Motivic homotopy type of a log scheme
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Vologodsky, Vadim
Given a log scheme X over the field of complex numbers Kato and Nakayama associated with X a topological space X_{log}. I will show that the homotopy type of X_{log} is motivic in the sense of Morel and Voevodsky. The talk is based on a work in progress with Nick Howell.
 Thank you very much for inviting me to speak here. It's a great pleasure. Well, we have collaborated with Arthur on one loan project, I think, for about three years, and this was, I think, the most fruitful time for me, mathematics. In addition to that, I learned from Arthur what a log scheme is and what a Baytich homologer of a log scheme. And in fact, this, the work I'm going to talk about is, well, grew up also from a question that we discussed a while ago with Arthur. And so the question is a full in. So I have four log schemes. And then I have various homology theories. Well, for example, I can look at a Tal homology or say, the RAM. OK, and so if my log scheme happens to be defined over a finite field, so then, so this Tal, et al. et la de homologie of X with QL coefficients, well, it's a vector space which is a cryptic field with an action of the Frobenius. And one can, co-merital, right? And one can check that for a reasonable scheme X, the eigenvalues are all real numbers. Of course, they are of different weight. But in particular, because they are real numbers, this gives me the weight filtration. And so the question is, so let's see whether I would use it. So OK, so can one define this filtration geometrically? Well, so I want a definition which would make sense for all these three theories. And well, more precise, well, so you can ask for rational beta homology or you can ask even whether it's well defined integrally, as for usual singular homology of algebraic varieties. So or well, so one can ask whether one can associate a motif. A Vodskiy motif. Motif to this X. Or maybe even a finer question, material homotopy type. Well, I want to tell you, so a little bit about this question. So what kind of homotopy type you expect to have? So this Vansper, so lies in the construction due to Katya Nakayama. So OK, so I will recall it here. Nakayama construction. So they defined beta homology. So what's OK? So that's beta homology of a Loxy Max. So Z coefficients, any coefficients. Well to be by definition homology of a certain topological space. So this is just usual. So this is X log. So that's what Arthur explained to me. So this is just a topological space. Well in many cases it's a. Yeah, yeah, sorry. This, yeah, my X is now over complex numbers, of course. Thank you. Thank you. So X is a log scheme over the complex numbers. And so this X, well it's just in general, well it's a topological space, but well in many cases it's a manifold with boundary. It has some additional structure. So what is it? So this is a topological space which is equipped with a map. It's proper map to X. And so the construction is the following. So well I have my log scheme. So I will denote by X underline then the line scheme. And M is the sheaf of manuets. So well, so I have the sheaf of manuets. I have a star that sits inside it and I have a map to O. Okay. So well the only thing I care about, in fact this, well it's true for the whole, for the entire talk, is the group completion, M group. So this is a sheaf of a billion groups. It has a star sitting in it and it has quotient which, well I assume I will look at fine log scheme so it's going to be a constructible sheaf for a tiled apology or if you wish you can consider version of it for the risky apology. In fact in my talk it will be more convenient to consider all these sheaves for the risky apology because at the end the object that, well, yeah, so let's stick to this. So this is, you can think of this, well, a good example of such lambda is just the constant shift supported on the device, for example. So you have such an exact sequence. And okay, so to a scheme and to such an extension I want to associate a topological space. So how I want to do this? So if I have a point, well I can sort of restrict my, so there is here a evaluation map, point x. And so from this extension I can derive an extension of the stock of this sheaf lambda at the point x and this will be m by c star. So this is just a finitely generated a billing group and this is c star. So it's extension. Okay, so now, yeah, it doesn't mean the stock, it means push out. So and here I have map from c star to s1 which is called the argument. So well, let's give a name to this map, let's call it gamma. So a look at all possible sections like this, we'll call it sigma x. So satisfying the property that if I compose gamma with sigma, I get argument. And so x look, yeah, which is the same exception so this is splitting of this extension. Well, push out of this extension. So and then this space as a set is just set of all pairs x and c. Your lambda is torsion free because you assume fs. It need not be torsion free. I will, I'm going to assume, in the second I will assume that it's fs but for the moment I don't even need this. So okay. So I consider all such splitings. Of course there is a map to x and I want to, so and the fibers are fiber over point x is a torsion over from lambda x to s1. So in the case of nice log scheme, a fast log scheme, this is just a real torsion. And so then there is a topology on, so any section of m group defines a function on x log is values in s1, namely it takes point x sigma to sigma of x applied to m. And I want these functions to be continuous. So I consider the weakest apologies such that the pullback of any continuous functions on x is continuous as well as these functions are continuous. Okay, so that's a topological space and I want to see whether, well I want to, the ultimate goal would be to realize this, the homotopy type, well to leave this homotopy type of this log space into a martyric homotopy category. So well, so that the possibility of such construction was suggested to me by a long time ago by Maxim and also by Nori, somehow independent at the same time. So they explained to me one single example which is kind of really the key example. So I want to tell you about this example. Okay, so what is, so this, the log structure, the log space that comes from, as a special fiber of a semi-stable degeneration. So it will be, this will be x0, this will be xt and so semi-stable degeneration. So it's sticked and moreover, so I will assume that it's, there are no like triple intersections, very, very simple. So on my picture there will be only two divisors, it can be generalized in the case of many. So I have these divisors, this is d12 and so then I have two line bundles on x0, namely I have l, well, i which is, which is o of di restricted to x0. So this is, so I have i is 1, 2 and so this li comes with trivialization. If I restrict it, here is another piece of notation, ui is the complement x0 minus di. So and li is, when restricted to li is canonical trivial. So this, these two structures, this, these two line bundles together with such trivialization can be organized, this is given such two line bundles, this is the same thing as given extension of z, z1 plus z, z2 by all star. This is, this is everything over x0. So, so I have one on x0, I consider x0 as a log scheme, well the log structure is just a pullback of the log structure and everything given by this divisor. So but well the same curious, this has this, this very simple description. So these are the constant shifts on d0, on di extended trivially to everything. So and I want to, in this case I want to look at x look and realize it as a element of the, a one homotopic category. So well what you do, so is the following. So well I need another piece of notation. So let's call li-circ. This will be gm-torser associated to li. So li is a line bundle, I take the total space, I remove the zero section. Okay, so and then so what I want to do, I want to consider the following complex, the following complex of, so I want to construct, well, weiwotsky motif and well in fact it will be even element of stable homotopic category and even unstable after a certain modification. So okay, so what is, what is this? So let's first define it as a motif. So well I want to do something very, very simple. I will take l-circ. So again this is, I regarded as a scheme, leaving over x0 and I want to restrict it to u1. Right, so sorry, I think, I want to restrict it to u2 of course because it's 3L on u1. And then the second thing that