 Okay, welcome back to the second half of the Shriver seminar with Professor Weber continuing his talk. Please go ahead. Thank you. So we have our object that we study this square zero matrices. And they be orbits in the square zero matrices and we have in mind that studying that we also study just matrix. Varieties, or you can say, be times be orbits in the matrices, rectangular matrices, and several invariants can be computed. Of course one can compute a fundamental class, a fundamental class in the vector space it's not. Interesting thing but of course you, you all know that then you take equivariant cohomology so you look at the fundamental classes in equivariant cohomology and this have several, several names like multi degrees or, or Tom, Tom polynomial, so many. Let's just stick to the multi degrees and I have to say that I really working on on that with my students. In fact, I didn't know that fundamental classes were already computed by the first different class cons interesting and then paper by Knudson and interesting. And it appears somewhere hidden in there, but one can point exactly the references and they also use the many cof results so fundamental classes, you should look to them to basically losing just in several these two out those. So I just concentrate on the transfer my personal classes or motive exchange classes, and this is a new that is not finished elliptic classes of Boris of the cover. To transfer my personal classes, I driver use the aloof approach that how to do it, how to compute it. So we are computing of them transfer my personal classes of orbits so orbits are itself are smooth, but the closure is singular. So one has to resolve the singularities of the boundary that means you, we look for the resolution of the closure, such that the boundary after resolving becomes a normal crossing divisor and then the formulas to compute the time just next slide the formulas will be present. And the same thing is for the motive exchange classes, motive exchange classes were defined by brahsada shulman in your current so it's case theoretic case theoretic, not a version but origin case theoretic origin of here's a group class something that lives in case theory. And in the equivariant version, it was studied by a Luffy, me culture shulman through and also me with Richard many and last year we also were interested but this is basically straight forward extension of them. And this is the first definition this classes are so total logical that making it equivariant making the equivariant version does some technicality, but, and finally there are elliptic classes that elliptic classes were defined in co homology by Boris offensive cover in a non equivariant setup. And this was in co homology with Richard, we have defined it in case theory, but they really live in the elliptic theory in the elliptic co homology the problem is that equivariant elliptic co homology have several versions and we are not sure which version one has to choose so. This is irrelevant because we work equivalently with assumption that fixed points are isolated so using localization everything lives over the fixed points so the version of equivariant elliptic theory is really. Here this is the theories about combinatorics of formulas not about the co homology theory and all these theories have a common. feature that you can compute all these using a resolution of singularities with. The assumption that the boundary has simple normal crossing. The transfer my question classes were defined by some factorial properties motivic charm classes can be defined by using mishatch modules but. In fact this. The definition of resolving is the fastest one. Then the definition is easy and to prove that things does not depend on the resolution is application of weak, weak factorization serum. The body of lip gober classes are defined in that way, in fact in the original paper of Boris of lip gober they exactly they used weak factorization to show that the definition does not depend on the resolution. And in our situation everything is easy because the resolution. The singularities that we study admit resolutions such that the fixed points of the torus acting. The whole borough group is acting but we compute the co homological invariance the torus and the fixed points of the torus are isolated on the resolution in the resolution. So to compute the invariant, for example, call it this invariant C. Orbit. Okay it's lives in the GLM or in the dust all in the upper triangular matrices anyway it's a vector space and the restriction to zero is isomorphism. In the localization CRM, you can, you can compute these classes just by summing up contribution that come from the fixed point of the resolution so this is the formula if you have a formula for the class. In the resolution here. You can compute the contribution of each fixed point and dividing by the other class of the fixed point other class of the tangent representation. You have this contribution and you get this unknown class for the singular. So this is really easy this is combinatorics of the fixed point plus information what you were information about weights of action at the tangent directions. So local, local computation so local computation what what's data is necessary to compute the local class upstairs on the resolution, the resolution so for simplicity assume that this locally look looks like a piece of CN. In fact, in our case this is exactly this space has a covering by affine spaces. And so let's assume that our normal crossing divisor is given by just you admit equation only of the type some some variables not equal to zero. So it's of dimension and and so the first K. K variables are different from zero and take a standard torus then you can specialize to do some sub to write. So to compute CSM class for the situation. This is the formula. So for the this coordinates that you assume that this is non zero you have to take one over weight. T is the weight of them. I coordinate. And for the other other directions here so the denominator is the other class. The numerator is the trend class. If you subtract from here that he will use additivity of the CSM classes. If you. Okay, so this is the calculus. So this is the trend class. Subtract the direct image of them, the image of them. Locus that this variable x, x, z equals zero so if you subtract it, you get this one. Anyway, so this is the formula and they're similar formula is for the multiple trend classes except that is in the case theory. So in the case theory of the point is the representation ring and you have extra variable y. So for the things that are without demanding that I coordinate is non zero you have this, this kind of contribution. And for the other, you can, you can see that if you subtract it from the from that other class you get the other other factor. So these are two kinds of factor that appeal. In the elliptic class the situation is a little bit more delicate because in the definition of Boris of the cover also matters them. Jacobian ideal. So these classes are defined with respect to a boundary that is a divisor it has to be Q divisor satisfying some properties. And these classes depend on this boundary divisor. If I write this device or in that shape that is one minus AI AI some small rational you can think about it AI as a small rational number. It's like you want to subtract this divisor, but the theory does not allow you to subtract this divisor so you subtract one minus something you can think it's epsilon. And then this is the formula of the same shape, you have one contribution of this part with restriction non zero restriction and the other part here. And in this theory, the data function Jacobi data function is used instead of as a Euler class and some normalization. So you have T is a variable and this data function is written in the multiplicative multiplicative