h stable is equal to h semi-stable so that the modelized space only consists of stable sheaves and for simplicity of the formulation I also assume that the canonical linear system contains an irreducible curve so that means there is a curve C which is irreducible so in just one piece which is a zero set of a holomorphic two form and in this case the assumption for the statement was that this virtual Euler characteristic of m so I may be right out m s h c1 c2 is equal to the coefficient of x to the virtual dimension of m of 8 times 1 over this eta bar of x squared to the 12 2 times this is total holomorphic Euler characteristic of s times 2 times eta of x to the 4 squared divided by the standard theta function to ks squared where maybe I should remind you that eta bar of x was product 1 minus x to the n over all positive integers and this theta function was just the sum over all n in the integers of x to the n squared so a standard the kobi-teta function and just to remind you also the virtual dimension of this modelized space was just the number 4 c2 minus c1 squared minus 3 times the holomorphic Euler characteristic okay so this is the statement so we will want to kind of check the conjecture in many cases but we are not going to prove it we will just be able to make computations that confirm it and so we want to check this conjecture and refine it maybe I can first look at the refinement so what would be a refinement so the Euler number is kind of the more or less the causes topological invariant that you can assign to a topological space so you can look at finite invariant and so one particular nice one one can look at is the chi-y genus so let me do for the chi-y genus so if you want to so refinement replace the Euler number by the chi-y genus so the chi-y genus is usually so if x is a smooth projective variety then the chi-y genus can be expressed in terms of hodge numbers so so we have these hodge numbers hpq of x which are say the dimension of hp of x and the holomorph the sheaf of holomorphic q forms and then the chi-y genus of x will be just the sum over p and q we have partially an alternating sum of these hodge numbers and I can also write this as this is the homology of this this it means we are looking at the holomorphic Euler characteristics of these sheaves and summing them up so this is the same as the sum over all q of the holomorphic minus y to the q the holomorphic Euler characteristic of x omega q of x so it's easy to check that if I put y equal to 1 then then I get the Euler characteristic and now we want to consider a virtual version of this and this is just by done by putting the letters here at the correct places maybe okay so just in case well no so just as an example if I take the chi-y genus of a k3 surface then this is 2 plus 20 y plus 2y squared and the Euler number as everybody knows is 24 which fits this so let's look at the virtual chi-y genus so this is just as follows and so assume that we are now dealing with our modelized place okay hope it's still alive it still works okay so we put it's clear in some sense clear about the holomorphic the virtual holomorphic p form should be this should be just the omega p there on m should be just the p exterior product of the dual of the virtual tension part and so we just replace this in our formulas and so then I get the virtual chi-y genus and I renormalize to make the formulas nicer so I divide by y to the virtual dimension divided by 2 times the sum over all p minus y to the p then I take the virtual holomorphic Euler characteristic of our modelized space with these holomorphic p-forms okay so this is the direct analog of this formula except that I have normalized it here and again it's easy to see for instance by this Riemann Roth formula that if I take this virtual holomorphic Euler characteristic and put y equal to 1 this is the virtual Euler and so now for the virtual chi-y genus we will write a very similar formula so we put now okay so it is surprisingly it is a finite sum that's not obvious from the definition so obviously by definition the sum is infinite because as it's the difference of two classes in the in the or in of two vector bundles there are infinitely many terms it's symmetric powers and alternating terms but you can prove that this beyond the virtual dimension it will actually be zero I mean this number will be zero it just follows from the computing with join numbers I mean from the Riemann Roth theorem but you know yeah it's not obvious from this no you it's actually somewhat surprising okay so anyway we put now we look again at this theta function theta uh now we before we had the the theta function as well as the theta zero value now we have the Jacobi theta function which also has elliptic variable and so this would be now some n in z x to the n squared y to the n and we have still our eta bar of before and then the formula where we actually okay and then we put psi s to have enough room of x y to be we take this power 3 minus kao for s plus ks squared times 1 over the product over all n bigger than zero 1 minus x to the 2n to the 10 1 minus x to the 2n y 1 minus x to the 2n y to the minus 1 we take this to the power kao for s through the holomorphic order characteristic and we multiply this by we have again