I want to do, I want to take l-circ2 and restrict it to u1. So when I write square brackets it means that I consider this as a motif. Just a plain scheme, so it makes it. Well and what I will have here is l-circ1 times l-circ2 over x0, restricted to d12. Okay, so and this d12 is intersection. So well, so my motif is going to be sort of the cone of this complex. Now I need to tell you what this is, this arrow. So and here is a reminder, so well this construction is possible because well in the category of wave-wod scheme motives you have identification between punctured tubular neighborhood of a smooth manifold is punctured normal bundle. And namely if you have so smooth y and z, this is closet, smooth closet, then there is a canonical map in from the motif of the punctured normal bundle to y minus z. This exists in differential geometry. This can be lived in the category. And well the map that I have here is exactly what comes from, is precisely that map. So here is the picture. So if I have, let's see how we get it. So okay, so here is my d1 for example and here I have this line bundle which is l1-circ. So and here I have this is d12. So I apply this construction, so I have this line bundle, this GM torsor, it leaves over the whole d12 and inside this the total space of l1, I have l1 minus, I have this fiber over d12. And I apply this construction to y being this GM torsor over d1 and z being its fiber over d12. Then you get precisely this first map and the second map is constructed similarly. So you do this. And do you put a sign or does that mean power? I don't think that there is any sign here, it must be symmetric at this moment. No but if you have several things that intersect each other and let us say go in a loop. No, no, in here. Well so I don't think that there is any sign in this picture so maybe, so let's, it seems to me that, it seems to me that at least here you don't need sign but maybe I'll buy sign, it's possible. So okay, so in fact this map that I used here exists even in a one, just in a one-homotopy category but only after suspension. So you can kind of do the same clue in except that what you'll get, you cannot get homotopy type of x log itself but you can get homotopy type of the suspension of x log. So at least you'll get something in the stable homotopic category. So the fact that this, that map exists after suspension is proven by Morellian way was. So okay, so and well one more remark about this whole picture. So in fact this ELI have additional structure if they come from this semi-stable degeneration. Namely EL1, Tensorel2 is trivial. Well strictly speaking this trivialization depends on the choice of a parameter here. Well I choose a T coordinate, right, because this is OD1 plus OD2 so special fiber, it's lifted from the base. And therefore, well if I have a map from this, from everything, this picture to this over x0 to complex numbers, right? So there is always, there is a map from these guys to complex numbers, from invertible complex numbers, right? So and therefore, so let's give a name to this, let's call it motif of x0m. So it's motif of this log scheme, just definition. And in fact in this case, this motif is not, is a material shift, material shift over cster. And it's fibers for example, well you can take fibers over, so every scheme here is a scheme over cster, so you can take fibers, you have to do this map to get what's called the, well, limit motif, right? So and the picture here is, I cannot really draw a picture of this gluing of this x-log but this x-log here, it actually, it maps to log space associated to the log point. So it maps to the circle. And I can draw a picture of a fiber of this over a point of the circle. So imagine that for example, you have the generation of elliptic curve. So okay, so here is my, so it generates interrational curve with two double points, right? So this is non-singular elliptic curve and this now it degenerates. So what you do, you sort of remove these two points and then you glue along puncture tube learning rules. And then you get something back, well, I'm going to pick it all into the original space. So I want to kind of extend this construction to all the schemes and that even if you have, in this semi-stable case, if you have many components and multiple intersections, it becomes really, well, direct the generalization is really unpleasant because, well, you can kind of write down a similar complex but, well, you have to do it, lift it in somehow on a digital level and then the square of the differential will not be zero. It will be the homotopy equivalent to zero and these homotopes are given by some double normal cone constructions and so it's going to be completely useless, though possible. You can't really do anything with that. So instead, one should look at the dual thing, one should look at the motivic homology of this motive and this has a very simple geometric description. I will show it to you at the end in the case of this semi-stable degeneration but first I want to formulate the main theorem in the kind of in abstract form. So the main theorem is the following. So while I need a bit, first I introduce a bit of a certain category. I will call it the category of log motives but, well, then you will be able, well, we can forget about this. I mean, so first I introduce a category that I will call the category of log motives. So and in order to do this, so let me just write down definition of usual category of what's the motives and then I will list two more kind of relations and get the category of log motives. So well, so I will work with log motives. So I will assume for simplicity that characteristic of my base field is zero. Well if it's p, you can do the same thing but you have to invert p in the coefficients. And then so, okay, so for Waiv-Wolcki motives you do the following. So you consider category of schemes, say, of finite type over a field and then you form an additive category. Just objects are schemes, all schemes are finite type and morphisms are linear combination of maps, no correspondences here. And then I want to take a complices, finite complices over this and then I take, I mod it out by certain relations, by subcategory. So this is a, well, if you can do it on triangulated level, past-ohomatopoeic category, then take verdequation or you can regard it as a differential graded category. So and what are objects of t? So objects, well, of t. So well, I will list them. Let me start here. So first object is, well, it's going to be class of objects, right? So it's x. So objects are complices of schemes, right? So this is first, this is called A1-homotopy. And then the second class of objects is, well, perhaps you have to, let me do it here. So this will, so this is the second class of objects. I will refer to it as CDH-topology. So this is a generating kind of sequences, covers. And so there will be two parts. So first, suppose I have open, the risk to open subset. And I have an ital map. This is ital. And so what are the conditions? One is that the map p in words of from x minus u to u to x minus u, this map p, the restriction, is an isomorphism of schemes. So that means that this is an ital map. And on the complement to u, it has a section. It's actually a bijection, right? So this is the risk to open. This is ital. So in particular, u and w cover x. But it's much, much, much more than that. And in this situation, I want relation. And this relation has the form x, u, w. And here I have the fiber product, u, w, x. So all these maps. And so this is part a and part b. So I want the following. I consider maps p from x prime to x. These are proper. And I assume that there exists a closed sub-scheme here. Such that over the complement, this map is an isomorphism. So p induces an isomorphism between p inverse of x minus z and x minus z. For example, blow up. So OK. And in this situation, I want to have the following relation that motif of x prime plus motif of z. And this maps to motif of x. And here I have motif of pre-image of z. And here you must put the sign somewhere. In those seconds. Yeah, I have to put sign. So either doesn't really matter where. So yeah, you're right. So I have to take the difference. Otherwise the square will be zero. So and right. This closed. This closed, right. So and also, maybe c is that the map from x is used to x. Must be an isomorphism. That's it. It follows from b because you can take z to be the. OK. OK, so but yeah, you're right. So many things in fact follow from others here. So and well, you can. So this is not. So you can have. So