convention. And there is extra variable age. So this is extra variable age and the variables of torus torus TI. So you have this kind of contribute this kind of contribution of the smooth point smooth direction and the direction associated to the device or you have to take power of this age. This power, depending on the coefficient of the divisor. And you can think what would happen if you take AI equals zero if you take a par AI equals zero, like divisor with coefficient one. And you get pulled. So these classes are not defined for the for the divisor with coefficients one but you can approach it. So in fact, this, these classes, you can think of them about as a function on the divisors, and this is the most convenient way of looking at them. Okay, so we know the local contributions and now we want to have a recursion for them for the for these classes in our context. So we suppose this is the GLN the or or just this upper triangular matrices here zero and suppose you have orbit. Somewhere here is this distinguished present active w. And then we have s I w. And it makes this orbit. It makes this one dimension bigger. Okay. And there is a relation between the resolution so suppose w is pi times minimal orbit and take a reduced to earth representing this permutation you'll get but some else on the resolution. So the relation between this is the further resolution is a twisted product over P1. So you have fiber x. So this is depending on the reduced to earth. And, okay. So I want to take this and this projects to PI over the, which is productive line, and there are two fibers over identity. You have, you have x pi. And over the permutation si you have also x pi. So this is a fiber bundle for the fibers are identical. Maybe I just raise. Yes, so it's in pi. P1. Two points. Fibers are identical but this is twisted twisted torus action. And we want to compute the characteristic class. So the characteristic class depends there's a vertical direction and there is horizontal direction. And also here vertical direction and horizontal direction and. Okay, there are some fixed points and we have to sum up over these six points. So, you have to take. Let's look at this. This part. So you have twisted action. So you have. I call it internal internal factor, because it's not lying in the boundary of this device or downstairs. So you have, you have to change the action so that means switch the variables and also compute the internal factor coming from the, from the P1 here. So you have to subtract this vertical fiber that you have to subtract and then. No, I'm sorry. In this one you don't do not subtract and in that one you the other other other parts. This one is. You have to twist the action but you have to subtract the device. So maybe I would move to the concrete series, the case theory and. That means. Motivate trend classes and change parts my personal classes. For the trend classes, you get the following that you can write it in this way that the more complicated, the more complicated class is obtained from the easier class by application, the operator and this operator is that take initial function divided. So this is all the class of them, the time and bundle. You want. And this the other one is them. The twisted twisted action that that's why you have switch variables. You can multiply by the trend class of the normal, normal direction. Okay, and alternatively, you can write it in the place or it's like left, like in the paper of recent paper of. I should tell it say it in the right there. Right or the Leonardo Michael child. Hiroshi Naruse and changing so we'll study the just the left, left the major operators for the grass mania and so we have this kind of operator but but acting on the level of the. The square zero matrices and they act in the same way. Okay, and exactly the same can be repeated for the motivate trend classes, you have. Here, the. You have here the factor corresponding to the internal points and the factor corresponding to the boundary points internal points, you have to switch the action of the torus and for the boundary points. You have to switch the action, and it's almost the isobaric divided difference except that there's opposite order of variables, but this is, this is that one and subtract F. That means subtract the the fight. The fight. Okay. So this is example. Maybe I skip this example. If we use this variables to the eyes of the torus conjugating, then we have to stay within this. upper triangular because otherwise you would have non isolated fixed point in them. The diagonal matrices are not fixed. But if you move to them extra variable that's acting just by scholar multiplication if you add another variable then you can consider these classes in the GLM whole algebra. And I should say that you can go beyond them beyond them. upper triangular matrices because this procedure of resolving orbits or resolving of the orbits does not require a particular like in the Bender parent paper. Okay. Maybe time is growing so maybe I can save from this formula. One can deduce them formulas for the transfer some classes and motivation classes of the Schubert. Varieties in the flag variety because just taking quotient, you have to just divide these classes by them. Tangent direction to the orbit if you, if you, we divide by them. We have to divide by this direction. And also there is another feature that maybe I should spend one or two minutes on that that. In fact, I got interested in this subject. Richard many presented me them. This is a previous literate class of what can call weight function. And for a long time this weight function was just something like what can represent something that they cooked up, trying to, to make weights of the certain function, living in the right space, and there was no geometric interpretation of this weight space like Schubert polynomial double Schubert polynomial for some time it was not known what what what geometry geometrically, what's the meaning geometrically. And you can interpret, I mean the outcome of our work with with Piotr Gniewski that this weight remaining weight function can be interpreted as a just same class of the this upper triangular matrices but living in this in this block. So, for example, if we take n equal to so we take block and times and minus one. So this is this block. This is a simple observation that to have a permutation, you have to just say, what are the images of and first n minus one elements, the last one is determined so it's enough to take this space instead. It's a smaller home. And in that way you get this weight functions weight function exactly corresponds to the, to the characteristic classes of orbits living in, living in that block. And finally, the electric classes are not too much time and I have not much to say that you have to be careful. The process depends on divisor. And if you move to the more complicated situation you have to change the divisor and here set some different coefficient, and that's the problem. In fact, there are many cases when you simplify situation you have several moves that can simplify just exchanging this right legs or the left legs, or just exchanging light with left with right. And also there are five cases and for each of these cases, one can find the formula that describes the elliptic classes, except that there is some choice of generators and with some some choice can be less messy than I suspect maybe I should say that you just associate divisor to every arc. And with this association. You see for example here. This is the first arrow, this is the second arrow. And then with this move this divisors are exchanged while here if you exchange the and and points of the arrows that the virus are not exchanged. So that's why you have to take it into account. Okay, so maybe I stop here. And showing further formulas. Alright, well thank you very much for a beautiful talk.