this eta of x to the 4 squared divided by this theta function I hope I have enough room and then I take this to the power the power ks squared okay this was a square so this is this formula as you see it's the same formula as before if we put y equal to 1 we get our previous I mean let me first write out the statement which is the same as before namely then the conjecture is that the high y genus of this modular space is the coefficient of x to the virtual dimension of m of this expression okay so this now will be an expression so it will be polynomial in y if I take the coefficient of x to the vb and so this is the same formula as before if we put y equal to 1 this will just be eta of x squared to the 12 as we had before and this theta and and this theta function here will also just become the theta zero value when y is equal put equal to 1 and so this generalizes the previous formula okay there will be there are further refinements one can just see where am I there would be so this is there would be further refinements of so just one example where we know that this conjecture is true so if s is the kc surface then it is known that and under our assumptions that stable is equal to semi stable and so on it's known that this modular space is a non-singular of the expected dimension and it is actually for instance by Yoshioca and others one knows that its deformation in equivalent to the Hilbert scheme of points on the surface of the same dimension so in particular the k y genus of this thing will be equal to that of the Hilbert scheme of points of the virtual dimension but by two because we know that the dimension of the Hilbert scheme is twice number of points and then ks squared is equal to zero and this formula here is a standard formula for the ah actually that's not true so yeah I should maybe have shouldn't have given this example so this is this is an example of the formula in the case when the formula doesn't apply because we had assumed that we had an irreducible canonical curve so therefore here kf is equal to two and here ks squared is equal to zero so we get a factor two here so we're actually wrong by a factor of two but still this is precisely what the formula the general formula predicts okay yeah yeah I mean so we have worked it out in rank three what the statement is I will maybe if I have time explain later I mean there's a somewhat more complicated formula there's some formula by some physicists in arbitrary rank but that contradicts our formula in rank three so therefore as far you know as far as we are concerned there's a formula only after rank three and okay well I mean yeah no no but that that's an arbitrary rank and and so the formula is always done in such a way that it contains such a factor which accounts for the Hilbert scheme for the kc surface okay yeah um so they're also further generalizations which I think will not speak about so one can also generalize this elliptic genus and to some extent to the cupordism class but um so there's a formula for the elliptic genus which is in expresses it the which is very similar to the dichrof more well in the well in the formula for the elliptic general for Hilbert schemes of points on kc surface but it's a bit complicated rather long to write so I will maybe not do that okay now maybe I want to try to explain how we try to verify these conjectures and so this is based on watch its formula so this is our principle tool so this computes intersection numbers on modelized basis of sheaves in terms of intersection numbers on Hilbert schemes of points and cyberquitney invariance so it should maybe set it computes virtual intersection so the integral over the virtual fundamental class of some homology class okay so maybe I can briefly recall the cyberquitney invariance just not much but just some make some statements about them so these are some c infinity invariance of four many folds of differential four many folds but in if the four many fold is an algebraic surface they're actually quite simple so in particular if s is a smooth projective surface and we make our assumption that the geometric genus of s is bigger than zero and the first petty number of s is zero they are rather easy to compute and I mean at least in many examples one can directly they are well known so anyway so the cyberquitney invariance can be viewed as a map as w from the second homology to the integers so class a is sent to the cyberquitney invariant of a I should maybe say that in gauge theory it's a bit different you associate a cyberquitney invariant to a spin c structure spin c structure would be a characteristic homology class so one which is say congruent model or two to the canonical class and so the relation of this to the standard ones is that our cyberquitney invariance are equal to the standard ones for two a minus case anyway in gauge theory but anyway this is just for simplicity of notation and so we have this map most of the time most of class the classes are mapped to zero and so if the cyberquitney invariant of a is non-zero then a is called the cyberquitney class and I just give some examples so so first if s is a k3 surface then there's only one cyberquitney class only cyberquitney class and this is actually the class zero in the second homology is the