this is not what's called, usually called, the category of Weyvotsky-Mozirs. It's also defined and studied by Weyvotsky. And it has a very complicated notation, this category, this quotient. So let me give it here. So it's h a1 cdh. And then here I take z of schemes. And I need Karubin completion of this. OK, so what is the relation of this category with the category of Weyvotsky-Mozirs? So there is a, well, at least there is a functor. So this z to Weyvotsky category, which is obtained exactly in the same way, except that you add transfers. So you start with, consider the same relations. But this category is different. You consider schemes and where maps are finite correspondences. And also only smooth scales. Don't think that if you do cdh the poetry, it makes any difference. You're speaking about the, OK, so you can do it. Yeah, OK. If you would also. So if you consider smooth schemes, then cdh the poetry, then b can be derived from a and other. OK, so now I want log motives. So what I do with log motives. How I, so let's just consider the same, the same similar category. But let's start with log. So by log schemes, so I consider, I mean f has finite saturated log schemes. OK, and then I want to take, so I want to consider quotient of z log scheme by t. Well, and what is, so what is, what is this t log? Well. So t log consists of, it's a full subcategory of objects. Well, I did not object of type 1 and 2, 2 and the following. So I need to do, to impose a bit more, a few more relations. So OK, so what is, what are the relations? So, well, maybe just before we explain the relation. So, I have here this, well, there is homotopy equivalence relation. So in this homotopy equivalence relation, x is allowed to be any log scheme, as well as here. So I want everything in this picture, x, well, this must be log schemes. What about a1? A1 here is just usual a1. But I will add axiom for log a1. And if you speak about closed immersion, perhaps you want the exact one. Yes, yes. So, OK, so what I want is the following axiom. So I have a1 log. So what is it? It's a, it's a, the underlying scheme is a1. And the log structure is given by one point. So that my m group is just corresponds to the line bundle of this point. So, and here I have, so log point, it's kind of the origin. And here I have gm. It has trivial log structure. So, and I want these maps to be isomorphism in my quotient category. But not only this. I want multiply this by x times a1 log. And again, x is any work scheme. So, and also this gm times x is valid. Well, there is one additional axiom. It's strange. It's just has to do, we already observed that, well, all the construction we have made so far depend only on m group, not on m. m is irrelevant. And I could consider the category of law scheme just by the category of extensions. They are in my category t. So, you require them to be 0. That's all. That's all. Yes. You want that a1 log is an a1 log. Is it a morphic or you, too? Yes, a1 log. Yes. Yes. So, so this are object in t. Like, for example, as this, those objects. So, and finally, 4 is very, very strange. Well, so it's very, very, is the following axiom. So, suppose I have any map p from x prime to x. These are log schemes. And the underlying map between usual schemes is an isomorphism. And suppose that it induces, the map induces on m group p up a star to m prime group. Is an isomorphism. Then I want the following relation that x prime goes to x. It must be in my category. So, this kind of way to say that everything depends on the n group. But again, so I could, you can consider just the category of pairs x plus this extension of star. So, now, very good. So, here is the theorem. Now, when you do block blow-ups, what seems when you do the operation of block blow-up, I think that the space, I don't remember now, the space x log doesn't change? No, it changes. OK, space x log, yeah, changes, but it's the same homework. So, here is the theorem. So, well, I have a category A1, CDH. So, this is my category. And here is the name for this. So, this is Z of schemes. So, there is, of course, a functor to this larger category. We are going to consider A1, CDH. And here I have Z log. OK, so, right. So, this functor is an occurrence of categories. So, what does it mean? So, what are, for example, functors from this category to, say, the category of complices? These are homologous theories, right, that satisfy these basic properties. The claim is that any homologous theory can be extended uniquely to log schemes provided it has all these properties. OK, so, let's see. So, in particular, well, of course, here you have map to functor to usual Weyvotsky category. And then you can compose the inverse and get Weyvotsky motif. Now, so, the construction, the proof is, it uses very little geometry, in fact. It's more or less linear algebra. And so, well, first of all, no, you'll have to prove. So, you have to prove two things, that this functor that I have here, the obvious functor, let's give a name to this functor. Let's call it. It's fully faithful, homotopically fully faithful, if it's dg, you think of this as dg functor. So, fully faithful. And b, so essentially subjective. Subjective. Well, and in fact, so easy step is that a implies b. And therefore, so all you have to do is to prove a. So, and that's because, and you can do it by induction on dimension. So, you need to show that any motif of any log scheme is in the image of this functor. So, have enough relations to express a motif of any log scheme in terms of motifs of usual scheme. So, but that's easy. So, you do it again by dimension of your log scheme. So, if it's zero dimensional, then, well, it's fine saturated. So, it means that my Manoid M group is just z to the n. And so, the corresponding, so if I have point and take its, well, its motif, for example, is motif of gm. And if I consider Cartesian product of any of any scheme of log point with itself, then it's, of course, it will be gm times gm. This follows from this axiom. And so, any, and using this property, any zero dimensional log scheme has the same motif as this product of this. So, now, what you're doing one dimensional case. So, you have some curve. Well, now, generically over an open set, the log structure is trivial. It just corresponds to trivial extension. M group is just trivial extension of z to the n by your star. And therefore, OK, so you know what to do over an open set. And then you can use this CDH axiom to see that this log motif of this curve is also in the image. So, what, what, what? Have you still got the formula here? No, I used because, because, so let's see. Well, if you look at this CDH axiom, how it looks like. So, you have some p inverse of z. And then you have z plus x prime. And this maps to x. So, what, so, so I want to show that this object, just, sorry, so that this object, so knowing that this object and this object are in the image and also this, I want to construct, conclude that this object is in the image. But then I need a map from x, from this square bracket x, to this guy. And I need to show that this map exists in the usual category of Wojewódzki, Wojewódzki category. It exists by definition. So, because it's exact triangle, there is a map from square bracket x to this guy, just shifted by one. And I need to show that this map actually exists. So, now, so what you do for? Do as soon as the log stack is a risky locally trivial. I use it as a risky locally trivial, right? So, I could, yeah. So, you don't have motives for et al. Then I need to work with et al, the poetry, rather than be risky, and well, at the end, I will get an et al Wojewódzki motive. Well, some applications that I had in mind have to do with this integral weight filtration, which exists only for realization of usual Wojewódzki motives, not et al. So, OK. So, what you do for A? So, what do you need? You have to, it's enough just by formal nonsense. It's enough to prove the theorem. It's enough to do the following. So, given any object of this category, usual category, A1, CDH, maybe I will just say the following. So, let me, because my time is short. So, I claim that it's enough, it suffices. I will sort of suppress the completely formal part. So, it only uses the fact that the category of Wojewódzki motives is rigid, duality. It suffices to show that the functor home Z of n from this usual category of CDH Z of schemes extends to larger category of this category, A1, CDH Z block. And what you do here, so you just sort of, you want kind of to define motivic homology. It's not quite motivic. It's home in this category with no transfer. So, I don't want to call it motivic homology. It would be motivic homology if I consider this category with transfer. So, you want to do it for any log scheme. And the idea is very simple. So, you have the same group. It can be evaluated, it can be considered as a motivic shift over the underlying scheme. That means simply that you can evaluate it over any scheme which is smooth over this guy. You simply pull the log structure. If I have a map like this, then you can pull the log structure, this log structure here, and take the same guy there. So, and then what you do, you just compute, you take homology of X, these coefficients in symmetric powers. These are operations in the category of motivic shift symmetric powers. Well, they're defined by Weyewodski. They're characterized by the properties that if you take X, that motif of a scheme X, the symmetric power of motif of a scheme X is the motif of the symmetric power of X. Is it M bar group or M group? Just M group. So, for example, what happens if you take N is 1? Well, when you pull at that, you don't make it a log structure or Y, you just take just N. You don't enlarge the units. I mean, usually when you pull back N. Yeah, and the Y units. You are going to enlarge the units. So, for example, what is first, well, motivic homology with N equal 1? This is just homology of M group. It's kind of the final look of the P-carve. So, OK. And now I want to finish by. How much of which category? This is the homology of what? So, well, you are. So, the base of this homology works. Yeah, this is Nisnevich homology of this motivic shift. So, well, I want to finish by a very explicit formula that I promised at the beginning, just very geometric, for the motivic homology of the limit motif in the case of semi-stable degeneration. Also for the tubular new problem. It will be just in the case of semi-stable degeneration. And it comes from this. I just, well, this formula is here. I just want to, I will make it explicit, which is trivial from this definition. So, the formula is the following. So, here is my setting. Well, now I will have, again, I have the log structure coming from semi-stable degeneration. But now I can have multiple intersections. But it's still semi-stable. Maybe that's not a good picture. So, this should be planes rather than lines. So, OK, probably I should not put it here. So, it's confusing. OK, semi-stable degeneration. And it's strict, right? So, I want really, that's, I want log structure for the risk typology. So, strict, strictly semi-stable degeneration. So, I have components, D1, D2, well, so, DI. DI, it belongs to some set. I have this GM torsors. So, first of all, I have LI, which is of DI, restricted to X0. And then I have the corresponding GM torsors. And I have UI, which is a complement of X0 to DI. So, OK, so now I will construct the material homologism. So, well, first I will produce a complex of smooth varieties of Rack Zero. The fibers will be algebraic tori. So, what are these varieties? So, step one, complex, very simple. Complex of smooth varieties. So, this will be product of all this LI. So, this is product over Rack Zero. It's a torsor over an algebraic tori of dimension. Now, what is the next term? So, you take sum. Let's consider it as a kind of, I want to, it's not really, it's, in fact, it's a, I want to consider kind of an additive category. I want to make sense of sum. So, it means that it's, you can think of this dejoint unit. So, of products of I in I, I is not equal to J. And here I will have L Cirque I times UI over R Zero. So, what is this map? Remember that LI is trivial over UI. Therefore, I have such a map. Just given by, well, it's section. Now, it's a long complex. And here is its last term. It's sum I over times intersection J not equal to I UG. It's again over X Zero. And so, all I use here, all the differentials, come from the sections. Remember that LI is trivial over UI. So, this gives me such a, again, this should be dejoint here. So, this is step one. It's very simple complex. Now, form define a complex of pre-sets on X Zero. Well, what you do? So, you call this Z dot. And then you take the following. So, if you have a scheme over X Zero, then, well, the value of my complex of pre-sets on X Zero on Y is the following. You take correspondences or X Zero from Y to this complex. So, well, informally, you just take sections of this complex of varieties over Y. And then, you make it homotopy invariant by applying the system construction. So, let's call this F and my material homology of X Zero M. Z is just CDH homology of this X Zero with coefficients in F. That's it. That's all I have to say. So, there is one, well, as you see, you can ask whether the log geometry here. And in fact, there is no log geometry. You see it's just linear algebra. And the log geometry appears if you want to prove, for example, that if you have a smooth log scheme, then it's motif. It's isomorphic to the motif of the open part where the log structure is trivial. This I don't know how to do. I know how to do it in the case of normal crossing situation. But in general, it really requires some geometry. Thank you very much. Was there any case with notation of the motif homology, did you say? Yeah, so here I defined only this one index, right? This index is absolute value of this set of indices, i. So if you want it defined for larger, for larger, and right, you have to just add in this whole construction empty divisor formula. And to define it for smaller, you don't need to define it for small by cancellation. It suffices to define this for sufficiently large numbers. This follows from the books. Yes. Just motif of homology. Yeah, so if you want it for, you have cancellation. So you can express motif homology of something with coefficients in z of n in terms of homology of the twist, this coefficients in z of n plus 1. This is the product of this GM? Yeah, yeah, product of this GM, for example. Yes? So I just was curious, what is n? Where is n? Shouldn't we have this, if we have a motif homology? Yeah, yeah, this twist here, right? No, no, this n means the log of monotone. Yeah, that's a very good question. It's this whole complex of schemes. It's least over GM. So if you have, it's exactly as in the picture I started with. This gives you the tubular neighborhood. If you want the vanishing cycles, you have to take the fiber over. Oh, OK. So it's a unipotent motif shape over GM. But how can you define the map to GM if you're given just the lid, you're in the fiber also? No, there is a, the map to GM comes from the trivialization of the tensile product of these fine bundles. So each of these smooth schemes over X0 admits a map to GM. For example. Because the other one, the product is. Yeah. I think this is implicit. When I said that you were going to start talking about how to define the weightful phase. Yeah? Does this come out easily from this? Yes. So there is a, well, yes, it comes thanks to work of Bandarka. So if Betty homology or the Ram homology or any Veyvodsky motif is equipped with integral weight filtration. Yes. Dr. Yavt, show that he's equivalent to. Rational. Yeah. To what? Yes. He has defined the weight filtration. Yes. But after, you have to show that he's equivalent to the others. Well, but here you don't have, well, a priori you don't have the others. What you have to show is that the basic homology of this log motif of this Veyvodsky motif I defined coincides with the homology of this, as defined by Kat and Nakayama. But this is obvious because Kat and Nakayama homology theory satisfies the list of all my axioms. And therefore, this factor from log motifs is uniquely determined by its restriction to usual motifs. And for usual schemes, there is nothing to prove. And sorry, last question. In characteristic P, this was everything in characteristic in the world. Yeah, you have to invert P. And always with Zariski? Yeah, for Zariski again. So the only reason why I work with Zariski is that I want Veyvodsky motif really for Nisnevich topology, as opposed to Veyvodsky motifs for Taltopolga. So it is enough to have Nisnevich locally, Nisnevich chance for the work start? Yes. Yes, yes, yes. Though I don't really know. No, maybe that way. There are others. Yes. You need to have your gel. That's not good enough. Any more questions? Thank you very much.