cyberquitney is one if this is if we have an irreducible canonical curve then we have only two cyberquitney classes there's zero and the canonical class and the cyberquitney invariant of zero is again one and the cyberquitney invariant of ks is minus one to the homomorphic order characteristic of s so everything is known and there are other statements for instance if we blow up a surface in a point we have a simple relation between the cyberquitney invariant so if s hat is the blow up of s in a point then the cyberquitney invariant of s hat are the cyberquitney classes of s hat so these are a set of all a and a plus e where e is the exceptional divisor of this blow up and a is the cyberquitney class on s so for this I mean a is the pullback from s of a on s okay there are other examples for instance they have a simple description for elliptic surfaces but maybe I will not so you can see at least it's very they are quite simply described in a quite simple way in particular if I assume that ks contains an irreducible curve then I know explicitly what the cyberquitney invariants are and this is the reason why I restrict in this presentation my attention to the case that it contains an irreducible curve otherwise we also have a formula but it will be more complicated some formula in terms of the cyberquitney invariant okay so this is the first part and now we have to kind of the mojizuki formula is unfortunately rather complicated so first yeah first we have to remember all these tautological chiefs that also were mentioned in noah arbilsfeld talk so we have a s a projective surface with our assumption so we look at this diagram assume we have here s and one times s and two so the Hilbert scheme of n one points and two points times s we can project it to s and we can project it to these Hilbert schemes so this I call p so then we have several things we have universal chiefs refer you have several universal chiefs so the first thing is the ideal chiefs so if we have c and s so the universal subscheme which is just can be described as the inside s times the Hilbert scheme of n points this pairs of a point in s and the subscheme such that the point flies in the subscheme so this incidence variety this is a maps this n and the fiber over every point here will be the subscheme parametrized this is the universal family and so we can look at its ideal sheaf is the ns which is so the ideal sheaf in s times sn so the functions in this ambient space which vanish on it this is the universal ideal sheaf and so we can use this to make some sheaves some ideal sheaves on this product here so if l is a line bundle on s we can put for the ii of l to be so and now I don't so in this case I actually I take this ideal sheaf so assuming the situation of the n i ideal sheaf so the factor I take here the projection to s times s and i for i is equal to one or two the pullback by that of the ideal sheaf of n i the of the universal subscheme and I tensorize this with the pullback of the line bundle so this will be some sheaf on so I pulled it back so this is some sheaf on this product and what is the property so this is a sheaf on s times sn1 times sn2 with the property that if I restrict it to the point corresponding to so so restricted to s times the point corresponding to two subschemes this will be the ideal sheaf of the ith subs of these two subschemes tensorized with f okay so this is one thing we can look at and we can also look at the tautological sheaves which were considered in our stock so we have here this universal family it maps to s though it lies after all in the product of s times sn1 can look at this this map was called pi now and I can look tautological sheaf on this thing would be the push forward with p the pullback of l in this diagram to first pull back this line bundle here to this thing we push it forward becomes a vector bundle of rank n and so so this is a vector bundle of rank n on this Hilbert scheme such that if I look at the fiber of this thing over the point corresponding to a subschemes z this will be h0 of this line bundle restricted to z or tensor oz is a vector span and we define then in the same way o1 of l and o2 of l to be the pullbacks to the product so these are vector bundles on sn1 times sn2 by pulling back from the two factors so oi of l is equal to p say sni up a star of this sheaf okay so we have all these sheaves on it we have the tautological sheaves pulled back from the two factors and same for the ideal sheaves and we want to express things in terms of all of them maybe we I should we first should talk about what kind of classes so I said that the mutual formula will compute intersection numbers on the on the moduli space in terms of intersection numbers on Hilbert scheme so let's first see what classes we want to compute so I assume again that we have a universal sheave then we can look at the two projection to s and to m and if you have a class in the homology of s we can pull it back to the product we can multiply it by a churn class of the universal sheave and we can push it forward so we put so for we put tau i of alpha equal to the push forward to m by m of say the ith churn class of the universal sheave times this class alpha so the pullback of the class alpha so this will be a homology