 Thank you very much for inviting me to speak here.  It's a great pleasure.  Well, we have collaborated with Arthur on one loan project, I think, for about three  years, and this was, I think, the most fruitful time for me, mathematics.  In addition to that, I learned from Arthur what a log scheme is and what a Baytich homologer  of a log scheme.  And in fact, this, the work I'm going to talk about is, well, grew up also from a question  that we discussed a while ago with Arthur.  And so the question is a full in.  So I have four log schemes.  And then I have various homology theories.  Well, for example, I can look at a Tal homology or say, the RAM.  OK, and so if my log scheme happens to be defined over a finite field, so then, so this  Tal, et al.  et la de homologie of X with QL coefficients, well, it's a vector space which is a cryptic  field with an action of the Frobenius.  And one can, co-merital, right?  And one can check that for a reasonable scheme X, the eigenvalues are all real numbers.  Of course, they are of different weight.  But in particular, because they are real numbers, this gives me the weight filtration.  And so the question is, so let's see whether I would use it.  So OK, so can one define this filtration geometrically?  Well, so I want a definition which would make sense for all these three theories.  And well, more precise, well, so you can ask for rational beta homology or you can ask  even whether it's well defined integrally, as for usual singular homology of algebraic  varieties.  So or well, so one can ask whether one can associate a motif.  A Vodskiy motif.  Motif to this X.  Or maybe even a finer question, material homotopy type.  Well, I want to tell you, so a little bit about this question.  So what kind of homotopy type you expect to have?  So this Vansper, so lies in the construction due to Katya Nakayama.  So OK, so I will recall it here.  Nakayama construction.  So they defined beta homology.  So what's OK?  So that's beta homology of a Loxy Max.  So Z coefficients, any coefficients.  Well to be by definition homology of a certain topological space.  So this is just usual.  So this is X log.  So that's what Arthur explained to me.  So this is just a topological space.  Well in many cases it's a.  Yeah, yeah, sorry.  This, yeah, my X is now over complex numbers, of course.  Thank you.  Thank you.  So X is a log scheme over the complex numbers.  And so this X, well it's just in general, well it's a topological space, but well in  many cases it's a manifold with boundary.  It has some additional structure.  So what is it?  So this is a topological space which is equipped with a map.  It's proper map to X.  And so the construction is the following.  So well I have my log scheme.  So I will denote by X underline then the line scheme.  And M is the sheaf of manuets.  So well, so I have the sheaf of manuets.  I have a star that sits inside it and I have a map to O.  Okay.  So well the only thing I care about, in fact this, well it's true for the whole, for the  entire talk, is the group completion, M group.  So this is a sheaf of a billion groups.  It has a star sitting in it and it has quotient which, well I assume I will look at fine log  scheme so it's going to be a constructible sheaf for a tiled apology or if you wish you  can consider version of it for the risky apology.  In fact in my talk it will be more convenient to consider all these sheaves for the risky  apology because at the end the object that, well, yeah, so let's stick to this.  So this is, you can think of this, well, a good example of such lambda is just the constant  shift supported on the device, for example.  So you have such an exact sequence.  And okay, so to a scheme and to such an extension I want to associate a topological space.  So how I want to do this?  So if I have a point, well I can sort of restrict my, so there is here a evaluation map, point  x.  And so from this extension I can derive an extension of the stock of this sheaf lambda  at the point x and this will be m by c star.  So this is just a finitely generated a billing group and this is c star.  So it's extension.  Okay, so now, yeah, it doesn't mean the stock, it means push out.  So and here I have map from c star to s1 which is called the argument.  So well, let's give a name to this map, let's call it gamma.  So a look at all possible sections like this, we'll call it sigma x.  So satisfying the property that if I compose gamma with sigma, I get argument.  And so x look, yeah, which is the same exception so this is splitting of this extension.  Well, push out of this extension.  So and then this space as a set is just set of all pairs x and c.  Your lambda is torsion free because you assume fs.  It need not be torsion free.  I will, I'm going to assume, in the second I will assume that it's fs but for the moment  I don't even need this.  So okay.  So I consider all such splitings.  Of course there is a map to x and I want to, so and the fibers are fiber over point x is  a torsion over from lambda x to s1.  So in the case of nice log scheme, a fast log scheme, this is just a real torsion.  And so then there is a topology on, so any section of m group defines a function on x  log is values in s1, namely it takes point x sigma to sigma of x applied to m.  And I want these functions to be continuous.  So I consider the weakest apologies such that the pullback of any continuous functions on  x is continuous as well as these functions are continuous.  Okay, so that's a topological space and I want to see whether, well I want to, the ultimate  goal would be to realize this, the homotopy type, well to leave this homotopy type of  this log space into a martyric homotopy category.  So well, so that the possibility of such construction was suggested to me by a long time ago by Maxim  and also by Nori, somehow independent at the same time.  So they explained to me one single example which is kind of really the key example.  So I want to tell you about this example.  Okay, so what is, so this, the log structure, the log space that comes from, as a special  fiber of a semi-stable degeneration.  So it will be, this will be x0, this will be xt and so semi-stable degeneration.  So it's sticked and moreover, so I will assume that it's, there are no like triple intersections,  very, very simple.  So on my picture there will be only two divisors, it can be generalized in the case of many.  So I have these divisors, this is d12 and so then I have two line bundles on x0, namely  I have l, well, i which is, which is o of di restricted to x0.  So this is, so I have i is 1, 2 and so this li comes with trivialization.  If I restrict it, here is another piece of notation, ui is the complement x0 minus di.  So and li is, when restricted to li is canonical trivial.  So this, these two structures, this, these two line bundles together with such trivialization  can be organized, this is given such two line bundles, this is the same thing as given extension  of z, z1 plus z, z2 by all star.  This is, this is everything over x0.  So, so I have one on x0, I consider x0 as a log scheme, well the log structure is just  a pullback of the log structure and everything given by this divisor.  So but well the same curious, this has this, this very simple description.  So these are the constant shifts on d0, on di extended trivially to everything.  So and I want to, in this case I want to look at x look and realize it as a element of the , a one homotopic category.  So well what you do, so is the following.  So well I need another piece of notation.  So let's call li-circ.  This will be gm-torser associated to li.  So li is a line bundle, I take the total space, I remove the zero section.  Okay, so and then so what I want to do, I want to consider the following complex, the  following complex of, so I want to construct, well, weiwotsky motif and well in fact it will  be even element of stable homotopic category and even unstable after a certain modification.  So okay, so what is, what is this?  So let's first define it as a motif.  So well I want to do something very, very simple.  I will take l-circ.  So again this is, I regarded as a scheme, leaving over x0 and I want to restrict it to  u1.  Right, so sorry, I think, I want to restrict it to u2 of course because it's 3L on u1.  And then the second thing that I want to do, I want to take l-circ2 and restrict it to  u1.  So when I write square brackets it means that I consider this as a motif.  Just a plain scheme, so it makes it.  Well and what I will have here is l-circ1 times l-circ2 over x0, restricted to d12.  Okay, so and this d12 is intersection.  So well, so my motif is going to be sort of the cone of this complex.  Now I need to tell you what this is, this arrow.  So and here is a reminder, so well this construction is possible because well in the category of  wave-wod scheme motives you have identification between punctured tubular neighborhood of  a smooth manifold is punctured normal bundle.  And namely if you have so smooth y and z, this is closet, smooth closet, then there  is a canonical map in from the motif of the punctured normal bundle to y minus z.  This exists in differential geometry.  This can be lived in the category.  And well the map that I have here is exactly what comes from, is precisely that map.  So here is the picture.  So if I have, let's see how we get it.  So okay, so here is my d1 for example and here I have this line bundle which is l1-circ.  So and here I have this is d12.  So I apply this construction, so I have this line bundle, this GM torsor, it leaves over  the whole d12 and inside this the total space of l1, I have l1 minus, I have this fiber  over d12.  And I apply this construction to y being this GM torsor over d1 and z being its fiber over  d12.  Then you get precisely this first map and the second map is constructed similarly.  So you do this.  And do you put a sign or does that mean power?  I don't think that there is any sign here, it must be symmetric at this moment.  No but if you have several things that intersect each other and let us say go in a loop.  No, no, in here.  Well so I don't think that there is any sign in this picture so maybe, so let's, it seems  to me that, it seems to me that at least here you don't need sign but maybe I'll buy sign,  it's possible.  So okay, so in fact this map that I used here exists even in a one, just in a one-homotopy  category but only after suspension.  So you can kind of do the same clue in except that what you'll get, you cannot get homotopy  type of x log itself but you can get homotopy type of the suspension of x log.  So at least you'll get something in the stable homotopic category.  So the fact that this, that map exists after suspension is proven by Morellian way was.  So okay, so and well one more remark about this whole picture.  So in fact this ELI have additional structure if they come from this semi-stable degeneration.  Namely EL1, Tensorel2 is trivial.  Well strictly speaking this trivialization depends on the choice of a parameter here.  Well I choose a T coordinate, right, because this is OD1 plus OD2 so special fiber, it's  lifted from the base.  And therefore, well if I have a map from this, from everything, this picture to this  over x0 to complex numbers, right?  So there is always, there is a map from these guys to complex numbers, from invertible complex  numbers, right?  So and therefore, so let's give a name to this, let's call it motif of x0m.  So it's motif of this log scheme, just definition.  And in fact in this case, this motif is not, is a material shift, material shift over cster.  And it's fibers for example, well you can take fibers over, so every scheme here is a  scheme over cster, so you can take fibers, you have to do this map to get what's called  the, well, limit motif, right?  So and the picture here is, I cannot really draw a picture of this gluing of this x-log  but this x-log here, it actually, it maps to log space associated to the log point.  So it maps to the circle.  And I can draw a picture of a fiber of this over a point of the circle.  So imagine that for example, you have the generation of elliptic curve.  So okay, so here is my, so it generates interrational curve with two double points, right?  So this is non-singular elliptic curve and this now it degenerates.  So what you do, you sort of remove these two points and then you glue along puncture tube  learning rules.  And then you get something back, well, I'm going to pick it all into the original space.  So I want to kind of extend this construction to all the schemes and that even if you have,  in this semi-stable case, if you have many components and multiple intersections, it becomes  really, well, direct the generalization is really unpleasant because, well, you can kind  of write down a similar complex but, well, you have to do it, lift it in somehow on a  digital level and then the square of the differential will not be zero.  It will be the homotopy equivalent to zero and these homotopes are given by some double  normal cone constructions and so it's going to be completely useless, though possible.  You can't really do anything with that.  So instead, one should look at the dual thing, one should look at the motivic homology of  this motive and this has a very simple geometric description.  I will show it to you at the end in the case of this semi-stable degeneration but first  I want to formulate the main theorem in the kind of in abstract form.  So the main theorem is the following.  So while I need a bit, first I introduce a bit of a certain category.  I will call it the category of log motives but, well, then you will be able, well, we  can forget about this.  I mean, so first I introduce a category that I will call the category of log motives.  So and in order to do this, so let me just write down definition of usual category of  what's the motives and then I will list two more kind of relations and get the category  of log motives.  So well, so I will work with log motives.  So I will assume for simplicity that characteristic of my base field is zero.  Well if it's p, you can do the same thing but you have to invert p in the coefficients.  And then so, okay, so for Waiv-Wolcki motives you do the following.  So you consider category of schemes, say, of finite type over a field and then you form  an additive category.  Just objects are schemes, all schemes are finite type and morphisms are linear combination  of maps, no correspondences here.  And then I want to take a complices, finite complices over this and then I take, I mod  it out by certain relations, by subcategory.  So this is a, well, if you can do it on triangulated level, past-ohomatopoeic category, then take  verdequation or you can regard it as a differential graded category.  So and what are objects of t?  So objects, well, of t.  So well, I will list them.  Let me start here.  So first object is, well, it's going to be class of objects, right?  So it's x.  So objects are complices of schemes, right?  So this is first, this is called A1-homotopy.  And then the second class of objects is, well, perhaps you have to, let me do it here.  So this will, so this is the second class of objects.  I will refer to it as CDH-topology.  So this is a generating kind of sequences, covers.  And so there will be two parts.  So first, suppose I have open, the risk to open subset.  And I have an ital map.  This is ital.  And so what are the conditions?  One is that the map p in words of from x minus u to u to x minus u, this map p, the restriction,  is an isomorphism of schemes.  So that means that this is an ital map.  And on the complement to u, it has a section.  It's actually a bijection, right?  So this is the risk to open.  This is ital.  So in particular, u and w cover x.  But it's much, much, much more than that.  And in this situation, I want relation.  And this relation has the form x, u, w.  And here I have the fiber product, u, w, x.  So all these maps.  And so this is part a and part b.  So I want the following.  I consider maps p from x prime to x.  These are proper.  And I assume that there exists a closed sub-scheme here.  Such that over the complement, this map is an isomorphism.  So p induces an isomorphism between p inverse of x minus z  and x minus z.  For example, blow up.  So OK.  And in this situation, I want to have the following relation  that motif of x prime plus motif of z.  And this maps to motif of x.  And here I have motif of pre-image of z.  And here you must put the sign somewhere.  In those seconds.  Yeah, I have to put sign.  So either doesn't really matter where.  So yeah, you're right.  So I have to take the difference.  Otherwise the square will be zero.  So and right.  This closed.  This closed, right.  So and also, maybe c is that the map from x is used to x.  Must be an isomorphism.  That's it.  It follows from b because you can take z to be the.  OK.  OK, so but yeah, you're right.  So many things in fact follow from others here.  So and well, you can.  So this is not.  So you can have.  So this is not what's called, usually called,  the category of Weyvotsky-Mozirs.  It's also defined and studied by Weyvotsky.  And it has a very complicated notation,  this category, this quotient.  So let me give it here.  So it's h a1 cdh.  And then here I take z of schemes.  