class in the two i because of the churn class minus four because of the dimension of s plus k of the moduli space and now we take any polynomial in these classes so i z because zero and alpha in the homology of s and for any such class we can compute the corresponding virtual intersection number Mochizuki's formula compute so this virtual intersection number for any such polynomial so on Hilbert schemes of points okay now the formula is a little bit complicated so and in fact in terms of you know these churn classes of these tautological sheaves and universal ideal sheaves that you have introduced explicitly we have the following so if we again look at this pi from s times sn1 times sn2 to sn1 times sn2 I'm sure that I actually should write the formula so then first if we have two sheaves e1 e2 sheaves on this product on this triple product and we can redefine this object q of e1 e2 which is the Euler class of minus the r pi push forward of the r home of e1 e2 okay so this is a certain homology class that one gets for any two sheaves like this and we also okay then yeah maybe should also introduce this although it will maybe so for us we also have a variable in this thing if e is a vector bundle say of a certain rank so then you could say that ck of e of s would be kind of for the moment defined as of this expression okay so how you think of this so in fact you treat this variable s as if it was a line bundle and e of s means you tensor by the line bundle and this gives a formula for so for the churn class of this if this is also the first churn class of the line bundle and this makes actually complete sense because actually s is a line bundle namely we are looking at a trivial line an equivalent line bundle whose which is trivial as a line bundle but is non non trivial equivalently and this s is actually its equivalent first churn class and this actually means we tensorize e with this equivalent line bundle and this is the formula for the equivalent churn classes so anyway but if one doesn't know what this is it's just this formula and so with this what with this we have four line bundles a1 a2 on s we can write psi a1 a2 n1 n2 s to be some internal expression remember that we had this p of e that we wanted to compute the integral over m of p of e and so we take this p of e so and replace the e by taking this first ideal sheaf tensorizing it by the line by the a1 minus s plus the second so basically we have replaced e by this thing we divide it by this q thing with the same contents and then we multiply by the Euler class of o1 of a1 so this was a tautological bundle times the Euler class of o2 of a2 plus 2s and then as a normalization we divide by some power of 2s so this will be some expression it will turn out so this lives according to our definition somehow we are in the homology of the this part of these two Hilbert scheme of points with q coefficients and so we have all these s's that we get a Laurent series in s okay so this is this expression and then we can as is like this we can integrate it over these Hilbert schemes of points so we put a of a1 a2 say n is that what I want yeah s to be the sum over all numbers such that in 1 and in 2 add up to n of the integral over the Hilbert scheme is the product of these two Hilbert schemes of this expression so this is just something we've written down so this will be according to what we have here this is some powers in each series in some Laurent series in s and so then the statement of Moffitt-Bietz's formula is that in terms of this we can compute our integral namely we assume again that all sheaves in ms h c1 c2 are stable we assume that the holomorphic order characteristic of every sheave in in the modellized space is positive and so this can always be achieved by the modellized if we twist the sheave if we twist every sheave in the modellized space by a line bundle this gives me an isomorphism of the modellized space to itself and so we can always twist in such a way that every sheave has positive homomorphic order characteristic and then for then for polynomial p as above we have that if we want to evaluate this over the virtual fundamental class this is equal to the sum over all ways how I can split up the first germ class here as a sum of two classes where a1 h so h is our ample class is smaller than a2 h and I multiply by the Zabek Wittner of a1 and I take the coefficient of s to the zero of this a thing here a a1 a2 and then n is replaced by the second germ class we have here minus a1 times a2 so which is a number because this is a class in second homology okay so this is the answer so one can for one thing so my time is essentially up so in particular we will be able to see that we can have this for instance the top germ class of the virtual tangent bundle can be expressed in this way and so one can apply to compute the virtual Euler number and so what we have achieved here is that we have a reasonably we started with a reasonably nice formula on this modular space of sheaves so of an intersection number on a modular space of sheaves and we have replaced it by a terrible formula on the Hilbert scheme of points and so that doesn't really seem like a big improvement but we will see next time that one can actually work with it okay thank you very much