And I need Karubin completion of this.  OK, so what is the relation of this category  with the category of Weyvotsky-Mozirs?  So there is a, well, at least there is a functor.  So this z to Weyvotsky category, which  is obtained exactly in the same way,  except that you add transfers.  So you start with, consider the same relations.  But this category is different.  You consider schemes and where maps are finite correspondences.  And also only smooth scales.  Don't think that if you do cdh the poetry,  it makes any difference.  You're speaking about the, OK, so you can do it.  Yeah, OK.  If you would also.  So if you consider smooth schemes, then cdh the poetry,  then b can be derived from a and other.  OK, so now I want log motives.  So what I do with log motives.  How I, so let's just consider the same,  the same similar category.  But let's start with log.  So by log schemes, so I consider,  I mean f has finite saturated log schemes.  OK, and then I want to take, so I want to consider  quotient of z log scheme by t.  Well, and what is, so what is, what is this t log?  Well.  So t log consists of, it's a full subcategory of objects.  Well, I did not object of type 1 and 2, 2 and the following.  So I need to do, to impose a bit more, a few more relations.  So OK, so what is, what are the relations?  So, well, maybe just before we explain the relation.  So, I have here this, well, there is  homotopy equivalence relation.  So in this homotopy equivalence relation,  x is allowed to be any log scheme, as well as here.  So I want everything in this picture, x, well,  this must be log schemes.  What about a1?  A1 here is just usual a1.  But I will add axiom for log a1.  And if you speak about closed immersion,  perhaps you want the exact one.  Yes, yes.  So, OK, so what I want is the following axiom.  So I have a1 log.  So what is it?  It's a, it's a, the underlying scheme is a1.  And the log structure is given by one point.  So that my m group is just corresponds to the line bundle  of this point.  So, and here I have, so log point,  it's kind of the origin.  And here I have gm.  It has trivial log structure.  So, and I want these maps to be isomorphism in my quotient  category.  But not only this.  I want multiply this by x times a1 log.  And again, x is any work scheme.  So, and also this gm times x is valid.  Well, there is one additional axiom.  It's strange.  It's just has to do, we already observed that,  well, all the construction we have made so far depend only  on m group, not on m.  m is irrelevant.  And I could consider the category of law scheme just  by the category of extensions.  They are in my category t.  So, you require them to be 0.  That's all.  That's all.  Yes.  You want that a1 log is an a1 log.  Is it a morphic or you, too?  Yes, a1 log.  Yes.  Yes.  So, so this are object in t.  Like, for example, as this, those objects.  So, and finally, 4 is very, very strange.  Well, so it's very, very, is the following axiom.  So, suppose I have any map p from x prime to x.  These are log schemes.  And the underlying map between usual schemes  is an isomorphism.  And suppose that it induces, the map  induces on m group p up a star to m prime group.  Is an isomorphism.  Then I want the following relation that x prime goes to x.  It must be in my category.  So, this kind of way to say that everything  depends on the n group.  But again, so I could, you can consider just the category  of pairs x plus this extension of star.  So, now, very good.  So, here is the theorem.  Now, when you do block blow-ups, what  seems when you do the operation of block blow-up,  I think that the space, I don't remember now, the space x log  doesn't change?  No, it changes.  OK, space x log, yeah, changes, but it's the same homework.  So, here is the theorem.  So, well, I have a category A1, CDH.  So, this is my category.  And here is the name for this.  So, this is Z of schemes.  So, there is, of course, a functor to this larger category.  We are going to consider A1, CDH.  And here I have Z log.  OK, so, right.  So, this functor is an occurrence of categories.  So, what does it mean?  So, what are, for example, functors from this category  to, say, the category of complices?  These are homologous theories, right,  that satisfy these basic properties.  The claim is that any homologous theory can be extended  uniquely to log schemes provided it has all these properties.  OK, so, let's see.  So, in particular, well, of course, here you  have map to functor to usual Weyvotsky category.  And then you can compose the inverse and get Weyvotsky  motif.  Now, so, the construction, the proof is,  it uses very little geometry, in fact.  It's more or less linear algebra.  And so, well, first of all, no, you'll have to prove.  So, you have to prove two things,  that this functor that I have here, the obvious functor,  let's give a name to this functor.  Let's call it.  It's fully faithful, homotopically fully faithful,  if it's dg, you think of this as dg functor.  So, fully faithful.  And b, so essentially subjective.  Subjective.  Well, and in fact, so easy step is that a implies b.  And therefore, so all you have to do is to prove a.  So, and that's because, and you can do it by induction  on dimension.  So, you need to show that any motif of any log scheme  is in the image of this functor.  So, have enough relations to express a motif of any log  scheme in terms of motifs of usual scheme.  So, but that's easy.  So, you do it again by dimension of your log scheme.  So, if it's zero dimensional, then, well,  it's fine saturated.  So, it means that my Manoid M group is just z to the n.  And so, the corresponding, so if I have point and take its,  well, its motif, for example, is motif of gm.  And if I consider Cartesian product of any of any scheme  of log point with itself, then it's, of course,  it will be gm times gm.  This follows from this axiom.  And so, any, and using this property,  any zero dimensional log scheme has the same motif  as this product of this.  So, now, what you're doing one dimensional case.  So, you have some curve.  Well, now, generically over an open set,  the log structure is trivial.  It just corresponds to trivial extension.  M group is just trivial extension of z to the n by your star.  And therefore, OK, so you know what to do over an open set.  And then you can use this CDH axiom to see that this log  motif of this curve is also in the image.  So, what, what, what?  Have you still got the formula here?  No, I used because, because, so let's see.  Well, if you look at this CDH axiom, how it looks like.  So, you have some p inverse of z.  And then you have z plus x prime.  And this maps to x.  So, what, so, so I want to show that this object, just,  sorry, so that this object, so knowing that this object  and this object are in the image and also this,  I want to construct, conclude that this object is in the image.  But then I need a map from x, from this square bracket x,  to this guy.  And I need to show that this map exists in the usual category  of Wojewódzki, Wojewódzki category.  It exists by definition.  So, because it's exact triangle, there is a map  from square bracket x to this guy, just shifted by one.  And I need to show that this map actually exists.  So, now, so what you do for?  Do as soon as the log stack is a risky locally trivial.  I use it as a risky locally trivial, right?  So, I could, yeah.  So, you don't have motives for et al.  Then I need to work with et al, the poetry,  rather than be risky, and well, at the end, I will get an et al  Wojewódzki motive.  Well, some applications that I had in mind have to do  with this integral weight filtration, which exists only  for realization of usual Wojewódzki motives, not et al.  So, OK.  So, what you do for A?  So, what do you need?  You have to, it's enough just by formal nonsense.  It's enough to prove the theorem.  It's enough to do the following.  So, given any object of this category, usual category,  A1, CDH, maybe I will just say the following.  So, let me, because my time is short.  So, I claim that it's enough, it suffices.  I will sort of suppress the completely formal part.  So, it only uses the fact that the category of Wojewódzki  motives is rigid, duality.  It suffices to show that the functor  home Z of n from this usual category of CDH Z of schemes  extends to larger category of this category, A1, CDH Z  block.  And what you do here, so you just sort of, you want kind  of to define motivic homology.  It's not quite motivic.  It's home in this category with no transfer.  So, I don't want to call it motivic homology.  It would be motivic homology if I consider this category  with transfer.  So, you want to do it for any log scheme.  And the idea is very simple.  So, you have the same group.  It can be evaluated, it can be considered as a motivic shift  over the underlying scheme.  That means simply that you can evaluate it over any scheme  which is smooth over this guy.  You simply pull the log structure.  If I have a map like this, then you  can pull the log structure, this log structure here,  and take the same guy there.  So, and then what you do, you just compute,  you take homology of X, these coefficients in symmetric powers.  These are operations in the category  of motivic shift symmetric powers.  Well, they're defined by Weyewodski.  They're characterized by the properties  that if you take X, that motif of a scheme X,  the symmetric power of motif of a scheme X  is the motif of the symmetric power of X.  Is it M bar group or M group?  Just M group.  So, for example, what happens if you take N is 1?  Well, when you pull at that, you don't make it a log structure  or Y, you just take just N.  You don't enlarge the units.  I mean, usually when you pull back N.  Yeah, and the Y units.  You are going to enlarge the units.  So, for example, what is first, well,  motivic homology with N equal 1?  This is just homology of M group.  It's kind of the final look of the P-carve.  So, OK.  And now I want to finish by.  How much of which category?  This is the homology of what?  So, well, you are.  So, the base of this homology works.  Yeah, this is Nisnevich homology of this motivic shift.  So, well, I want to finish by a very explicit formula  that I promised at the beginning, just very geometric,  for the motivic homology of the limit motif in the case  of semi-stable degeneration.  Also for the tubular new problem.  It will be just in the case of semi-stable degeneration.  And it comes from this.  I just, well, this formula is here.  I just want to, I will make it explicit, which is trivial  from this definition.  So, the formula is the following.  So, here is my setting.  Well, now I will have, again, I have the log structure  coming from semi-stable degeneration.  But now I can have multiple intersections.  But it's still semi-stable.  Maybe that's not a good picture.  So, this should be planes rather than lines.  So, OK, probably I should not put it here.  So, it's confusing.  OK, semi-stable degeneration.  And it's strict, right?  So, I want really, that's, I want log structure  for the risk typology.  So, strict, strictly semi-stable degeneration.  So, I have components, D1, D2, well, so, DI.  DI, it belongs to some set.  I have this GM torsors.  So, first of all, I have LI, which is of DI,  restricted to X0.  And then I have the corresponding GM torsors.  And I have UI, which is a complement of X0 to DI.  So, OK, so now I will construct the material homologism.  So, well, first I will produce a complex of smooth varieties  of Rack Zero.  The fibers will be algebraic tori.  So, what are these varieties?  So, step one, complex, very simple.  Complex of smooth varieties.  So, this will be product of all this LI.  So, this is product over Rack Zero.  It's a torsor over an algebraic tori of dimension.  Now, what is the next term?  So, you take sum.  Let's consider it as a kind of, I want to, it's not really,  it's, in fact, it's a, I want to consider kind of an additive  category.  I want to make sense of sum.  So, it means that it's, you can think of this dejoint unit.  So, of products of I in I, I is not equal to J.  And here I will have L Cirque I times UI over R Zero.  So, what is this map?  Remember that LI is trivial over UI.  Therefore, I have such a map.  Just given by, well, it's section.  Now, it's a long complex.  And here is its last term.  It's sum I over times intersection J not equal to I UG.  It's again over X Zero.  And so, all I use here, all the differentials,  come from the sections.  Remember that LI is trivial over UI.  So, this gives me such a, again, this should be dejoint here.  So, this is step one.  It's very simple complex.  Now, form define a complex of pre-sets on X Zero.  Well, what you do?  So, you call this Z dot.  And then you take the following.  So, if you have a scheme over X Zero,  then, well, the value of my complex of pre-sets on X Zero  on Y is the following.  You take correspondences or X Zero from Y to this complex.  So, well, informally, you just take sections  of this complex of varieties over Y. And then, you make it  homotopy invariant by applying the system construction.  So, let's call this F and my material homology of X Zero  M. Z is just CDH homology of this X Zero with coefficients  in F. That's it.  That's all I have to say.  So, there is one, well, as you see,  you can ask whether the log geometry here.  And in fact, there is no log geometry.  You see it's just linear algebra.  And the log geometry appears if you want to prove,  for example, that if you have a smooth log scheme,  then it's motif.  It's isomorphic to the motif of the open part  where the log structure is trivial.  This I don't know how to do.  I know how to do it in the case of normal crossing situation.  But in general, it really requires some geometry.  Thank you very much.  Was there any case with notation of the motif homology, did you say?  Yeah, so here I defined only this one index, right?  This index is absolute value of this set of indices, i.  So if you want it defined for larger, for larger,  and right, you have to just add in this whole construction  empty divisor formula.  And to define it for smaller, you don't need to define it  for small by cancellation.  It suffices to define this for sufficiently large numbers.  This follows from the books.  Yes.  Just motif of homology.  Yeah, so if you want it for, you have cancellation.  So you can express motif homology of something  with coefficients in z of n in terms of homology of the twist,  this coefficients in z of n plus 1.  This is the product of this GM?  Yeah, yeah, product of this GM, for example.  Yes?  So I just was curious, what is n?  Where is n?  Shouldn't we have this, if we have a motif homology?  Yeah, yeah, this twist here, right?  No, no, this n means the log of monotone.  Yeah, that's a very good question.  It's this whole complex of schemes.  It's least over GM.  So if you have, it's exactly as in the picture I started with.  This gives you the tubular neighborhood.  If you want the vanishing cycles,  you have to take the fiber over.  Oh, OK.  So it's a unipotent motif shape over GM.  But how can you define the map to GM  if you're given just the lid, you're in the fiber also?  No, there is a, the map to GM comes  from the trivialization of the tensile product  of these fine bundles.  So each of these smooth schemes over X0 admits a map to GM.  For example.  Because the other one, the product is.  Yeah.  I think this is implicit.  When I said that you were going to start talking about how  to define the weightful phase.  Yeah?  Does this come out easily from this?  Yes.  So there is a, well, yes, it comes thanks to work of Bandarka.  So if Betty homology or the Ram homology or any  Veyvodsky motif is equipped with integral weight filtration.  Yes.  Dr. Yavt, show that he's equivalent to.  Rational.  Yeah.  To what?  Yes.  He has defined the weight filtration.  Yes.  But after, you have to show that he's equivalent to the others.  Well, but here you don't have, well,  a priori you don't have the others.  What you have to show is that the basic homology  of this log motif of this Veyvodsky motif I defined  coincides with the homology of this, as defined by Kat  and Nakayama.  But this is obvious because Kat and Nakayama homology theory  satisfies the list of all my axioms.  And therefore, this factor from log motifs  is uniquely determined by its restriction to usual motifs.  And for usual schemes, there is nothing to prove.  And sorry, last question.  In characteristic P, this was everything in characteristic  in the world.  Yeah, you have to invert P.  And always with Zariski?  Yeah, for Zariski again.  So the only reason why I work with Zariski  is that I want Veyvodsky motif really for Nisnevich  topology, as opposed to Veyvodsky motifs for Taltopolga.  So it is enough to have Nisnevich locally,  Nisnevich chance for the work start?  Yes.  Yes, yes, yes.  Though I don't really know.  No, maybe that way.  There are others.  Yes.  You need to have your gel.  That's not good enough.  Any more questions?  Thank you very much.
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